From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:14:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3EWaJ024329 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3EUgk021105 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:30 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-TEN.MIT.EDU [18.244.5.10]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3EPe2011146 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:14:29 -0400 (EDT) Message-Id: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 21 Sep 2005 21:14:20 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1"; format=flowed X-Spam-Score: 1.08 X-Spam-Level: * (1.08) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8M3EWaJ024329 Status: RO X-Status: A Content-Length: 417 X-IMAPbase: 1127956025 131 X-UID: 1 X-Keywords: NonJunk I found it interesting that the three proof methods: induction, strong induction, and the well ordering principle, can be used to prove the same theorems. I would like to understand why this is so. The equivalence between induction and strong induction is relatively simple, but I wouldn´t know how to prove that the well ordering principle can be used in place of the other methods. Sergio Bacallado Group 1 From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:17:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3HXaJ024524 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3HVgk023397 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:32 -0400 (EDT) Received: from mopspeak.mit.edu (EASTCAMPUS-SIX-SIXTY-SEVEN.MIT.EDU [18.238.5.156]) (authenticated bits=0) (User authenticated as jehan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3HRg9012075 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 23:17:28 -0400 (EDT) Message-Id: <5.2.1.1.2.20050921231343.00bb83f8@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 21 Sep 2005 23:17:31 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.046 X-Spam-Level: * (1.046) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 187 X-UID: 2 X-Keywords: NonJunk I found the well ordered proof interesting, because we've been using it so much without even thinking about it. Even though it seems useless, it does have an important purpose. Jehan From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:33:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3X4aJ026901 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:33:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3X2gk005276 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:33:03 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3WuDO014644 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 21 Sep 2005 23:32:56 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8M3WuBI010631; Wed, 21 Sep 2005 23:32:56 -0400 Received: from RANDOM-ONE-FORTY-FOUR.MIT.EDU (RANDOM-ONE-FORTY-FOUR.MIT.EDU [18.243.5.144]) (User authenticated as lye@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 21 Sep 2005 23:32:56 -0400 Message-ID: <20050921233256.1iyrb4m3kuq880sw@webmail.mit.edu> Date: Wed, 21 Sep 2005 23:32:56 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: required email comments, week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 263 X-UID: 3 X-Keywords: NonJunk Week 3, Section 4: The Well Ordering Principle p.12: "Well Ordering proofs typically involve proof by contradiction, so using it is not always the best approach." Are there proofs that can be done only by contradiction and not induction? (Or not as easily.) From meyer@imap.theory.csail.mit.edu Wed Sep 21 23:57:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M3vbaJ028119 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M3vagk025327 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:36 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M3vXKt018713 for <6042-probs@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 23:57:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8M3vXxh006080; Wed, 21 Sep 2005 23:57:33 -0400 Received: from 18.218.1.221 ([18.218.1.221]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 21 Sep 2005 23:57:33 -0400 Message-ID: <20050921235733.499kbjv219gk04ws@webmail.mit.edu> Date: Wed, 21 Sep 2005 23:57:33 -0400 From: Neil M Dowgun To: 6042-probs@theory.csail.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -1.3 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 657 X-UID: 4 X-Keywords: NonJunk I completely understand everything in this week's readings, although I had to reread the false horse proof (section 2.5) before I really got how we were trying to prove it. It occured to me that if you take any set of elements for whom any pair of elements share a certain trait, then every element in that set must have that trait. Is this called by any certain name? It seems obvious, but so does the Well Ordering Principle. Also, I look forward to learning the difference between choosability and colorability. And I found "The country Inductia, whose unit of currency is the Strong" to be hilariously unimaginative. Keep up the good work. Neil Dowgun From meyer@imap.theory.csail.mit.edu Thu Sep 22 01:10:28 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M5ASaJ003872 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:28 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M5ARTW006405 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:27 -0400 (EDT) Received: from [18.239.7.198] (MACGREGOR-SEVEN-O-NINE.MIT.EDU [18.239.7.198]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M5AJ3a002326 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:10:19 -0400 (EDT) Message-ID: <43323CB8.7030406@mit.edu> Date: Thu, 22 Sep 2005 01:10:16 -0400 From: Irene Zhang User-Agent: Mozilla Thunderbird 1.0RC1 (Windows/20041201) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 69 X-UID: 5 X-Keywords: I woul like the proof of product of primes explained again in class. From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:17:49 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6HnaJ013713 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:49 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6Hn4k007672 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:49 -0400 (EDT) Received: from [18.241.6.223] (NEW-FOUR-SEVENTY-EIGHT.MIT.EDU [18.241.6.223]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M6Ha0v003901 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:17:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <0621A36B-A6F2-4FA7-93E8-E3D3FB7FBEFD@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Hamidou Soumare Subject: Week 3 Email Comments Date: Thu, 22 Sep 2005 02:25:57 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 223 X-UID: 6 X-Keywords: NonJunk I found the passage on the well ordering principle to be the most surprising because i dont see how such a simple principle can be used to provide proofs that are shorter than using proof by induction. Hamidou Soumare From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:28:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6ShaJ015053 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6SfuR009330 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:41 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M6SZrQ004609 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 02:28:35 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8M6SYeO018484; Thu, 22 Sep 2005 02:28:34 -0400 Received: from MACGREGOR-TWO-FIFTEEN.MIT.EDU (MACGREGOR-TWO-FIFTEEN.MIT.EDU [18.239.5.215]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 02:28:34 -0400 Message-ID: <20050922022834.uxo9bn34oi4g40go@webmail.mit.edu> Date: Thu, 22 Sep 2005 02:28:34 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: Week 3 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 317 X-UID: 7 X-Keywords: NonJunk Reading comments for this week on induction reading: The thing that I found hardest in this weeks reading were the concepts of strong induction, presented on pages 9 and 10, and the product of primes proof on page 10. I would appreciate going over the definition of strong induction and this proof in class. -Dave From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:28:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MESfaJ025546 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 10:28:41 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MESeWi026370; Thu, 22 Sep 2005 10:28:40 -0400 (EDT) Received: from [18.96.6.34] (SIMMONS-TWO-EIGHTY-NINE.MIT.EDU [18.96.6.34]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MESXpP011947 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Thu, 22 Sep 2005 10:28:33 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Lecture Notes 3 Comments Date: Thu, 22 Sep 2005 10:30:29 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 363 X-UID: 8 X-Keywords: NonJunk After reading the notes I still can't differentiate readily between when or how to use induction and strong induction. I can tell you what the difference is, but the application of each is baffling to me. Further I can't figure out when there is an error in the logic of an induction proof, some added time on that would be very beneficial. -Harrison Hall From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:37:07 2005 Message-ID: <4331D283.7020202@csail.mit.edu> Date: Wed, 21 Sep 2005 17:37:07 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 550 X-UID: 9 X-Keywords: No, I don't think it would "make it more clear": it's better illustrated with examples. Let me know if you still disagree after Friday lecture. Regards, A Akari Kameyama wrote: > In other words, any theorem that can be proved with strong induction > could also be proved with ordinary > induction (using a slightly more complicated indcution hypothesis). > -page 9 > > An example of proving the same thing with ordinary then strong (or vice > versa) would help to make it more clear how one can be better than the > other. > --Apple-Ma From meyer@imap.theory.csail.mit.edu Wed Sep 21 02:11:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8L6BLaJ025131 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8L6BKNi023056 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:20 -0400 (EDT) Received: from [18.234.1.87] (DP-EIGHTY-SEVEN.MIT.EDU [18.234.1.87]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8L6BCNT026169 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 02:11:12 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-1-799499547 From: Akari Kameyama Subject: Week 3 Comments Date: Wed, 21 Sep 2005 02:11:11 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.695 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 2608 X-UID: 10 X-Keywords: NonJunk --Apple-Mail-1-799499547 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed In other words, any theorem that can be proved with strong induction could also be proved with ordinary induction (using a slightly more complicated indcution hypothesis). -page 9 An example of proving the same thing with ordinary then strong (or vice versa) would help to make it more clear how one can be better than the other. --Apple-Mail-1-799499547 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
=A0In other=A0words, any theorem that can be proved with = strong induction could also be proved with ordinary=A0
induction (using a = slightly more complicated indcution hypothesis).=A0 =A0-page 9

An example of = proving the same thing with ordinary then strong (or vice versa) would = help to make it more clear how one can be better than the = other.
= --Apple-Ma From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:34:16 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8LLYGaJ006626 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 17:34:16 -0400 Received: (qmail 30545 invoked from network); 21 Sep 2005 21:34:14 -0000 Received: from unknown (HELO ?128.30.49.31?) (armeyer10@128.30.49.31 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 21 Sep 2005 21:34:14 -0000 Message-ID: <4331D1D6.1080403@csail.mit.edu> Date: Wed, 21 Sep 2005 17:34:14 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold Subject: week 3 Notes, p.7 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 167 X-UID: 11 X-Keywords: Any further comment now you've seen one? regards, A. Amanda Seybold wrote: > I would like to hear more about these other induction hypotheses > mentioned on page 7. From meyer@imap.theory.csail.mit.edu Wed Sep 21 21:35:20 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M1ZKaJ006925 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:20 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M1ZJmK001314 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:19 -0400 (EDT) Received: from [127.0.0.1] (NEXT-FOUR-EIGHTY-EIGHT.MIT.EDU [18.242.6.233]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M1ZB0w025235 for <6042-probs@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:35:16 -0400 (EDT) Message-ID: <43320A4C.6010508@mit.edu> Date: Wed, 21 Sep 2005 21:35:08 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 135 X-UID: 12 X-Keywords: NonJunk I'm confused on how the Well Ordering Principle (pg. 12) is actually useful. I get the idea, but what do I use it with? -chris wong From meyer@imap.theory.csail.mit.edu Wed Sep 21 19:47:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8LNlsaJ021544 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8LNlqGh009789 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:52 -0400 (EDT) Received: from [18.113.6.208] (ROGERS-FOUR-SIXTY-THREE.MIT.EDU [18.113.6.208]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8LNlkNO006187 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 19:47:47 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Sheldon Chan Subject: Comments on Reading - Week 3 Date: Wed, 21 Sep 2005 19:47:45 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 149 X-UID: 13 X-Keywords: NonJunk I find the concept of strong induction to still be confusing. From what I've gathered, it's simply an induction proof that has more assumptions. From meyer@imap.theory.csail.mit.edu Wed Sep 21 22:00:54 2005 Return-Path: Received: from smtp012.mail.yahoo.com (smtp012.mail.yahoo.com [216.136.173.32]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8M20raJ011895 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 22:00:54 -0400 Received: (qmail 4121 invoked from network); 22 Sep 2005 02:00:53 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp012.mail.yahoo.com with SMTP; 22 Sep 2005 02:00:52 -0000 Message-ID: <43321058.4010507@csail.mit.edu> Date: Wed, 21 Sep 2005 22:00:56 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 233 X-UID: 14 X-Keywords: yes that's right; so what's confusing you? regards, A. Sheldon Chan wrote: > I find the concept of strong induction to still be confusing. From > what I've gathered, it's simply an induction proof that has more > assumptions. From meyer@imap.theory.csail.mit.edu Wed Sep 21 21:59:56 2005 Return-Path: Received: from smtp016.mail.yahoo.com (smtp016.mail.yahoo.com [216.136.174.113]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8M1xtaJ011781 for <6042-staff@theory.csail.mit.edu>; Wed, 21 Sep 2005 21:59:55 -0400 Received: (qmail 29449 invoked from network); 22 Sep 2005 01:59:55 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp016.mail.yahoo.com with SMTP; 22 Sep 2005 01:59:54 -0000 Message-ID: <4332101D.3040004@csail.mit.edu> Date: Wed, 21 Sep 2005 21:59:57 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Wong CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <43320A4C.6010508@mit.edu> In-Reply-To: <43320A4C.6010508@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 292 X-UID: 15 X-Keywords: wait till lecture& class probs on fri for some more examples. ask me or your TA again after if that's not enough. regards, A. Chris Wong wrote: > I'm confused on how the Well Ordering Principle (pg. 12) is actually > useful. I get the idea, but what do I use it with? > > -chris wong > From meyer@imap.theory.csail.mit.edu Wed Sep 21 22:13:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M2DnaJ016804 for <6042-staff@theory.csail.MIT.EDU>; Wed, 21 Sep 2005 22:13:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M2Dmht002686; Wed, 21 Sep 2005 22:13:48 -0400 (EDT) Received: from m56-129-17.mit.edu (M56-129-17.MIT.EDU [18.56.0.46]) (authenticated bits=56) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M2Dktj000246 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 21 Sep 2005 22:13:47 -0400 (EDT) Received: from localhost (sheldons@localhost) by m56-129-17.mit.edu (8.12.9) with ESMTP id j8M2Dkqp008288; Wed, 21 Sep 2005 22:13:46 -0400 Date: Wed, 21 Sep 2005 22:13:45 -0400 (EDT) From: Sheldon Chan To: "Prof. Albert R. Meyer" cc: <6042-staff@theory.csail.MIT.EDU> Subject: Re: Comments on Reading - Week 3 In-Reply-To: <43321058.4010507@csail.mit.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 780 X-UID: 16 X-Keywords: Perhaps its simply an expectation that the "strong" in the name implies that it should be more direct and supportive evidence in the proof. It makes sense making assumptions allows us to more easily prove something, but counterintuitive in producing a sound proof, since it could leave more room for faulty assumptions. I think it would be good if there were more examples or a clarification of the criteria for such assumptions made in a strong induction proof. Thanks, Sheldon On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > yes that's right; so what's confusing you? > regards, A. > > Sheldon Chan wrote: > > > I find the concept of strong induction to still be confusing. From > > what I've gathered, it's simply an induction proof that has more > > assumptions. > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 01:32:35 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M5WZaJ007009 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 01:32:35 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M5WXTW006762; Thu, 22 Sep 2005 01:32:33 -0400 (EDT) Received: from [192.168.1.101] (c-65-96-167-243.hsd1.ma.comcast.net [65.96.167.243]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8M5WP0v002688; Thu, 22 Sep 2005 01:32:25 -0400 (EDT) Message-ID: <433241E3.2050008@csail.mit.edu> Date: Thu, 22 Sep 2005 01:32:19 -0400 From: sayan mitra User-Agent: Mozilla Thunderbird 1.0.6-1.1.fc4 (X11/20050720) X-Accept-Language: en-us, en MIME-Version: 1.0 CC: Sheldon Chan Subject: Re: Comments on Reading - Week 3 References: <43321058.4010507@csail.mit.edu> In-Reply-To: <43321058.4010507@csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.297 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 717 X-UID: 17 X-Keywords: NonJunk Compare the two induction rules, leaving aside the base cases: Regular induction: [P(n) => P(n+1) ] => \forall n P(n) Strong induction: [P(1),...,P(n) => P(n+1)] => \forall n P(n) Now the antecedent of the implication inside the [ ]'s is stronger for strong induction, making the [ ] implication itself weaker, right ? Therefore, if you look at the overall statement, the antecedent of the strong induction principle is weaker, and so, the strong induction as a rule (the outer implication) is stronger. Makes sense ? -Sayan > Sheldon Chan wrote: > >> I find the concept of strong induction to still be confusing. From >> what I've gathered, it's simply an induction proof that has more >> assumptions. > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:37:54 2005 Message-ID: <4332C1C2.8010709@csail.mit.edu> Date: Thu, 22 Sep 2005 10:37:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 866 X-UID: 18 X-Keywords: Yes, could use simple induction to PROVE that P(0),...,P(n) hold if you've proved P(0) and P(k) ==> P(k+1) for all k Section 3, page 10: > I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? > > |Arup| > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:41:10 2005 Message-ID: <4332C289.2080605@csail.mit.edu> Date: Thu, 22 Sep 2005 10:41:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1046 X-UID: 19 X-Keywords: not sure what kind of further "clarification" you're looking for, but you'll see several more examples Friday. Let me know if you're not satisfied after you've seen them. regards, A. Sheldon Chan wrote: >Perhaps its simply an expectation that the "strong" in the name implies >that it should be more direct and supportive evidence in the proof. >It makes sense making assumptions allows us to more easily prove >something, but counterintuitive in producing a sound proof, since it could >leave more room for faulty assumptions. I think it would be good if >there were more examples or a clarification of the criteria for such >assumptions made in a strong induction proof. > >Thanks, > >Sheldon > >On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > > > >>yes that's right; so what's confusing you? >>regards, A. >> >>Sheldon Chan wrote: >> >> >> >>>I find the concept of strong induction to still be confusing. From >>>what I've gathered, it's simply an induction proof that has more >>>assumptions. >>> >>> >> >> > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:00:30 2005 Message-ID: <4332C711.2000700@csail.mit.edu> Date: Thu, 22 Sep 2005 11:00:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-staff@theory.csail.mit.edu Subject: Re: Email comments for reading References: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> In-Reply-To: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 642 X-UID: 20 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Katherine A Romer wrote: >"...there's a routine way to transform any proof using the Well Ordering >Principle into a proof using Strong Induction, and vice-versa." (p.11) >I'd like to see an example in lecture of how one of the theorems we've proved >using induction could be proved using the Well Ordering Principle--I'm having a >hard time seeing how this might be done. > >Katherine Romer > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:38:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MEciaJ028852 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MEchWi007792 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:43 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MEcaBT016270 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 10:38:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8MEcao7001649; Thu, 22 Sep 2005 10:38:36 -0400 Received: from BURTON-FIFTY-THREE.MIT.EDU (BURTON-FIFTY-THREE.MIT.EDU [18.247.5.53]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 10:38:36 -0400 Message-ID: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Date: Thu, 22 Sep 2005 10:38:36 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: Email comments for reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 360 X-UID: 21 X-Keywords: "...there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa." (p.11) I'd like to see an example in lecture of how one of the theorems we've proved using induction could be proved using the Well Ordering Principle--I'm having a hard time seeing how this might be done. Katherine Romer From meyer@imap.theory.csail.mit.edu Wed Sep 21 12:12:46 2005 21 12: 12:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8LGCkaJ007328 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8LGCjJU007908 for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:45 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8LGCbKt026857 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 21 Sep 2005 12:12:38 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 21 Sep 2005 12:11:37 -0400 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -1.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 87 X-UID: 22 X-Keywords: I would like to hear more about these other induction hypotheses mentioned on page 7. From meyer@imap.theory.csail.mit.edu Wed Sep 21 17:38:52 2005 BCC: 6042-staff@theory.csail.mit.edu Message-ID: <4331D2EB.2090000@csail.mit.edu> Date: Wed, 21 Sep 2005 17:38:51 -0400 From: "Albert R. Meyer" User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Akari Kameyama Subject: Re: Week 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 560 X-UID: 23 X-Keywords: No, I don't think it would "make it more clear": I think it's better illustrated with examples. Let me know if you still disagree after Friday lecture. Regards, A Akari Kameyama wrote: > In other words, any theorem that can be proved with strong induction > could also be proved with ordinary > induction (using a slightly more complicated indcution hypothesis). > -page 9 > > An example of proving the same thing with ordinary then strong (or vice > versa) would help to make it more clear how one can be better than the > other. > --Apple-Ma From meyer@imap.theory.csail.mit.edu Thu Sep 22 05:45:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M9jWaJ010784 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M9jV3p005920 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:31 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-SEVEN-FORTY-FOUR.MIT.EDU [18.238.5.233]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M9jRUC012432 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 05:45:28 -0400 (EDT) Message-ID: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments Week 3 Date: Thu, 22 Sep 2005 05:45:30 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 201 X-UID: 24 X-Keywords: NonJunk I'm a bit confused about the difference between simple induction and strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 qualifies as strong induction rather than simple. --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 22 02:54:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8M6snaJ019565 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8M6sluR026305 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:48 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-SIXTY-ONE.MIT.EDU [18.247.6.6]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8M6sOOn006135 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 02:54:35 -0400 (EDT) Message-Id: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 22 Sep 2005 02:54:24 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: Reading Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.165 X-Spam-Level: ** (2.165) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 233 X-UID: 25 X-Keywords: NonJunk Section 3, page 10: I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? |Arup| From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:16:04 2005 Message-ID: <4332CA84.4050107@csail.mit.edu> Date: Thu, 22 Sep 2005 11:15:16 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20050921210935.032f39d0@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 704 X-UID: 26 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Sergio Bacallado wrote: > I found it interesting that the three proof methods: induction, strong > induction, and the well ordering principle, can be used to prove the > same theorems. I would like to understand why this is so. The > equivalence between induction and strong induction is relatively > simple, but I wouldn´t know how to prove that the well ordering > principle can be used in place of the other methods. > > Sergio Bacallado > Group 1 > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 13:55:55 2005 Message-ID: <4332F032.9090004@csail.mit.edu> Date: Thu, 22 Sep 2005 13:56:02 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-staff@theory.csail.mit.edu Subject: Re: Lecture Notes 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 576 X-UID: 27 X-Keywords: If you're still feeling shaky about using and/or debugging induction proofs, I suggest you take it up in office hours with your TA at the next opportunity. regards, A. Harrison King Hall wrote: > After reading the notes I still can't differentiate readily between when > or how to use induction and strong induction. I can tell you what the > difference is, but the application of each is baffling to me. Further I > can't figure out when there is an error in the logic of an induction > proof, some added time on that would be very beneficial. > -Harrison Hall > From meyer@imap.theory.csail.mit.edu Thu Sep 22 14:28:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MISnaJ003988 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MISmbH022504 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:48 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-TWO-TWELVE.MIT.EDU [18.244.5.212]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MISjf5018067 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:28:46 -0400 (EDT) Message-Id: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Thu, 22 Sep 2005 14:28:31 -0400 To: 6042-probs@theory.csail.mit.edu From: Kamil Sindi Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 800 X-UID: 28 X-Keywords: NonJunk "2.5 A Faulty Induction Proof False Theorem. All horses are the same color. Notice that no n is mentioned in this assertion, so we're going to have to reformulate it in a way that makes an n explicit. In particular, we'll (falsely) prove that False Theorem 2.4. In every set of n 1 horses, all are the same color. This a statement about all integers n 1 rather 0, so it's natural to use a slight variation on induction: prove P(1) in the base case and then prove that P(n) implies P(n + 1) for all n 1 in the inductive step. This is a perfectly valid variant of induction and is not the problem with the proof below." I would like to see more mathematical faulty inductions and see why certain assertions by induction are false; I didn't think the "horse theorems" were good examples. From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:33:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJXBaJ017447 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJX9bH022421 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:09 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MJX5wB013648 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 15:33:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8MJX5ic016466; Thu, 22 Sep 2005 15:33:05 -0400 Received: from DKE-ONE-TWENTY-ONE.MIT.EDU (DKE-ONE-TWENTY-ONE.MIT.EDU [18.233.1.121]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 15:33:05 -0400 Message-ID: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> Date: Thu, 22 Sep 2005 15:33:05 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -0.031 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 342 X-UID: 29 X-Keywords: NonJunk I thought the reading was interesting, but there was one portion that was a little confusing. On page 5, the paragraph under equation (3) does the equivalent of a lot of fast talk and hand-waving, and while if you go through it it is most certainly correct, if one or two steps were actually written out it would have helped. -Paul Groudas From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:38:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJcHaJ019068 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJcGbH027575 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:16 -0400 (EDT) Received: from ajshafer (AJSHAFER.MIT.EDU [18.247.4.109]) (authenticated bits=0) (User authenticated as ajshafer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MJcCMB015618 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:38:12 -0400 (EDT) Message-ID: <000201c5bfad$2acafa40$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Thu, 22 Sep 2005 15:38:10 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2670 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2670 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 436 X-UID: 30 X-Keywords: NonJunk I have a question about the horse color problem. What if you start with the base case of n=2? Can you from there begin the same logic and show that all horses are again the same color? What would be the problem in this proof? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:03:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MK3naJ022727 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MK3lbH021838 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:48 -0400 (EDT) Received: from m12-182-13.mit.edu (M12-182-13.MIT.EDU [18.19.0.44]) (authenticated bits=56) (User authenticated as bens@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MK3f7v025558 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 16:03:42 -0400 (EDT) Received: (from bens@localhost) by m12-182-13.mit.edu (8.12.9) id j8MK3fBM015703; Thu, 22 Sep 2005 16:03:41 -0400 Subject: Comments From: Benjamin Schwartz To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 22 Sep 2005 16:03:41 -0400 Message-Id: <1127419421.15420.1.camel@m12-182-13.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 262 X-UID: 31 X-Keywords: On page 10, the comment that strong induction is not more powerful than weak induction is surprising. There is a conflict between logic, which says that every true statement is provable in infinitely many ways, and intuition, which demands One True Proof. -Ben From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:18:07 2005 Message-ID: <43331182.4000009@csail.mit.edu> Date: Thu, 22 Sep 2005 16:18:10 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Sindi CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> In-Reply-To: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1020 X-UID: 32 X-Keywords: you'll see quite a few more false proofs, including by induction,as the term progresses. But why was the "horses" example not good? Regards, A. Kamil Sindi wrote: > "2.5 A Faulty Induction Proof > > False Theorem. All horses are the same color. > Notice that no n is mentioned in this assertion, so we're going to > have to reformulate it in a way > that makes an n explicit. In particular, we'll (falsely) prove that > > False Theorem 2.4. In every set of n 1 horses, all are the same color. > This a statement about all integers n 1 rather 0, so it's natural to > use a slight variation on > induction: prove P(1) in the base case and then prove that P(n) > implies P(n + 1) for all n 1 in > the inductive step. This is a perfectly valid variant of induction and > is not the problem with the > proof below." > > I would like to see more mathematical faulty inductions and see why > certain assertions by induction are false; I didn't think the "horse > theorems" were good examples. > > > From - Thu From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:57:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJvPaJ021444 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:57:25 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJvNbc008734 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:57:24 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJvNOD021232; Thu, 22 Sep 2005 15:57:23 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8MJvF3a011824; Thu, 22 Sep 2005 15:57:16 -0400 (EDT) Message-ID: <43330C83.6020107@mit.edu> Date: Thu, 22 Sep 2005 15:56:51 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Andrew Shafer Subject: Re: Week 3 Comments References: <000201c5bfad$2acafa40$6d04f712@ajshafer> In-Reply-To: <000201c5bfad$2acafa40$6d04f712@ajshafer> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 791 X-UID: 33 X-Keywords: Great question! The answer is, there is nothing wrong with the proof! But think about what that proves: "IF any set of 2 horses is the same color, THEN all horses are the same color." Since the IF part is false, the statement as a whole is vacuously true, regardless of what comes in the THEN part. Andrew Shafer wrote: > I have a question about the horse color problem. > > What if you start with the base case of n=2? Can you from there begin > the same logic and show that all horses are again the same color? > What would be the problem in this proof? > > -Andrew > ---------------------------- > Illegitmitatum Non Carborundum Est > Andrew Shafer, MIT Blog > http://shaferandrew.blogspot.com > Si hoc legere scis numium eruditionis habes. > ---------------------------- > From meyer@imap.theory.csail.mit.edu Thu Sep 22 14:40:47 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MIegaJ007051 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 14:40:43 -0400 Received: (qmail 64372 invoked from network); 22 Sep 2005 17:55:53 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 17:55:53 -0000 Message-ID: <4332F032.9090004@csail.mit.edu> Date: Thu, 22 Sep 2005 13:56:02 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-staff@theory.csail.mit.edu Subject: Re: Lecture Notes 3 Comments References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 575 X-UID: 34 X-Keywords: If you're still feeling shaky about using and/or debugging induction proofs, I suggest you take it up in office hours with your TA at the next opportunity. regards, A. Harrison King Hall wrote: > After reading the notes I still can't differentiate readily between when > or how to use induction and strong induction. I can tell you what the > difference is, but the application of each is baffling to me. Further I > can't figure out when there is an error in the logic of an induction > proof, some added time on that would be very beneficial. > -Harrison Hall > From meyer@imap.theory.csail.mit.edu Thu Sep 22 15:54:39 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MJsdaJ021206 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:54:39 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJsbbc004073 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 15:54:38 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MJrFOD021032; Thu, 22 Sep 2005 15:53:15 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8MJr70v009280; Thu, 22 Sep 2005 15:53:08 -0400 (EDT) Message-ID: <43330B8B.3070807@mit.edu> Date: Thu, 22 Sep 2005 15:52:43 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Kamil Sindi Subject: Re: Week 3 Comments References: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> In-Reply-To: <6.2.3.4.2.20050922142401.0253c7b0@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 439 X-UID: 35 X-Keywords: Did you think they weren't good examples because the flaws were too obvious? If so, I think you're in great shape - that means you have a good grasp of induction. If it was something else, please let us know it was about the examples that bothered you. DS > I would like to see more mathematical faulty inductions and see why > certain assertions by induction are false; I didn't think the "horse > theorems" were good examples. From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:22:28 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MKMRaJ025372 for ; Thu, 22 Sep 2005 16:22:27 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIXaN-0003G0-Me for meyer@csail.mit.edu; Thu, 22 Sep 2005 16:22:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MKMQbH009881 for ; Thu, 22 Sep 2005 16:22:26 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MKMNmt002614 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for ; Thu, 22 Sep 2005 16:22:24 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: In-Reply-To: <4331D1D6.1080403@csail.mit.edu> References: <4331D1D6.1080403@csail.mit.edu> Date: Thu, 22 Sep 2005 16:21:24 -0400 To: "Albert R. Meyer" From: Amanda Seybold Subject: Re: week 3 Notes, p.7 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO X-Status: A Content-Length: 292 X-UID: 36 X-Keywords: It would be nice to see more? Perhaps ones with more than one variable, so one has to figure out what to assign n to. >Any further comment now you've seen one? >regards, A. > >Amanda Seybold wrote: >>I would like to hear more about these other induction hypotheses >>mentioned on page 7. From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:25:32 2005 Message-ID: <4333133D.2030402@csail.mit.edu> Date: Thu, 22 Sep 2005 16:25:33 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Amanda Seybold CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 Notes, p.7 References: <4331D1D6.1080403@csail.mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 464 X-UID: 37 X-Keywords: you'll see more tmw, and lots during the term: induction is a pretty basic proof technique we use all the time. regards, A Amanda Seybold wrote: > It would be nice to see more? Perhaps ones with more than one > variable, so one has to figure out what to assign n to. > >> Any further comment now you've seen one? >> regards, A. >> >> Amanda Seybold wrote: >> >>> I would like to hear more about these other induction hypotheses >>> mentioned on page 7. >> From meyer@imap.theory.csail.mit.edu Thu Sep 22 16:58:54 2005 Message-ID: <43331B11.9020408@csail.mit.edu> Date: Thu, 22 Sep 2005 16:58:57 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Paul Groudas CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> In-Reply-To: <20050922153305.irm4vf76xf0os4sg@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 475 X-UID: 38 X-Keywords: I agree, and I just revised a bit. See if new version helps. Thx for the feedback. regards, A. Paul Groudas wrote: >I thought the reading was interesting, but there was one portion that was a >little confusing. On page 5, the paragraph under equation (3) does the >equivalent of a lot of fast talk and hand-waving, and while if you go through >it it is most certainly correct, if one or two steps were actually written out >it would have helped. > >-Paul Groudas > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:41:11 2005 Return-Path: Received: from smtp013.mail.yahoo.com (smtp013.mail.yahoo.com [216.136.173.57]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEfAaJ029152 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:41:10 -0400 Received: (qmail 65890 invoked from network); 22 Sep 2005 14:41:09 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp013.mail.yahoo.com with SMTP; 22 Sep 2005 14:41:09 -0000 Message-ID: <4332C289.2080605@csail.mit.edu> Date: Thu, 22 Sep 2005 10:41:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sheldon Chan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Comments on Reading - Week 3 References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1045 X-UID: 39 X-Keywords: not sure what kind of further "clarification" you're looking for, but you'll see several more examples Friday. Let me know if you're not satisfied after you've seen them. regards, A. Sheldon Chan wrote: >Perhaps its simply an expectation that the "strong" in the name implies >that it should be more direct and supportive evidence in the proof. >It makes sense making assumptions allows us to more easily prove >something, but counterintuitive in producing a sound proof, since it could >leave more room for faulty assumptions. I think it would be good if >there were more examples or a clarification of the criteria for such >assumptions made in a strong induction proof. > >Thanks, > >Sheldon > >On Wed, 21 Sep 2005, Prof. Albert R. Meyer wrote: > > > >>yes that's right; so what's confusing you? >>regards, A. >> >>Sheldon Chan wrote: >> >> >> >>>I find the concept of strong induction to still be confusing. From >>>what I've gathered, it's simply an induction proof that has more >>>assumptions. >>> >>> >> >> > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:45:30 2005 Return-Path: Received: from smtp107.mail.sc5.yahoo.com (smtp107.mail.sc5.yahoo.com [66.163.169.227]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEjSaJ029551 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:45:28 -0400 Received: (qmail 9436 invoked from network); 22 Sep 2005 14:29:22 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp107.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 14:29:21 -0000 Message-ID: <4332BFC5.1000706@csail.mit.edu> Date: Thu, 22 Sep 2005 10:29:25 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments Week 3 References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> In-Reply-To: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 545 X-UID: 40 X-Keywords: The k and m in the proof are only known to be SMALLER than n+2 (in fact, they cannot equal n+2). Strong induction let's us assume both k & m are products of primes. Simple induction would only let us assume that n+2 was a product of primes, which isn't much use in attacking n+3. Make sense now? Regards, A. Chieu Nguyen wrote: > I'm a bit confused about the difference between simple induction and strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 qualifies as strong induction rather than simple. > > --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 22 10:54:07 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MEs6aJ030723 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 10:54:07 -0400 Received: (qmail 95926 invoked from network); 22 Sep 2005 14:37:52 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 22 Sep 2005 14:37:51 -0000 Message-ID: <4332C1C2.8010709@csail.mit.edu> Date: Thu, 22 Sep 2005 10:37:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Arup Sarma CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> In-Reply-To: <6.2.3.4.2.20050922025422.04c53d58@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 865 X-UID: 41 X-Keywords: Yes, could use simple induction to PROVE that P(0),...,P(n) hold if you've proved P(0) and P(k) ==> P(k+1) for all k Section 3, page 10: > I found this difficult to understand, because I'm still not sure what the difference between simple and strong induction is. Doesn't simple induction imply that P(0), P(1), ..., P(n) are true as well? > > |Arup| > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:00:31 2005 Return-Path: Received: from smtp017.mail.yahoo.com (smtp017.mail.yahoo.com [216.136.174.114]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8MF0UaJ031242 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 11:00:30 -0400 Received: (qmail 33082 invoked from network); 22 Sep 2005 15:00:30 -0000 Received: from unknown (HELO ?192.168.1.104?) (armeyer10@71.245.230.197 with plain) by smtp017.mail.yahoo.com with SMTP; 22 Sep 2005 15:00:29 -0000 Message-ID: <4332C711.2000700@csail.mit.edu> Date: Thu, 22 Sep 2005 11:00:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-staff@theory.csail.mit.edu Subject: Re: Email comments for reading References: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> In-Reply-To: <20050922103836.tsdry2whhynwccok@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 641 X-UID: 42 X-Keywords: we used to teach this, and students found it more confusing than helpful. Since it's not important, except foundationally, I now leave out the details. If you're really curious, I can sketch it for you after class some time. regards, A. Katherine A Romer wrote: >"...there's a routine way to transform any proof using the Well Ordering >Principle into a proof using Strong Induction, and vice-versa." (p.11) >I'd like to see an example in lecture of how one of the theorems we've proved >using induction could be proved using the Well Ordering Principle--I'm having a >hard time seeing how this might be done. > >Katherine Romer > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 11:01:14 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MF1EaJ031327 for ; Thu, 22 Sep 2005 11:01:14 -0400 Received: from theory.lcs.mit.edu ([128.30.51.92] helo=theory.csail.mit.edu) by rozz.csail.mit.edu with esmtp (Exim 4.50) id 1EISZW-0005RL-1J for meyer@csail.mit.edu; Thu, 22 Sep 2005 11:01:14 -0400 Received: from lyrebird.csail.mit.edu (lyrebird.csail.mit.edu [128.30.51.62]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MF1DaJ031324 for ; Thu, 22 Sep 2005 11:01:13 -0400 Received: from lyrebird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by lyrebird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8MF1DDW010858 for ; Thu, 22 Sep 2005 11:01:13 -0400 Received: (from ronitt@localhost) by lyrebird.csail.mit.edu (8.12.11/8.12.11/Submit) id j8MF1DGi010857 for meyer@csail.mit.edu; Thu, 22 Sep 2005 11:01:13 -0400 Date: Thu, 22 Sep 2005 11:01:13 -0400 From: Ronitt Rubinfeld Message-Id: <200509221501.j8MF1DGi010857@lyrebird.csail.mit.edu> To: meyer@csail.mit.edu Subject: Re: ps3 comments X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 4009 X-UID: 43 X-Keywords: Hi Albert, I have been working on ps3 in the week4 directory. It doesn't latex well, because I still have the problem including the pdf diagram. However, if you use latex (instead of pdflatex, which doesn't work at all), you can get something without the figure. Here are some comments on questions/modifications that came up: Relations course notes: in section 5 of the spring 04 notes, lemma 5.3 says that every partial order is reflexive and transitive, but I think it would be good to specify that this lemma only applies to weak partial orders. problem 1: I had them do both weak and strict partial orders in part (b). problem 2: for part (a), should we have them prove it is a maximum length chain? for part (b), changed wording, so that hopefully it is clear that they don't have to prove that the antichain is maximum. Do they have the math background to prove that it is at least as big as 2^n/n? problem 4 didn't change anything, since will cover DAG's in class. best, Ronitt From meyer@csail.mit.edu Wed Sep 21 09:33:54 2005 Date: Wed, 21 Sep 2005 09:33:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Ronitt Rubinfeld Subject: ps3 comments References: <200509211033.j8LAXi38023080@lyrebird.csail.mit.edu> In-Reply-To: <200509211033.j8LAXi38023080@lyrebird.csail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=5.0 tests=AWL,BAYES_00 autolearn=ham version=2.63 Prob 1 is ok if reworded to say "which are p.o.'s? which are equiv rels? which are equiv rels on their domain?" because I'd prefer to focus on these kinds of relations we use and deemphasize pure relational axioms/properties in in isolation. For this reason, I wouldn't use Prob 2 (unless we can rework it into a false proof like the one that was in lecture). Prob 3(b) seems very tricky TO PROVE as Eric pointed out -- he said it requires Sperner's Lemma or similar cleverness to show you can' t beat the size n choose n/2 antichain of subsets of size n/2 (though I suspect there might be a more elementary induction argument lurking around there). But I like your idea to derive a 2^n/n lower bound, which is not a bad bound: using Stirling's formula I get 2^n/sqrt{2 pi n}. I like prob 4, though for 4(c) I'd give them the power of 2 chain and ask that they prove it's maximal. Then for 4(d), I'd say: "let c be the length of the power of 2 chain. By Dilworth Lemma, there must be an antichain of length 2^n/c. Describe one." But these probs are all theoretical, and it would be nice to have something reflecting possible applications, eg, scheduling. Take a look at F05 pset5, prob 5. It's writeup uses a lot of DAG/graph terminology, but shouldn't be hard to express purely in partial order language. If you go with this one, I'd suggest dropping Prob 3 above, which seems unmotivated. I'll work on finding an intro-to-graphs problem or two. If you agree to go with probs 1, 4, and F04 ps5, prob 5, that should do it for a reasonable ps4. regards, A. P.S. please delete latex-macros-ronitt from devel/week3 (baffled me for a while why my edits to course.dat were not working.) Much better to link to the single copy of the macros in pub/staff/ than make your own that can get out of sync. Ronitt Rubinfeld wrote: >on problem 3 of the proposed homework, >there is a partial order on subsets of {1..n}. >couldn't we get them to lower bound the >size of a max length antichain by dilworth's >thm by 2^n/(n+1) (since there is no chain of length > n+1) > >maybe problem 3 and problem 4 are too similar... >in that case, I would probably go with problem 4. > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 17:18:48 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MLImaJ004507 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:48 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MLIkag027351 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:46 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MLIhV7022281 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 17:18:44 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8E59DBA2-C7A8-4296-9EE0-5AD3A6AB043C@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: week 3 reading Date: Thu, 22 Sep 2005 17:17:46 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 310 X-UID: 44 X-Keywords: NonJunk I thought the reading on induction and strong induction were pretty clear, but I'm still not sure how the Well Ordering Principle is used to prove things. More examples would've been nice. It would've been cool also to see one thing proved three different ways, just to compare and contrast. ~Crystal From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:07:27 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MM7RaJ014538 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:27 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MM7Pag001883 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:25 -0400 (EDT) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MM7Njq005029 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 18:07:24 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Message-Id: Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rian Hunter Subject: Assigned Reading Date: Thu, 22 Sep 2005 18:07:23 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.475 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8MM7RaJ014538 Status: RO X-Status: A Content-Length: 2457 X-UID: 45 X-Keywords: NonJunk From page 12 of the reading: "4 The Well Ordering Principle Another proof method closely related to induction depends on the Well Ordering Principle. Every nonempty subset of natural numbers has a smallest element. The Well Ordering Principle looks nothing like the induction axiom, and it may seem obvious but useless. But as for obvious, note that this axiom would be false if the set of non-negative integers, N, were replaced by, say, the set, Z, of all integers, or the set, Q+, of positive rational numbers. Neither of these sets has a least element. So the Well Ordering Principle does capture something special about the natural numbers. As for useless, it turns out that there’s a routine way to transform any proof using the Well Or- dering Principle into a proof using Strong Induction, and vice-versa. (We won’t take the time to describe the transformations, though they are not hard.) So Well Ordering could have been used instead of induction in all the previous proofs. In fact, looking back, we implicitly relied on the Well Ordering Principle in the proof in Week 2 Notes that √2 is irrational. That proof assumed that any rational number, q, could be written as a fraction in lowest terms, that is, q = m/n where m and n are integers with no common factors. How do we know this is always possible? First, we can assume m ≥ 0 (otherwise, replace m/n by −m/(−n)), so the set of natural numbers, m, such that q = m/n for some integer, n, is not empty. Therefore, by Well Ordering, there must be a least natural number, m0 , such that q = m0 /n0 for some integer, n0 . Now if m0 and n0 had a common factor, p > 1, then (m0 /p)/(n0 /p) would be another way to express q as a quotient of integers. But since 0 ≤ (m0 /p) < m0 . this contradicts the minimality of m0 . We’ve using the Well-ordering Principle on the sly from early on! Mathematicians often use Well Ordering because it often leads to shorter proofs than induction. On the other hand, Well Ordering proofs typically involve proof by contradiction, so using it is not always the best approach. The choice of method is really a matter of style—but style does matter." This seemed to be the most difficult passage to grasp simply because it does not concretely explain what "The Well Ordering Principle" is, because of this I would personally like to have this discussed or elaborating upon in lecture. Thanks. Rian Hunter From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:22:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMMiaJ015609 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMMgag011659 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:43 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMMdBv008253 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:22:39 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MMMdU0028571; Thu, 22 Sep 2005 18:22:39 -0400 Received: from NEXT-TWO-SEVENTEEN.MIT.EDU (NEXT-TWO-SEVENTEEN.MIT.EDU [18.242.5.217]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:22:39 -0400 Message-ID: <20050922182239.wlp6xdcmx8mc4ko4@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:22:39 -0400 From: cbossard@MIT.EDU To: "6042-probs@theory.lcs.mit.edu" <6042-probs@theory.lcs.MIT.EDU> Subject: Week 3 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.563 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 419 X-UID: 46 X-Keywords: NonJunk passage: Section 4 The Well Ordering Principle pg 12 The principle seems obvious and easy enough but I would like to see more examples on how it could be used or even an example on how it could be used instead of induction Also I am having trouble understanding being vacuously true from page 11. It seems like a contradiction that when the hypothesis for something is false the whole thing is true. Cynthia Bossard From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:53:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMrraJ020654 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMrqag000617 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMroZQ014107 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:53:50 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MMrn53001025; Thu, 22 Sep 2005 18:53:49 -0400 Received: from MDMURRAY.MIT.EDU (MDMURRAY.MIT.EDU [18.234.0.116]) (User authenticated as a_lopez@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:53:49 -0400 Message-ID: <20050922185349.n11mg3bfhr4gwsw8@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:53:49 -0400 From: a_lopez@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: week 3 readings MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 1.031 X-Spam-Level: * (1.031) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 749 X-UID: 47 X-Keywords: NonJunk "Despite the name, strong induction is actually no more powerful than ordinary induction. In other words, any theorem that can be proved with strong induction could also be proved with ordinary induction (using a slightly more complicated indcution hypothesis)." I found this very surprising, I have seen induction and strong induction before and always thought that strong induction was more powerful, that is, that some problems could be solved using strong induction but not ordinary induction. I have never actually seen a problem be given two different solutions: one with strong induction and another with ordinary induction. I wonder how much stronger the hypothesis is in ordinary induction than it is in strong induction. Adriana Lopez From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:56:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMuiaJ020819 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMugag002198 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:42 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMue7V014659 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 18:56:41 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8MMueKp006855; Thu, 22 Sep 2005 18:56:40 -0400 Received: from EASTCAMPUS-NINE-FORTY-EIGHT.MIT.EDU (EASTCAMPUS-NINE-FORTY-EIGHT.MIT.EDU [18.238.6.181]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 18:56:40 -0400 Message-ID: <20050922185640.8zrcp3xy8hcs004w@webmail.mit.edu> Date: Thu, 22 Sep 2005 18:56:40 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 216 X-UID: 48 X-Keywords: I thought the section on simple induction was just fine. I am still having trouble understanding the well ordering principle, and might like to see a different example of strong induction in class. -Rob Crowell From meyer@imap.theory.csail.mit.edu Thu Sep 22 18:57:59 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MMvxaJ020892 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MMvvag003027 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:57 -0400 (EDT) Received: from [18.243.2.35] (GALVATRON.MIT.EDU [18.243.2.35]) (authenticated bits=0) (User authenticated as zev@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MMvEUG014749 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 18:57:35 -0400 (EDT) Message-ID: <433336C9.7020102@mit.edu> Date: Thu, 22 Sep 2005 18:57:13 -0400 From: Zev Benjamin User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments X-Enigmail-Version: 0.91.0.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -2.086 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 480 X-UID: 49 X-Keywords: NonJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 I found the strong induction examples (3.2 and 3.3 on pages 10 and 11) a bit confusing. The idea of strong induction was cleared up somewhat on the problem set, however. Zev -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.1 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD8DBQFDMzbJlO3j8HLL0+4RAsMfAKDB95MzFZ0PTvenM6U2S+dlGjktqwCg/a8m nwfl5hh1oHFh8b0QTtHFEtc= =Itmf -----END PGP SIGNATURE----- From meyer@imap.theory.csail.mit.edu Thu Sep 22 19:15:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8MNFWaJ021962 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8MNFVag012949 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:31 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8MNFTjO017569 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 19:15:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8MNFTSO004043; Thu, 22 Sep 2005 19:15:29 -0400 Received: from SENIOR-TWO-O-SIX.MIT.EDU (SENIOR-TWO-O-SIX.MIT.EDU [18.244.5.206]) (User authenticated as berke@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 19:15:29 -0400 Message-ID: <20050922191529.rynea7i9aflwgcg4@webmail.mit.edu> Date: Thu, 22 Sep 2005 19:15:29 -0400 From: Allison Berke To: 6042-probs@theory.lcs.MIT.EDU Subject: reading questions MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.518 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 297 X-UID: 50 X-Keywords: NonJunk I found the section on the well-ordering principle (page 12) most interesting. I'm concurrently enrolled in 18.100b, and we've covered some proofs that use this principle, but never had it directly named. I'd be interested to see what other kinds of proofs can use this principle as a main tenet. From meyer@imap.theory.csail.mit.edu Thu Sep 22 19:33:44 2005 Message-ID: <43333F5B.4040601@csail.mit.edu> Date: Thu, 22 Sep 2005 19:33:47 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Rian Hunter CC: 6042-staff@theory.csail.mit.edu Subject: Re: Assigned Reading References: In-Reply-To: Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 584 X-UID: 51 X-Keywords: " it does not concretely explain what "The Well Ordering Principle" is" I thought the box labeled "The Well Ordering Principle" was about as concrete as it gets. Can you explain what kind of further statement of the Principle you're looking for? Rian Hunter wrote: > From page 12 of the reading: .... > > This seemed to be the most difficult passage to grasp simply because > it does not concretely explain what "The Well Ordering Principle" is, > because of this I would personally like to have this discussed or > elaborating upon in lecture. Thanks. > > Rian Hunter From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:05:34 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N05YaJ030875 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:34 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N05X4C009109 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:33 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N05W3E028720 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:32 -0400 (EDT) Received: from ebrum.mit.edu (NEW-SIX-SIXTY-THREE.MIT.EDU [18.241.7.152]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N05Q0w019037 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:05:27 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922200232.02503eb0@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 20:05:25 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: -0.685 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 267 X-UID: 52 X-Keywords: NonJunk $Label4 Theorem 2.2 was an interesting point for me, I was aware that Fibonacci numbers had weird properties but this was even beyond my expectations. By the way, the theorem states that For all n greater than or equal to 2, F_0^2+F_1^2+...+F_n^2=F_n+F_(n+1) best yasin From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:43:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N0h7aJ003599 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N0h5QQ000091 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:05 -0400 (EDT) Received: from [18.239.5.240] (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N0h1DB000578 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 20:43:03 -0400 (EDT) Message-ID: <43334F94.6010402@mit.edu> Date: Thu, 22 Sep 2005 20:43:00 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 3 Reading Assignment Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.909 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 341 X-UID: 53 X-Keywords: NonJunk Hi, After reading the week's lecture notes, I found the /well-ordering principle/ the most confusing topic to date. I didn't fully understand its connection with Induction and I was confused with the Tutor problem related to this principle. I would appreciate it if you could possibly explain this topic in more detail. Thanks, Lohith From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:04:13 2005 Message-ID: <43335495.7060805@csail.mit.edu> Date: Thu, 22 Sep 2005 21:04:21 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-staff@theory.csail.mit.edu Subject: [Fwd: Re: Reading Comments Week 3] Content-Type: text/plain; charset=iso-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1190 X-UID: 54 X-Keywords: -------- Original Message -------- Subject: Re: Reading Comments Week 3 Date: Thu, 22 Sep 2005 20:20:47 -0400 From: Chieu Nguyen To: Prof. Albert R. Meyer References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> <4332BFC5.1000706@csail.mit.edu> Dear Prof. Meyer, That was very helpful. Thanks for clearing things up. --Chieu ----- Original Message ----- From: "Prof. Albert R. Meyer" To: "Chieu Nguyen" Cc: <6042-staff@theory.csail.mit.edu> Sent: Thursday, September 22, 2005 10:29 AM Subject: Re: Reading Comments Week 3 > The k and m in the proof are only known to be SMALLER than n+2 (in fact, > they cannot equal n+2). Strong induction let's us assume both k & m are > products of primes. Simple induction would only let us assume that n+2 > was > a product of primes, which isn't much use in attacking n+3. > > Make sense now? > > Regards, A. > > Chieu Nguyen wrote: > > > I'm a bit confused about the difference between simple induction and > strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 > qualifies as strong induction rather than simple. > > > > --Chieu > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 20:20:52 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N0KpaJ032742 for ; Thu, 22 Sep 2005 20:20:51 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIbJ5-0006R6-KS for meyer@csail.mit.edu; Thu, 22 Sep 2005 20:20:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N0KoQQ018217 for ; Thu, 22 Sep 2005 20:20:50 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-SEVEN-FORTY-FOUR.MIT.EDU [18.238.5.233]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N0KjQ6027500 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for ; Thu, 22 Sep 2005 20:20:46 -0400 (EDT) Message-ID: <002401c5bfd4$a6432f40$e905ee12@addressisp.com> From: "Chieu Nguyen" To: "Prof. Albert R. Meyer" References: <00f601c5bf5a$5ee931f0$e905ee12@addressisp.com> <4332BFC5.1000706@csail.mit.edu> Subject: Re: Reading Comments Week 3 Date: Thu, 22 Sep 2005 20:20:47 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=response X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -1.882 X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 897 X-UID: 55 X-Keywords: NonJunk $Forwarded Dear Prof. Meyer, That was very helpful. Thanks for clearing things up. --Chieu ----- Original Message ----- From: "Prof. Albert R. Meyer" To: "Chieu Nguyen" Cc: <6042-staff@theory.csail.mit.edu> Sent: Thursday, September 22, 2005 10:29 AM Subject: Re: Reading Comments Week 3 > The k and m in the proof are only known to be SMALLER than n+2 (in fact, > they cannot equal n+2). Strong induction let's us assume both k & m are > products of primes. Simple induction would only let us assume that n+2 > was > a product of primes, which isn't much use in attacking n+3. > > Make sense now? > > Regards, A. > > Chieu Nguyen wrote: > > > I'm a bit confused about the difference between simple induction and > strong induction. I'm not sure how the proof of Lemma 3.1 on page 10 > qualifies as strong induction rather than simple. > > > > --Chieu > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:25:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N1PraJ009865 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N1PqQQ023263 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:52 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N1PnAC006779 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:25:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8N1Pnas029585; Thu, 22 Sep 2005 21:25:49 -0400 Received: from BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (User authenticated as hectorb@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:25:49 -0400 Message-ID: <20050922212549.kxmlkccxu1kcs80g@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:25:49 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 3 Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 315 X-UID: 56 X-Keywords: NonJunk "...there?s a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. So Well Ordering could have been used instead of induction in all the previous proofs." I would like to see some more examples of this, it's confusing to me. -Hector Beltran From meyer@imap.theory.csail.mit.edu Thu Sep 22 21:27:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N1RAaJ009907 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N1R9QQ023912 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:09 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N1R6up006951 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 21:27:06 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N1R69C021223; Thu, 22 Sep 2005 21:27:06 -0400 Received: from NEXT-FIVE-O-THREE.MIT.EDU (NEXT-FIVE-O-THREE.MIT.EDU [18.242.6.248]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:27:06 -0400 Message-ID: <20050922212706.54lp2oenk9444o08@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:27:06 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: E-mail Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.804 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 308 X-UID: 57 X-Keywords: NonJunk Reference: Section 3.3 Pg 11 I clearly understood normal induction, however I did have some difficulty with strong induction (especially this problem) and would like to discuss it in more detail in class. I did understand it after close reading but I was hoping we could go over it in class. -Rahul Shroff From meyer@imap.theory.csail.mit.edu Thu Sep 22 22:12:55 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N2CtaJ014084 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:55 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2CrQQ020487 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:53 -0400 (EDT) Received: from [18.227.1.178] ([18.227.1.178]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N2CmjR013844 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 22:12:50 -0400 (EDT) Message-ID: <43336488.5090707@mit.edu> Disposition-Notification-To: Scot Frank Date: Thu, 22 Sep 2005 22:12:24 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.577 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 265 X-UID: 58 X-Keywords: NonJunk $MDNSent Hi, Even after reading about Strong Inductions twice, I did not feel the example was explained clearly in class. Perhaps it is my own fault but I was not able to make more sense of it even from the reading. I think I will come to office hours. Thanks, Scot From meyer@imap.theory.csail.mit.edu Thu Sep 22 22:17:10 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N2HAaJ014512 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:10 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2H94C027571 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:10 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N2H93E001590 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:09 -0400 (EDT) Received: from [18.234.0.116] (MDMURRAY.MIT.EDU [18.234.0.116]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N2H63a025969 for <6042-probs@theory.csail.mit.edu>; Thu, 22 Sep 2005 22:17:06 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-34-958248826 From: Michael Murray Subject: E-mail question Date: Thu, 22 Sep 2005 22:17:00 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.523 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 7896 X-UID: 59 X-Keywords: NonJunk --Apple-Mail-34-958248826 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed "First, we can assume m 0 (otherwise, replace m/n by =E2=88=92m/(=E2=88=92= n)), =20 so the set of natural numbers, m, such that q =3D m/n for some integer, n, is not empty. Therefore, by =20= Well Ordering, there must be a least natural number, m0, such that q =3D m0/n0 for some integer, =20= n0. Now if m0 and n0 had a common factor, p > 1, then (m0/p)/(n0/p) would be another way to =20 express q as a quotient of integers. But since 0 (m0/p) < m0. this contradicts the minimality =20 of m0." I did not fully understand this paragraph because i do not =20 understand how we reach a contradiction. This was the part in the =20 notes that I found most confusing. MIchael Murray --Apple-Mail-34-958248826 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
"First, we can assume m =C2=A0 0 (otherwise, replace m/n by =E2=88=92m/(=E2=88= =92n)), so the set of natural numbers,
m, such that q =3D = m/n for some integer, n, is not empty. Therefore, by Well Ordering, = there must
be a least natural number, = m0, such that q =3D = m0/n0 = for some integer, = n0. Now if = m0 = and = n0 = had
a common factor, p > 1, then = (m0/p)/(n0/p) would be = another way to express q as a quotient of
integers. But = since 0 =C2=A0 = (m0/p) < = m0. this contradicts = the minimality of m0."

I did not fully = understand this paragraph=C2=A0 because i do not understand how we reach = a contradiction.=C2=A0 This was the part in the notes that I found most = confusing.

MIchael = Murray



= --Apple-Mail-34-958248826-- From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:12:33 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3CXaJ018919 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:33 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3CVk8023598 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:31 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3CTs1023533 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 22 Sep 2005 23:12:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N3CTsd028402; Thu, 22 Sep 2005 23:12:29 -0400 Received: from comm575-0401-dhcp142.bu.edu (comm575-0401-dhcp142.bu.edu [168.122.167.142]) (User authenticated as brevzin@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 23:12:29 -0400 Message-ID: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Date: Thu, 22 Sep 2005 23:12:29 -0400 From: Boris E Revzin To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 449 X-UID: 60 X-Keywords: NonJunk Regarding the induction lecture, it seems excessive to me to have to write out the entire induction template. By showing a base case and P(n)->P(n+1), it is pretty clear to anyone reading the proof that it is by induction. I just want to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all that is necessary, oftentimes writing out the template will take more time than writing out the actual proof... Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:15:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3FwaJ019046 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:58 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3Fuk8025673 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:57 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3FrDc024086 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:15:53 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8N3FrgJ018437; Thu, 22 Sep 2005 23:15:53 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 21:15:53 -0600 Message-ID: <20050922211553.u61uh3a33cw0g8k4@webmail.mit.edu> Date: Thu, 22 Sep 2005 21:15:53 -0600 From: John Marrero To: 6042-probs@theory.csail.MIT.EDU Subject: Reading Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 459 X-UID: 61 X-Keywords: NonJunk Page 10, "Strong Induction" This is the only thing I had a problem with, since it seems so similar to simple induction. I found it somewhat difficult to see the difference between the two. If there's an extra example that can be seen in lecture, I'm sure it'd clarify things! Page 12, "The Well Ordering Principle" Interesting; I wonder if there's a principle that says there's a greatest element in a nonempty subset? Just made me think. - John Marrero From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:36:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3aQaJ022235 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3aOk8006811 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:25 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3aMN0027189 for <6042-probs@theory.csail.MIT.EDU>; Thu, 22 Sep 2005 23:36:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8N3aMYA021006; Thu, 22 Sep 2005 23:36:22 -0400 Received: from EASTCAMPUS-NINE-FIFTY-THREE.MIT.EDU (EASTCAMPUS-NINE-FIFTY-THREE.MIT.EDU [18.238.6.186]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 22 Sep 2005 23:36:22 -0400 Message-ID: <20050922233622.h94vhwbllhugw8oc@webmail.mit.edu> Date: Thu, 22 Sep 2005 23:36:22 -0400 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: `Required reading comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 201 X-UID: 62 X-Keywords: NonJunk Page 12, well ordering principle (section 4) I am pretty sure that I underrstood everything, but a little bit more work on the well ordering principle could potentially be helpful. -Forrest Green From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:49:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3nkaJ025685 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3nik8014142 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:44 -0400 (EDT) Received: from JOSHDESKTOP.mit.edu (BURTON-ONE.MIT.EDU [18.247.5.1]) (authenticated bits=0) (User authenticated as jjmonzon@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3nb5j029156 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:49:40 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 23:49:50 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Joshua Jen C. Monzon" Subject: reading comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.117 X-Spam-Level: * (1.117) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 578 X-UID: 63 X-Keywords: NonJunk I found the whole topic of induction to be such an elegant yet sound way to prove stuff. For example the courtyard problem amazed me since without knowing about induction I would still be brute forcing my way to tile the whole courtyard with L-shaped tiles. I'm only a bit confused on reasoning and claims which would make a proof using induction wrong. It could be made much clearer when induction would not be valid. Josh Monzon Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:53:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N3rhaJ025922 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N3rgk8016571 for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:42 -0400 (EDT) Received: from hzhou.mit.edu (BURTON-ONE-O-ONE.MIT.EDU [18.247.5.101]) (authenticated bits=0) (User authenticated as hzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N3rTT1029809 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 22 Sep 2005 23:53:38 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922235226.02a6c168@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 22 Sep 2005 23:53:31 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 150 X-UID: 64 X-Keywords: NonJunk I would like to see more practices in Induction in class, and more particular cases when induction fails (like the horse example in class) - Steve From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:57:05 2005 Message-ID: <43337D19.3030007@csail.mit.edu> Date: Thu, 22 Sep 2005 23:57:13 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Boris E Revzin CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> In-Reply-To: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 743 X-UID: 65 X-Keywords: we'll relax the template discipline in a week or so, but for now and through ps2, please string along with us and use the template. A surprising number of students in the class need the template when they begin learning induction proofs. regards, A. Boris E Revzin wrote: > Regarding the induction lecture, it seems excessive to me to have to write out > the entire induction template. By showing a base case and P(n)->P(n+1), it is > pretty clear to anyone reading the proof that it is by induction. I just want > to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all > that is necessary, oftentimes writing out the template will take more time than > writing out the actual proof... > > Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:02:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N42vaJ028277 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N42tk8021784 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:55 -0400 (EDT) Received: from TNG.mit.edu (BURTON-SIX-THIRTY-SEVEN.MIT.EDU [18.247.7.126]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N42pYh001291 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:02:53 -0400 (EDT) Message-Id: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 23 Sep 2005 00:02:58 -0400 To: 6042-probs@theory.csail.mit.edu From: Tony Ng Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 398 X-UID: 66 X-Keywords: NonJunk The "All Horses are the Same Color", starting on page 9, proof kind of bothered me. The flaw with the proof is that it is not true for n = 1. However, is it legitimate to try to prove some add horse function (which doesn't seem really related to mathematics and numbers)? I can imagine someone clever figuring out some proof that satisfy the n=1 case and prove all horses are the same color. From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:12:46 2005 Message-ID: <433380C7.9000400@csail.mit.edu> Date: Fri, 23 Sep 2005 00:12:55 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Tony Ng CC: 6042-staff@theory.csail.mit.edu Subject: Re: Week 3 Comments References: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922235948.01f56998@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 711 X-UID: 67 X-Keywords: I don't know what you mean when you say "try to prove some add horse function." Propositions get proved; functions don't get proved, they get applied. But you seem to be saying that a clever person could prove something that wasn't true. Do you really mean that? Regards, A. Tony Ng wrote: > The "All Horses are the Same Color", starting on page 9, proof kind of > bothered me. The flaw with the proof is that it is not true for n = 1. > However, is it legitimate to try to prove some add horse function (which > doesn't seem really related to mathematics and numbers)? I can imagine > someone clever figuring out some proof that satisfy the n=1 case and > prove all horses are the same color. > From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:06:23 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N46NaJ029504 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:06:23 -0400 Received: (qmail 46094 invoked from network); 23 Sep 2005 04:06:22 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 04:06:22 -0000 Message-ID: <43337F48.10907@csail.mit.edu> Date: Fri, 23 Sep 2005 00:06:32 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 918 X-UID: 68 X-Keywords: Induction is always valid :-). What you have to watch out for are proofs that claim to be by induction but do it wrong: they skip some cases or make a mistake in proving the cases. We'll discuss additional mistakes in induction proofs as the term progresses. regards, A. Joshua Jen C. Monzon wrote: > I found the whole topic of induction to be such an elegant yet sound way > to prove stuff. For example the courtyard problem amazed me since > without knowing about induction I would still be brute forcing my way to > tile the whole courtyard with L-shaped tiles. I'm only a bit confused on > reasoning and claims which would make a proof using induction wrong. It > could be made much clearer when induction would not be valid. > > Josh Monzon > > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:59:30 2005 Return-Path: Received: from smtp104.mail.sc5.yahoo.com (smtp104.mail.sc5.yahoo.com [66.163.169.223]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N3xTaJ028047 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 23:59:30 -0400 Received: (qmail 33870 invoked from network); 23 Sep 2005 03:59:28 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp104.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 03:59:28 -0000 Message-ID: <43337DAA.7030202@csail.mit.edu> Date: Thu, 22 Sep 2005 23:59:38 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Michael Murray CC: 6042-staff@theory.csail.mit.edu Subject: Re: E-mail question References: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> In-Reply-To: <9B3CB3F4-7DBB-4BE2-ACA3-E68CD73454BF@mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 878 X-UID: 69 X-Keywords: we let m0 be the smallest numerator in a fraction for and rational number, q, and then we found a smaller one -- that's the contradiction. Does that help? regards, A. Michael Murray wrote: > "First, we can assume m 0 (otherwise, replace m/n by −m/(−n)), so the > set of natural numbers, > m, such that q = m/n for some integer, n, is not empty. Therefore, by > Well Ordering, there must > be a least natural number, m0, such that q = m0/n0 for some integer, n0. > Now if m0 and n0 had > a common factor, p > 1, then (m0/p)/(n0/p) would be another way to > express q as a quotient of > integers. But since 0 (m0/p) < m0. this contradicts the minimality of m0." > > I did not fully understand this paragraph because i do not understand > how we reach a contradiction. This was the part in the notes that I > found most confusing. > > MIchael Murray > > > From meyer@imap.theory.csail.mit.edu Thu Sep 22 23:57:06 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N3v6aJ027853 for <6042-staff@theory.csail.mit.edu>; Thu, 22 Sep 2005 23:57:06 -0400 Received: (qmail 69752 invoked from network); 23 Sep 2005 03:57:05 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 03:57:04 -0000 Message-ID: <43337D19.3030007@csail.mit.edu> Date: Thu, 22 Sep 2005 23:57:13 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Boris E Revzin CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading Comments References: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> In-Reply-To: <20050922231229.fko9r1zte424ocw8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 742 X-UID: 70 X-Keywords: we'll relax the template discipline in a week or so, but for now and through ps2, please string along with us and use the template. A surprising number of students in the class need the template when they begin learning induction proofs. regards, A. Boris E Revzin wrote: > Regarding the induction lecture, it seems excessive to me to have to write out > the entire induction template. By showing a base case and P(n)->P(n+1), it is > pretty clear to anyone reading the proof that it is by induction. I just want > to know if it's enough to show P(0), P(n)->P(n+1), QED. I just think that's all > that is necessary, oftentimes writing out the template will take more time than > writing out the actual proof... > > Boris "Barry" Revzin From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:42:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4gqaJ002453 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4gpk8013572 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:51 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4gntJ007021 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 00:42:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N4gnm1006146; Fri, 23 Sep 2005 00:42:49 -0400 Received: from MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU (MACGREGOR-SEVEN-FORTY-TWO.MIT.EDU [18.239.7.231]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 00:42:49 -0400 Message-ID: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Date: Fri, 23 Sep 2005 00:42:49 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: week 3 comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 170 X-UID: 71 X-Keywords: NonJunk Proofs by induction are completely foriegn to me. Even after class, I still have no clue how to figure out where th flaws in the tutor quesiton #3 is. -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:48:34 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4mYaJ002720 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:34 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4mXgh014819 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:33 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4mWTI004839 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:33 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N4mP0v028194 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:48:30 -0400 (EDT) Message-Id: <200509230448.j8N4mP0v028194@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 00:48:22 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_00AB_01C5BFD8.82A1CC50" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW/+gZWJmeNNMF0R8G6OUMmTW+Gnw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 1.957 X-Spam-Level: * (1.957) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2231 X-UID: 72 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_00AB_01C5BFD8.82A1CC50 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Page 12's discussion of the Well Ordering Principle is quite fascinating. How do you prove the principle, in a general sense? Straightforward subsets (like even numbers) are pretty obviously true, but what about more complicated sets? Chris Yang ------=_NextPart_000_00AB_01C5BFD8.82A1CC50 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Page 12’s discussion of the Well Ordering = Principle is quite fascinating.  How do you prove the principle, in a general = sense?  Straightforward subsets (like even numbers) are pretty obviously true, = but what about more complicated sets?

 

Chris Yang

------=_NextPart_000_00AB_01C5BFD8.82A1CC50-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:49:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4nHaJ002740 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4nGk8017911 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:16 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4n3Wn007703 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 00:49:08 -0400 (EDT) Message-Id: <200509230449.j8N4n3Wn007703@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: reading comments week 3 Date: Fri, 23 Sep 2005 00:49:01 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFD8.9A7CD2C0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW/+h3wb19sjKSST4G8C60eQyp81Q== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2545 X-UID: 73 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFD8.9A7CD2C0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit For me the most confusing part of the reading was the Well Ordering Principle. Particularly, how it was presented as a parallel method to induction, whereas said method merely depends on it. Additionally the tutor problems seemed to reinforce its presentation as a regular proof method rather than a tool used by one. Peter Bilodeau ------=_NextPart_000_0000_01C5BFD8.9A7CD2C0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

 

For me the most confusing part of the reading was the = Well Ordering Principle.  Particularly, how it was presented as a = parallel method to induction, whereas said method merely depends on it.  Additionally = the tutor problems seemed to reinforce its presentation as a regular proof method = rather than a tool used by one.

 

Peter Bilodeau

------=_NextPart_000_0000_01C5BFD8.9A7CD2C0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:56:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N4uKaJ003207 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:20 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N4uJk8022635 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:19 -0400 (EDT) Received: from [10.0.1.2] (c-24-218-218-5.hsd1.ma.comcast.net [24.218.218.5]) (authenticated bits=0) (User authenticated as rian@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N4uHdS008479 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 00:56:17 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) In-Reply-To: <43333F5B.4040601@csail.mit.edu> References: <43333F5B.4040601@csail.mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <9BF45317-CBDD-41CA-B1B0-314DC9819D1E@mit.edu> Content-Transfer-Encoding: 7bit From: Rian Hunter Subject: Re: Assigned Reading Date: Fri, 23 Sep 2005 00:56:15 -0400 To: 6042-probs@theory.csail.mit.edu X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1337 X-UID: 74 X-Keywords: NonJunk $Forwarded Actually I guess the reading tricked me. I was used to the proof methods being explained how to be used step by step, while the Well Ordering Principle is just a simple statement. I thought the statement in the box "Every nonempty subset..." was a statement that could be proved using the Well Ordering Principle; I wasn't expecting that it was the Well Ordering Principle itself. Actually it's unclear to me whether or not the Well Ordering Principle is a proof method itself or just a principle used in other proof methods (proof by contradiction for example), but after re-reading and looking at the context it seems that it isn't. -rian On Sep 22, 2005, at 7:33 PM, Prof. Albert R. Meyer wrote: > " it does not concretely explain what "The Well Ordering Principle" > is" I thought the box labeled "The Well Ordering Principle" was > about as concrete as it gets. Can you explain what kind of further > statement of the Principle you're looking for? > > Rian Hunter wrote: > > >> From page 12 of the reading: .... >> >> This seemed to be the most difficult passage to grasp simply >> because it does not concretely explain what "The Well Ordering >> Principle" is, because of this I would personally like to have >> this discussed or elaborating upon in lecture. Thanks. >> >> Rian Hunter >> > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:03:31 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N53VaJ004283 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:03:31 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N53Vbd017229 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:03:31 -0400 (EDT) Received: from white-meteo.mit.edu (WHITE-METEO.MIT.EDU [18.243.0.221]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N53S3a001033 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:03:28 -0400 (EDT) Received: (from rnjacobs@localhost) by white-meteo.mit.edu (8.12.9) id j8N53Sqd027218; Fri, 23 Sep 2005 01:03:28 -0400 Message-Id: <200509230503.j8N53Sqd027218@white-meteo.mit.edu> To: 6042-probs@theory.lcs.mit.edu Subject: reading questions due sep 23 Date: Fri, 23 Sep 2005 01:03:28 -0400 From: r n jacobs X-Spam-Score: 1.729 X-Spam-Level: * (1.729) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 225 X-UID: 75 X-Keywords: NonJunk I found myself confused by the explanation of Strong Induction. Although maybe the fibbonaci proof helped that. (The definition made sense, but it took far to long for me to figure out the answer to the fibbonaci misproof) From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:12:35 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5CZaJ004634 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:35 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5CYk8001347 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:34 -0400 (EDT) Received: from AMMAR (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5CUZL010074 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:12:31 -0400 (EDT) Message-ID: <002b01c5bffd$66bd72d0$d406f112@AMMAR> From: "Adnaan" To: <6042-probs@theory.lcs.mit.edu> Subject: email comments for 24/9 Date: Fri, 23 Sep 2005 01:12:30 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0028_01C5BFDB.DEC6F110" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -0.595 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1176 X-UID: 76 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C5BFDB.DEC6F110 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Hi,=20 I found the reading hard especially the "Well order reading" part. I = didn't understand how it is related to induction Adnaan Jiwaji ------=_NextPart_000_0028_01C5BFDB.DEC6F110 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Hi,
 
I found the reading hard especially the = "Well order=20 reading" part. I didn't understand how it is related to = induction
 
Adnaan = Jiwaji
------=_NextPart_000_0028_01C5BFDB.DEC6F110-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:27:00 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5R0aJ005427 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:27:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5Qxk8008147 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:26:59 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-FORTY-SEVEN.MIT.EDU [18.241.6.92]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5Qrrn011667 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:26:55 -0400 (EDT) Message-Id: <6.2.1.2.2.20050923012004.039a2e20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 23 Sep 2005 01:26:52 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: Week 3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 328 X-UID: 77 X-Keywords: NonJunk Passage: 2.4 Courtyard Tiling Page: 6 I found this passage most surprising because it used induction to prove a geometric predicate. I knew that induction could be used to prove mathematical predicates but I did not think it was possible to use induction to prove a geometric predicate. Sincerely, Timothy M. Mwangi From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:28:15 2005 Message-ID: <43339279.90703@csail.mit.edu> Date: Fri, 23 Sep 2005 01:28:25 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 comments References: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> In-Reply-To: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 274 X-UID: 78 X-Keywords: I encourage you to go to TA office hours and go over this. regards,A. Daniel A Gutierrez wrote: > Proofs by induction are completely foriegn to me. Even after class, I still > have no clue how to figure out where th flaws in the tutor quesiton #3 is. > -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:26:15 2005 Message-ID: <43339201.3070700@csail.mit.edu> Date: Fri, 23 Sep 2005 01:26:25 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-staff@theory.csail.mit.edu Subject: [Fwd: Re: Assigned Reading] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1588 X-UID: 79 X-Keywords: -------- Original Message -------- Subject: Re: Assigned Reading Date: Fri, 23 Sep 2005 00:56:15 -0400 From: Rian Hunter To: 6042-probs@theory.csail.mit.edu References: <43333F5B.4040601@csail.mit.edu> Actually I guess the reading tricked me. I was used to the proof methods being explained how to be used step by step, while the Well Ordering Principle is just a simple statement. I thought the statement in the box "Every nonempty subset..." was a statement that could be proved using the Well Ordering Principle; I wasn't expecting that it was the Well Ordering Principle itself. Actually it's unclear to me whether or not the Well Ordering Principle is a proof method itself or just a principle used in other proof methods (proof by contradiction for example), but after re-reading and looking at the context it seems that it isn't. -rian On Sep 22, 2005, at 7:33 PM, Prof. Albert R. Meyer wrote: > " it does not concretely explain what "The Well Ordering Principle" > is" I thought the box labeled "The Well Ordering Principle" was > about as concrete as it gets. Can you explain what kind of further > statement of the Principle you're looking for? > > Rian Hunter wrote: > > >> From page 12 of the reading: .... >> >> This seemed to be the most difficult passage to grasp simply >> because it does not concretely explain what "The Well Ordering >> Principle" is, because of this I would personally like to have >> this discussed or elaborating upon in lecture. Thanks. >> >> Rian Hunter >> > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 00:06:22 2005 Message-ID: <43337F48.10907@csail.mit.edu> Date: Fri, 23 Sep 2005 00:06:32 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-staff@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050922234608.02076ba0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 919 X-UID: 80 X-Keywords: Induction is always valid :-). What you have to watch out for are proofs that claim to be by induction but do it wrong: they skip some cases or make a mistake in proving the cases. We'll discuss additional mistakes in induction proofs as the term progresses. regards, A. Joshua Jen C. Monzon wrote: > I found the whole topic of induction to be such an elegant yet sound way > to prove stuff. For example the courtyard problem amazed me since > without knowing about induction I would still be brute forcing my way to > tile the whole courtyard with L-shaped tiles. I'm only a bit confused on > reasoning and claims which would make a proof using induction wrong. It > could be made much clearer when induction would not be valid. > > Josh Monzon > > > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:28:18 2005 Return-Path: Received: from smtp101.mail.sc5.yahoo.com (smtp101.mail.sc5.yahoo.com [216.136.174.139]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8N5SGaJ005485 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:28:16 -0400 Received: (qmail 59927 invoked from network); 23 Sep 2005 05:28:15 -0000 Received: from unknown (HELO ?192.168.1.101?) (armeyer10@71.245.230.197 with plain) by smtp101.mail.sc5.yahoo.com with SMTP; 23 Sep 2005 05:28:15 -0000 Message-ID: <43339279.90703@csail.mit.edu> Date: Fri, 23 Sep 2005 01:28:25 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-staff@theory.csail.mit.edu Subject: Re: week 3 comments References: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> In-Reply-To: <20050923004249.3u9ogggf52r4sgg0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 273 X-UID: 81 X-Keywords: I encourage you to go to TA office hours and go over this. regards,A. Daniel A Gutierrez wrote: > Proofs by induction are completely foriegn to me. Even after class, I still > have no clue how to figure out where th flaws in the tutor quesiton #3 is. > -Daniel Gutierrez From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:36:08 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5a8aJ005889 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:08 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5a6k8012554 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:07 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5a4Pt013770 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:36:04 -0400 (EDT) Message-ID: <43339446.1070908@mit.edu> Date: Fri, 23 Sep 2005 01:36:06 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: (no subject) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 264 X-UID: 82 X-Keywords: NonJunk From this week's notes, I found the proof that we went over last class re: the tiling (Section 2.4, page 8) to be the most surprising, since it was a very elegant and easy to prove answer to what I thought initially was going to be a hard to prove conjecture. From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:36:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5aqaJ005906 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:52 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5apk8012968 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:51 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5ajP8013932 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 01:36:46 -0400 (EDT) Message-Id: <200509230536.j8N5ajP8013932@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 01:36:34 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFDF.3C0C2BD0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 thread-index: AcXAAMIwN13vJ5m8Spus8/pOCtkcnw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 3341 X-UID: 83 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFDF.3C0C2BD0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Well Ordering is well.not understandable. If we study it tomorrow (later today) in class, perhaps I'll get it. Now the passage just looks like a bunch of words that I do not comprehend. Could you provide an example, or two? --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_0000_01C5BFDF.3C0C2BD0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Well Ordering is well…not understandable. If we = study it tomorrow (later today) in class, perhaps I’ll get it. Now the = passage just looks like a bunch of words that I do not comprehend. Could you = provide an example, or two?

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

------=_NextPart_000_0000_01C5BFDF.3C0C2BD0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:46:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5kraJ007386 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5kqk8017589 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5knKh016117 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:46:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N5knrx018267; Fri, 23 Sep 2005 01:46:49 -0400 Received: from NEW-THREE-SIXTY-SEVEN.MIT.EDU (NEW-THREE-SIXTY-SEVEN.MIT.EDU [18.241.6.112]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 01:46:49 -0400 Message-ID: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> Date: Fri, 23 Sep 2005 01:46:49 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_3wt9ryflgz40" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.329 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 767 X-UID: 84 X-Keywords: NonJunk This message is in MIME format. --=_3wt9ryflgz40 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit Sorry there isn't as much this time, its just rather more familiar material. --=_3wt9ryflgz40 Content-Type: text/plain; charset=ISO-8859-1; name="week3comment.txt" Content-Disposition: attachment; filename="week3comment.txt" Content-Transfer-Encoding: 7bit Induction was covered in about as much detail as a highschool can cover it. I'm familiar with it, although its been a while. The in class problems were much easier for me than those of previous lectures. Strong induction is new, but its not hard to grasp. So, I think I understand pretty well whats going on with induction. --=_3wt9ryflgz40-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 01:53:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N5rTaJ007760 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:29 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N5rBk8020639 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:11 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5r8dY017163 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 01:53:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N5r86m012850; Fri, 23 Sep 2005 01:53:08 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 01:53:08 -0400 Message-ID: <20050923015308.8xqusatei7vowwow@webmail.mit.edu> Date: Fri, 23 Sep 2005 01:53:08 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP2 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 129 X-UID: 85 X-Keywords: NonJunk I would like to see an application of the well ordering principle in lecture (possibly in a proof or solution to a problem) Lana From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:00:17 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N60HaJ008158 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:00:17 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N60DaW022733 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:00:16 -0400 (EDT) Received: from [18.238.5.210] (EASTCAMPUS-SEVEN-TWENTY-ONE.MIT.EDU [18.238.5.210]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N5tn2I017355 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 01:55:49 -0400 (EDT) Message-ID: <433398EC.90508@mit.edu> Date: Fri, 23 Sep 2005 01:55:56 -0400 From: Alison Cichowlas User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: reading comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.537 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 234 X-UID: 86 X-Keywords: NonJunk I found the discussion of the Well-Ordering principle (last page, p. 12) most surprising. Although the statement itself makes perfect sense, it's really interesting how many other things depend on this seemingly-obvious principle. From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:27:02 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6R2aJ010644 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:27:02 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6R1Cg019151 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:27:01 -0400 (EDT) Received: from [127.0.0.1] (EASTCAMPUS-EIGHT-NINETY-EIGHT.MIT.EDU [18.238.6.131]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N6Qt3c003956 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:26:58 -0400 (EDT) Message-ID: <4333A02A.8090800@mit.edu> Date: Fri, 23 Sep 2005 02:26:50 -0400 From: Isaac Charny User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Lecture notes 3 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 176 X-UID: 87 X-Keywords: NonJunk On page 12 of the notes, you discuss the well ordering principle (section 4). Please discuss this further in class, as well as applications of this principle. ~Isaac Charny From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:31:26 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6VQaJ010928 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:26 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6VPad010737 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:25 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6VOdU007124 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:25 -0400 (EDT) Received: from MACGREGOR-TWO-EIGHTEEN.MIT.EDU (MACGREGOR-TWO-EIGHTEEN.MIT.EDU [18.239.5.218]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N6VL0v001607 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:31:21 -0400 (EDT) Subject: Reading Comments #3 From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 02:31:16 -0400 Message-Id: <1127457077.6912.5.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 402 X-UID: 88 X-Keywords: NonJunk One thing I found that was particularly surprising about the reading was the discussion of the Well Ordering Principle on page 12. While (aside from the described caveats) it does seem obvious, I found it particularly interesting that the Well Ordering Principle could be used to replace strong induction. Though it's only out of curiosity, it might be good to see an example or two of this. -- Matt From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:32:21 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6WLaJ010963 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:21 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6WKaQ006639 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:20 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6WCDb019621 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 02:32:13 -0400 (EDT) Message-Id: <200509230632.j8N6WCDb019621@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 02:32:09 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5BFE6.FFAC9190" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXACEy/bhMaT0tDTvuaHAYgTNLwvg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.833 X-Spam-Level: * (1.833) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2018 X-UID: 89 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5BFE6.FFAC9190 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Could you please go over 2.5 A Faulty Induction Proof (pages 8-9 of ln3). I'm not sure I understand it. ------=_NextPart_000_0000_01C5BFE6.FFAC9190 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Could you please go over 2.5 A Faulty = Induction Proof (pages 8-9 of ln3). I’m not sure I understand = it.

------=_NextPart_000_0000_01C5BFE6.FFAC9190-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:38:03 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6c3aJ011162 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:38:03 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6c1aQ009102 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:38:02 -0400 (EDT) Received: from grumpy-fuzzball.mit.edu (GRUMPY-FUZZBALL.MIT.EDU [18.7.16.79]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6bxXl019980 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 02:37:59 -0400 (EDT) Received: (from medrano@localhost) by grumpy-fuzzball.mit.edu (8.12.9) id j8N6bxAY023978; Fri, 23 Sep 2005 02:37:59 -0400 (EDT) Date: Fri, 23 Sep 2005 02:37:59 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: Induction Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 197 X-UID: 90 X-Keywords: NonJunk Induction is more formulaic than deductive proofs and thus much easier to grasp. There is not much else I can say about it. The reading was straightforward and easy to understand. Jesus Medrano From meyer@imap.theory.csail.mit.edu Fri Sep 23 02:57:52 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N6vpaJ013208 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N6voaQ017326 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:50 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY.MIT.EDU (MACGREGOR-FOUR-EIGHTY.MIT.EDU [18.239.6.225]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N6vfxu021004 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 02:57:48 -0400 (EDT) Subject: tp3 reading From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 02:57:36 -0400 Message-Id: <1127458656.18591.1.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.227 X-Spam-Level: *** (3.227) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 147 X-UID: 91 X-Keywords: NonJunk While the well-ordering principle (p. 12) is fairly straight forward, I am wondering if we can see some more examples of what it can be used for. From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:06:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N76VaJ013853 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N76UaQ021064 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:30 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-ONE-O-THREE.MIT.EDU [18.242.5.103]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N76QnE021478 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:06:28 -0400 (EDT) Message-Id: <6.2.3.4.2.20050923030000.01c13d90@po10@mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 23 Sep 2005 03:06:24 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: Comments for Reading Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.544 X-Spam-Level: * (1.544) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 383 X-UID: 92 X-Keywords: NonJunk Hi Prof Meyer and Rubinfeld, I found sections 3.2 and 3.3 on pages 10-12 to be confusing. I didn't grasp the usage of strong induction in these proofs, though I had some vague idea of how they came into play (the P(n+2) for the primes and argument: Case (n+1 > 35): etc ... were very confusing). If you could go into detail about them, then that would be great. Thanks, Kevin From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:29:26 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7TQaJ015468 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:26 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7TPaQ000593 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:25 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7TMBD022472 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 03:29:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8N7TM4m017698; Fri, 23 Sep 2005 03:29:22 -0400 Received: from SIMMONS-SEVEN-TWENTY-NINE.MIT.EDU (SIMMONS-SEVEN-TWENTY-NINE.MIT.EDU [18.96.7.218]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 03:29:22 -0400 Message-ID: <20050923032922.urcb7e9e0zk0wcoc@webmail.mit.edu> Date: Fri, 23 Sep 2005 03:29:22 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: week 3 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 291 X-UID: 93 X-Keywords: NonJunk I would like to have the principle of strong induction, specifically the Product of Primes example (pp. 10-11) explained more fully. I am a bit confused as to why you can assume more with the strong induction principle, and also when to use strong induction as opposed to regular induction. From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:51:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7paaJ019295 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7pZaQ009148 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:35 -0400 (EDT) Received: from [18.240.5.41] (MCCORMICK-FORTY-ONE.MIT.EDU [18.240.5.41]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7pXat023357 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:51:33 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Transfer-Encoding: 7bit Message-Id: <66636106d07b6a5641402dcc17e625a2@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: Comment on Induction Reading Date: Fri, 23 Sep 2005 03:51:31 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 236 X-UID: 94 X-Keywords: NonJunk I thought the sections on Strong Induction were confusing and the example given didn't make sense to me. I would appreciate more time spent on this in class, especially as it was a large part of the tutor problems. Rachel Shearer From meyer@imap.theory.csail.mit.edu Fri Sep 23 03:58:10 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N7wAaJ019523 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:10 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N7w9aQ011735 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:09 -0400 (EDT) Received: from [18.242.5.26] (NEXT-TWENTY-SIX.MIT.EDU [18.242.5.26]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N7vwva023615 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 03:58:06 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: reading comment Date: Fri, 23 Sep 2005 03:57:56 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.074 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 105 X-UID: 95 X-Keywords: NonJunk I'm still a little confused on the Well Ordering Principle on page 12 of the reading and how it works. From meyer@imap.theory.csail.mit.edu Fri Sep 23 04:11:33 2005 Return-Path: Received: from grand-central-station.mit.edu (GRAND-CENTRAL-STATION.MIT.EDU [18.7.21.82]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N8BWaJ020744 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:33 -0400 Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by grand-central-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N8BWCg021007 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:32 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8N8BT3a006750 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 04:11:29 -0400 (EDT) Subject: Comments for Course Notes, Week 3 (Induction) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 23 Sep 2005 04:11:28 -0400 Message-Id: <1127463088.5662.12.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 265 X-UID: 96 X-Keywords: NonJunk I found section 4, The Well Ordering Principle, (located on page 12) very surprising. I was surprised that Strong Induction can be transformed into the Well Ordering Principle (and vice versa). I would be interested in seeing how this is done. Moira Racich From meyer@imap.theory.csail.mit.edu Fri Sep 23 04:21:32 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N8LWaJ021609 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:32 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N8LUaQ021053 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:30 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N8LNgA024466 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 04:21:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8N8LNUR028297; Fri, 23 Sep 2005 04:21:23 -0400 Received: from EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU (EASTCAMPUS-NINE-THIRTY-THREE.MIT.EDU [18.238.6.166]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 04:21:23 -0400 Message-ID: <20050923042123.0f7ianeqbykg4cog@webmail.mit.edu> Date: Fri, 23 Sep 2005 04:21:23 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: Email comments, Week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 437 X-UID: 97 X-Keywords: NonJunk On page 12, the notes say: "...it turns out that there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. (We won't take the time to describe the transformations...)" This surprised me at first, and it is definitely intruiging to me. When I have the time, if it isn't covered in class, I think I'll look it up or try it myself. From meyer@imap.theory.csail.mit.edu Fri Sep 23 05:17:22 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8N9HMaJ026642 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:22 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8N9HKR8012198 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:21 -0400 (EDT) Received: from el-ternero.mit.edu (SIMMONS-THREE-TWELVE.MIT.EDU [18.96.6.57]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8N9HGCS026528 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 05:17:18 -0400 (EDT) Message-Id: <6.2.3.4.2.20050923051508.02eeaea0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 23 Sep 2005 05:17:14 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: Reading #3 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 331 X-UID: 98 X-Keywords: NonJunk The difference between strong induction and simple induction is not too clear. If possible, it would be great to have a brief explanation of the proof of Lemma 3.1 Every integer greater than 1 is a product of primes and why if this is implied why can't we prove that there are infinite number of primes? Thanks, Manuel Rivas From meyer@imap.theory.csail.mit.edu Fri Sep 23 07:51:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NBpgaJ009624 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NBpUR8015621 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:30 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NBpRMm006476 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 07:51:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NBpRZ7021318; Fri, 23 Sep 2005 07:51:27 -0400 Received: from PSK-THIRTEEN.MIT.EDU (PSK-THIRTEEN.MIT.EDU [18.217.1.13]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 07:51:27 -0400 Message-ID: <20050923075127.hm55454ciipwg0cg@webmail.mit.edu> Date: Fri, 23 Sep 2005 07:51:27 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: email comments 9-23 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.358 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 388 X-UID: 99 X-Keywords: NonJunk I felt this week's reading was fairly straightforward, so I don't have much to really complain about. It might be nice to see the same proposition get proven by both strong induction and regular induction, since on p11 (product of primes) it says that if you can do one you can do the other ... just to compare, I guess, see what situation is best to use which one on, etc. -Ryan Young From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:03:37 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NC3baJ010186 for <6042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 08:03:37 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NC0Liv019135; Fri, 23 Sep 2005 08:03:35 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NBvWdV007257; Fri, 23 Sep 2005 07:57:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NBvWbR021862; Fri, 23 Sep 2005 07:57:32 -0400 Received: from WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU (WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU [18.139.6.237]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 07:57:32 -0400 Message-ID: <20050923075732.0ri6h7bw77wookws@webmail.mit.edu> Date: Fri, 23 Sep 2005 07:57:32 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: Reading assignment -- Week 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 562 X-UID: 100 X-Keywords: NonJunk Hi, The proof of Theorem 2.2 on pages 5-6 (in the "Fibonacci Identity" section) surprised me. Before the proof you do some work until you reach a useful assertion (equation 5), and then you begin the proof in the opposite direction, using your new assertion as a link between P(n) and P(n+1). For the beginner like me, it was good to see this backwards "scratch work," because a lot of times proofs look so clean and clever but are not logical flows that would have occurred to me in the direction in which they're presented in texts, papers, etc. Nancy Keuss From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:39:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NCdVaJ019321 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NCdTgZ012318 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:29 -0400 (EDT) Received: from [18.237.0.82] (TDCIP82.MIT.EDU [18.237.0.82]) (authenticated bits=0) (User authenticated as letrec@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NCdRje014777 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 08:39:27 -0400 (EDT) Message-ID: <4333F77E.4070805@mit.edu> Date: Fri, 23 Sep 2005 08:39:26 -0400 From: Alton Torregano User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: pp 11 of 12: Strong Induction X-Enigmail-Version: 0.90.0.0 X-Enigmail-Supports: pgp-inline, pgp-mime Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.203 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 453 X-UID: 101 X-Keywords: NonJunk The section on Inductia proved a little difficult as I'm generally unexposed to proof by induction, by cases. Though the proof makes sense, I feel that maybe strong induction hides alot of the machinery and therefore makes it harder to process. Also, I was confused as to whether or not the cases need be made explicit (formally) before use in the inductive step. If possible, I would like for this to be explained more fully in the next lecture. From meyer@imap.theory.csail.mit.edu Fri Sep 23 08:44:28 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NCiSaJ019989 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 08:44:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NCiOgZ015356; Fri, 23 Sep 2005 08:44:24 -0400 (EDT) Received: from w92-130-webmail-4.mit.edu (W92-130-WEBMAIL-4.MIT.EDU [18.7.22.135]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NCiINK015818; Fri, 23 Sep 2005 08:44:18 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8NCiHX3032458; Fri, 23 Sep 2005 08:44:17 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 08:44:17 -0400 Message-ID: <20050923084417.6p3anl8wc5ws8k08@webmail.mit.edu> Date: Fri, 23 Sep 2005 08:44:17 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: 18.062J Reading Assignment 3 MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.226 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 275 X-UID: 102 X-Keywords: 18.062J Reading Assignment 3 On pg. 10 of the reading, the proof that every integer greater than 1 is a product of primes confused me. This reminded me of the false proof that every horse has the same color. I don't see how P(0) is true implies P(1) is true. ~Thu Duong From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:44:34 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.202]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDiYaJ027805 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:44:34 -0400 Received: by xproxy.gmail.com with SMTP id t12so118039wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 06:44:28 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=A+dKziQ0h8wHcu8lFOrgyT4iqypclIDvt35vMd1vjbSB4zE6xlbMFB4SRSEW5AMGE9knBTscOli1R0SfauoXKLMEGIqRaKGhOzPxOMgP+ZM5plypATXHoxDUUBW0mxMDmTWzTQv21i8mPvpm/LI12OxEST//fVALw0W9Wronu60= Received: by 10.70.8.10 with SMTP id 10mr878399wxh; Fri, 23 Sep 2005 06:44:27 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 23 Sep 2005 06:44:27 -0700 (PDT) Message-ID: <4f2613e50509230644463a87ed@mail.gmail.com> Date: Fri, 23 Sep 2005 09:44:27 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8NDiYaJ027805 Status: RO Content-Length: 823 X-UID: 103 X-Keywords: NonJunk I don't really undertand the horses of the same color theorem. I understand that the proof fails to show that all horses are of the same color since it operates on two different sets, and fails to show that the first set implies the other. Howver, my problem lies with the assumption that the first n horses are the same color and then that the last n horses are of the same color on pages 7 and 8 of the course notes. It seems analagous to assuming that the charachteristic is true for an n size set and then assuming it it true for the n+1 element and through these assumption, proving that the assumed charachteristic was true for all elements to begin with when, in fact, it was just assumed to be true. In summation, it seems as though an assumption is being proven with another assumption, which bothers me. -zozer From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:45:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDjPaJ028581 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDjOgZ003264 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:24 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NDjKJI006239 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 09:45:20 -0400 (EDT) Message-Id: <200509231345.j8NDjKJI006239@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 09:45:11 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.502 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 109 X-UID: 104 X-Keywords: NonJunk In section 4, I understand the Well Ordering Principle, but what is its significance to induction? - David From meyer@imap.theory.csail.mit.edu Fri Sep 23 09:49:45 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NDnjaJ029643 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:49:45 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDnhaD020199 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:49:43 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NDmrtp016795 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:48:53 -0400 (EDT) Received: from [18.245.5.65] (BAKER-SIXTY-FIVE.MIT.EDU [18.245.5.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NDmp0v016185 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 09:48:51 -0400 (EDT) Message-ID: <43342422.4040309@mit.edu> Date: Fri, 23 Sep 2005 09:49:54 -0600 From: Vera Carr User-Agent: Mozilla Thunderbird 1.0.2 (Windows/20050317) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Induction Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.775 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 284 X-UID: 105 X-Keywords: NonJunk Strong Induction pg 9 Having learned a bit about induction before I initially thought Strong Induction would be more difficult to do because you're working towards a more specific goal. However, seeing its applications I was surprised at how much simpler and more powerful it is. From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:32:20 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEWJaJ006241 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:19 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEWIgZ016696 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:18 -0400 (EDT) Received: from silencer (SILUET.MIT.EDU [18.234.0.112]) (authenticated bits=0) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEWEi0025629 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:32:15 -0400 (EDT) Message-Id: <200509231432.j8NEWEi0025629@outgoing.mit.edu> From: "Anton Katz" To: <6042-probs@theory.lcs.mit.edu> Subject: slightly unclear Date: Fri, 23 Sep 2005 10:32:10 -0400 Organization: Massachusetts Institute of Technology MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0028_01C5C02A.0E8BF5A0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXAS5UjcrZJ0cLeSk+dn4YSWAQa8g== X-Spam-Score: 0.2 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 4054 X-UID: 106 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C5C02A.0E8BF5A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Hi, Two topics are still slightly unclear for me: Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction? Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don't fully understand where exactly the proof becomes false. Thank you in advance, Anton. ------=_NextPart_000_0028_01C5C02A.0E8BF5A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Hi,

 

Two topics are still slightly unclear for = me:

Well-ordering and how is it possible to transfer way = of proof to another way. Is it done by math or is there a template to be = used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using = induction?

 

Second topic is the false proof, I understand that = the problem with the horses proof was that we did not take into account = different horses at P(1) both in the tutorial problems and in the proof in the reading I = don’t fully understand where exactly the proof becomes = false.

 

Thank you in advance,

 

Anton.

 

------=_NextPart_000_0028_01C5C02A.0E8BF5A0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:45:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEjlaJ007782 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEjjgZ029968 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:45 -0400 (EDT) Received: from [18.113.6.244] (ROGERS-FOUR-NINETY-NINE.MIT.EDU [18.113.6.244]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEjgoo002091 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 10:45:42 -0400 (EDT) Message-ID: <43341516.9040905@mit.edu> Date: Fri, 23 Sep 2005 10:45:42 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 0.8 (Windows/20040913) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 3 Comments Content-Type: multipart/alternative; boundary="------------030804090609060605060201" X-Spam-Score: -2.059 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 1672 X-UID: 107 X-Keywords: NonJunk This is a multi-part message in MIME format. --------------030804090609060605060201 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Section 3.1 I'd like to see this discussed in lecture, as I don't think we've seen it yet and it'd be good to illuminate the difference between itself and regular induction. Incidentally, I also think that these readings and tutor problems should be announced in lecture as well :) -Pete -- Pete Kruskall 28 The Fenway Boston, MA 02215 ::::::::::::::::::::::::::::: 617.536.9925 :::Sigma Nu::::: 508.843.5861 ::::Cell Phone:: ::::http://tege.mit.edu:::::: --------------030804090609060605060201 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Section 3.1
   I'd like to see this discussed in lecture, as I don't think we've seen it yet and it'd be good to illuminate the difference between itself and regular induction.  Incidentally, I also think that these readings and tutor problems should be announced in lecture as well :)

-Pete
 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

:::::::::::::::::::::::::::::
617.536.9925 :::Sigma Nu:::::
508.843.5861 ::::Cell Phone::
::::http://tege.mit.edu::::::
--------------030804090609060605060201-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:46:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEksaJ007860 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEkqgZ001129 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:53 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEkkhZ002563 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:46:46 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NEkki9031645; Fri, 23 Sep 2005 10:46:46 -0400 Received: from BURTON-FOUR-THIRTY-EIGHT.MIT.EDU (BURTON-FOUR-THIRTY-EIGHT.MIT.EDU [18.247.6.183]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 10:46:46 -0400 Message-ID: <20050923104646.p180njih9bn28so4@webmail.mit.edu> Date: Fri, 23 Sep 2005 10:46:46 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading 3: Induction MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.236 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 353 X-UID: 108 X-Keywords: NonJunk I was wondering if it would be possible to go over Strong Induction more thoroughly in class. While I understand that in principle, it is a more general method of induction, I do not exactly understand how it can be proved and, well, exactly how one goes about using strong induction. - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:48:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEmsaJ008459 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:54 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEmqgZ002947 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:52 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEmnip003458 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 10:48:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NEmnRi031973; Fri, 23 Sep 2005 10:48:49 -0400 Received: from BURTON-FOUR-THIRTY-EIGHT.MIT.EDU (BURTON-FOUR-THIRTY-EIGHT.MIT.EDU [18.247.6.183]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 10:48:49 -0400 Message-ID: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> Date: Fri, 23 Sep 2005 10:48:49 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading 3: Induction MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.543 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 552 X-UID: 109 X-Keywords: NonJunk I forgot to include in my last e-mail: I refer to the passage that defines Strong Induction at the top of page 10. Specifically, the part that confuses me is: "The only change from the ordinary induction principle is that strong induction allows you to assume more stuff in the inductive step of your proof! ...In a strong induction argument, you may assume that P(0), P(1), ..., and P(n) are all true when you go to prove P(n + 1)." Why are you allowed to assume all these cases? - Karena -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From meyer@imap.theory.csail.mit.edu Fri Sep 23 10:51:06 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NEp6aJ009052 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:06 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NEp4gZ005057 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:04 -0400 (EDT) Received: from Charizard (NEW-TWO-SIXTY-SIX.MIT.EDU [18.241.6.11]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NEp1vk004328 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 10:51:01 -0400 (EDT) Message-Id: <200509231451.j8NEp1vk004328@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: 6.042 reading email Date: Fri, 23 Sep 2005 10:50:57 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0010_01C5C02C.AE570640" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXATjTD8vh5ACiSQ5C6Q+arJVUQcA== X-Spam-Score: 2.247 X-Spam-Level: ** (2.247) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 5265 X-UID: 110 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0010_01C5C02C.AE570640 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Dear .042, It would be great if you give some more examples on the Well Ordered Problems theorem- in particular, how to convert it from Strong and back. On paper it seems reasonable, but it'll sink in better with more examples. Thanks! _Yaser ------=_NextPart_000_0010_01C5C02C.AE570640 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Dear = .042,

 

It would be great if you give = some more examples on the Well Ordered Problems theorem- in particular, how to = convert it from Strong and back. On paper it seems reasonable, but it’ll sink = in better with more examples.

 

Thanks!

 

_Yaser

------=_NextPart_000_0010_01C5C02C.AE570640-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:01:13 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF1CaJ012412 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:01:13 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF0cgZ013969 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:00:38 -0400 (EDT) Received: from w92-130-webmail-2.mit.edu (W92-130-WEBMAIL-2.MIT.EDU [18.7.22.132]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF0XE5008616 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:00:33 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8NF0XT0001624; Fri, 23 Sep 2005 11:00:33 -0400 Received: from KS-ONE-TWENTY-EIGHT.MIT.EDU (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 11:00:33 -0400 Message-ID: <20050923110033.vv23cwplepw00wgc@webmail.mit.edu> Date: Fri, 23 Sep 2005 11:00:33 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: email comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 185 X-UID: 111 X-Keywords: NonJunk The part I found most confusing was the difference between simple and strong induction. I understand both, but I am confused about how to distinguish between the two like on the pset. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:01:43 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF1haJ012432 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:01:43 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF10gZ014315 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:01:01 -0400 (EDT) Received: from w92-130-webmail-1.mit.edu (W92-130-WEBMAIL-1.MIT.EDU [18.7.22.131]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF0v8w008815 for <6.042-probs@theory.csail.MIT.EDU>; Fri, 23 Sep 2005 11:00:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8NF0v0s015240; Fri, 23 Sep 2005 11:00:57 -0400 Received: from RLE-13-099.MIT.EDU (RLE-13-099.MIT.EDU [18.62.13.99]) (User authenticated as kushan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 23 Sep 2005 11:00:57 -0400 Message-ID: <20050923110057.yqvtqs8i88o0ks4o@webmail.mit.edu> Date: Fri, 23 Sep 2005 11:00:57 -0400 From: Kushan K Surana To: 6.042-probs@theory.csail.MIT.EDU Subject: Reading assignment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.559 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 87 X-UID: 112 X-Keywords: NonJunk I am unclear about the well ordering principle. How is it relevant for proofs? Kushan From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:04:11 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF4BaJ012567 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:04:11 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF2ugZ015795 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:02:56 -0400 (EDT) Received: from PRAVEENPAMIDI.mit.edu (KS-ONE-TWENTY-EIGHT.MIT.EDU [18.235.1.128]) (authenticated bits=0) (User authenticated as mukkala@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF2qCu009651 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 11:02:54 -0400 (EDT) Message-Id: <6.2.1.2.2.20050923110050.01e50f50@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 23 Sep 2005 11:02:56 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 2.267 X-Spam-Level: ** (2.267) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 271 X-UID: 113 X-Keywords: NonJunk In section 3.2 of the reading, I don't understand the proof entirely. The step that troubles me is the one where we just jump from "k-2 is a natural number less than n+1" to "(k-2) +2 is a product of primes by induction hypothesis." I could use more explanation. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:04:14 2005 Return-Path: Received: from rozz.csail.mit.edu (mail@rozz.csail.mit.edu [128.30.2.16]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NF4EaJ012572 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:04:14 -0400 Received: from biscayne-one-station.mit.edu ([18.7.7.80]) by rozz.csail.mit.edu with esmtps (TLSv1:AES256-SHA:256) (Exim 4.50) id 1EIp5y-0002ue-BU for 6042-probs@theory.lcs.MIT.EDU; Fri, 23 Sep 2005 11:04:14 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NF3sgZ016553 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:03:54 -0400 (EDT) Received: from m56-129-2.mit.edu (M56-129-2.MIT.EDU [18.56.0.31]) (authenticated bits=56) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF3kvg010038 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:03:47 -0400 (EDT) Received: (from juang@localhost) by m56-129-2.mit.edu (8.12.9) id j8NF3k54024750; Fri, 23 Sep 2005 11:03:46 -0400 (EDT) Subject: Reading comments From: Jason K Juang To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Date: Fri, 23 Sep 2005 11:03:45 -0400 Message-Id: <1127487825.24672.1.camel@m56-129-2.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Content-Transfer-Encoding: 7bit Status: RO Content-Length: 264 X-UID: 114 X-Keywords: NonJunk On page 10: "The only change from the ordinary induction principle is that strong induction allows you to assume more stuff in the inductive step of your proof!" So then why do we bother making a distinction between simple induction and strong induction? Jason. From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:10:24 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFAOaJ013397 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:10:24 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NFALOS021319 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:10:23 -0400 (EDT) Received: from m56-129-19.mit.edu (M56-129-19.MIT.EDU [18.56.0.48]) (authenticated bits=56) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF8aRR012117 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 11:08:37 -0400 (EDT) Received: (from jstritar@localhost) by m56-129-19.mit.edu (8.12.9) id j8NF8aMv028897; Fri, 23 Sep 2005 11:08:36 -0400 Subject: Comments From: Jonathan E Stritar To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Fri, 23 Sep 2005 11:08:36 -0400 Message-Id: <1127488116.28781.1.camel@m56-129-19.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 278 X-UID: 115 X-Keywords: NonJunk Referring to 2.4 Courtyard Tiling (page 6). I think this is really cool. Its interesting how with induction you can prove that a certain solution exists even though it does not show you exactly how to get there (in this case how you would lay the L shaped tiles). Jon Stritar From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:11:29 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFBSaJ013871 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:11:28 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NFALPg021319 for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:11:27 -0400 (EDT) Received: from Drache (NEW-ONE-EIGHTY-NINE.MIT.EDU [18.241.5.189]) (authenticated bits=0) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NF7VCx011688 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:07:32 -0400 (EDT) Message-Id: <200509231507.j8NF7VCx011688@outgoing.mit.edu> From: "Natalia Chernenko" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 11:07:25 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0003_01C5C02E.FBCA87B0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXAUFwbpPeJFVRLSbyy6PxLm4BY/A== X-Spam-Score: 1.155 X-Spam-Level: * (1.155) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2593 X-UID: 116 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0003_01C5C02E.FBCA87B0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit The most interesting part of the reading was the tiling problem (section 2.4). I really like visual techniques and it was fun to see a mathematical proof of a graphic layout. I am still a bit confused as to the difference between strong induction and simple induction, though. Natalia Chernenko ------=_NextPart_000_0003_01C5C02E.FBCA87B0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

The most interesting part of the reading was the = tiling problem (section 2.4). I really like visual techniques and it was fun to = see a mathematical proof of a graphic layout.

 

I am still a bit confused as to the difference = between strong induction and simple induction, though. =

 

Natalia Chernenko

------=_NextPart_000_0003_01C5C02E.FBCA87B0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 16:53:43 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.199]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NKrhaJ006138 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:53:43 -0400 Received: by xproxy.gmail.com with SMTP id t5so179888wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:mime-version:content-type; b=L0mhzDsLYHNz935TsABX8nVV6UhuWfqN3iMlcjYRiXeib3AJyj/LGp5kPxspv8FoSwbniuKTWQuQpLN6D0qljXAkiBU5rm45Dv0Zl2K17zE1a0fX7h2RvhbJ7QI2hT8xzRQfyiEb9k6XeC3hzqXow7ztU8mFevT01nrLcy0JTsQ= Received: by 10.70.31.20 with SMTP id e20mr1211195wxe; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 23 Sep 2005 13:53:38 -0700 (PDT) Message-ID: <3c33fe380509231353f100e4c@mail.gmail.com> Date: Fri, 23 Sep 2005 16:53:38 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: Reading Comments - 9/23 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_11355_12228901.1127508818431" Status: RO Content-Length: 2466 X-UID: 117 X-Keywords: NonJunk ------=_Part_11355_12228901.1127508818431 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi, My comments on this week's reading below. I'm sorry for the tardiness, I didn't check the course website to see when the reading comments were due. = I learned my lesson and will check the website more often in the future. Thanks for your understanding. The part I found most difficult in the reading was in the last section, using the Well Ordering Principle to transform a proof into Proof by Strong Induction. Taken from the reading: "As for useless, it turns out that there's a routine way to transform any proof using the Well Ordering Principle into a proof using Strong Induction, and vice-versa. (We won't take the time to describe the transformations, though they are not hard.)" (page 12). I don't quite understand how this transformation can take place, as when doing Proof by Induction, the base case we choose is usually the least element (like 0). I think I'd be cool to see some examples of applying the Principle in Proofs by Strong Induction. Thanks, Shreyes Seshasai Group 7 ------=_Part_11355_12228901.1127508818431 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi,

My comments on this week's reading below.  I'm sorry for the tardiness, I didn't check the course website to see when the reading comments were due.  I learned my lesson and will check the website more often in the future.  Thanks for your understanding.

The part I found most difficult in the reading was in the last section, using the Well Ordering Principle to transform a proof into Proof by Strong Induction.  Taken from the reading: "As for useless, it turns out that there's a routine way to transform any proof using the Well Ordering
Principle into a proof using Strong Induction, and vice-versa. (We won't ta= ke the time to
describe the transformations, though they are not hard.)" (page 12). I don't quite understand how this transformation can take place, as when doing Proof by Induction, the base case we choose is usually the least element (like 0).  I think I'd be cool to see some examples of applying the Principle in Proofs by Strong Induction.

Thanks,
Shreyes Seshasai
Group 7


------=_Part_11355_12228901.1127508818431-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 16:55:07 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NKt7aJ006267 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:55:07 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NKsqew020139 for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:54:52 -0400 (EDT) Received: from aston (BOOKX.MIT.EDU [18.241.0.191]) (authenticated bits=0) (User authenticated as aston@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8NKsoPC023095 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 23 Sep 2005 16:54:50 -0400 (EDT) Message-Id: <200509232054.j8NKsoPC023095@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Comments on Reading Date: Fri, 23 Sep 2005 16:55:44 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0043_01C5C05F.A3DAF8B0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXAgSpy3FbpZA9xSPi0TpwbBqKYaQ== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 2227 X-UID: 118 X-Keywords: This is a multi-part message in MIME format. ------=_NextPart_000_0043_01C5C05F.A3DAF8B0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found the comment on page 11 about the ability to prove anything with ordinary induction that you can with strong induction extremely interesting. I'm not so clear, though, on how that's possible. - Aston ------=_NextPart_000_0043_01C5C05F.A3DAF8B0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I found the comment on page 11 about the ability to = prove anything with ordinary induction that you can with strong induction = extremely interesting. I’m not so clear, though, on how that’s = possible.

 

         =    - Aston

------=_NextPart_000_0043_01C5C05F.A3DAF8B0-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:25:07 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHP7aJ002589 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6id003556 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6HR026006; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHP20v027960; Fri, 23 Sep 2005 13:25:03 -0400 (EDT) Message-ID: <43343A52.6060506@mit.edu> Date: Fri, 23 Sep 2005 13:24:34 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason K Juang Subject: Re: Reading comments References: <1127487825.24672.1.camel@m56-129-2.mit.edu> In-Reply-To: <1127487825.24672.1.camel@m56-129-2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 748 X-UID: 119 X-Keywords: My simple response would be that when you present a proof, you shouldn't state extraneous assumptions. There's nothing wrong with it logically, but it's not good style. For example, "a=b, therefore, a^2 = b^2." is good, but "a=b, and b=c, therefore, a^2 = b^2." is not really ideal. Distinguishing between strong and simple induction allows us to follow this "minimal-assumption" convention. Let me know if you're still confused. DS Jason K Juang wrote: >On page 10: "The only change from the ordinary induction principle is >that strong induction allows you to assume more stuff in the inductive >step of your proof!" > >So then why do we bother making a distinction between simple induction >and strong induction? > >Jason. > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:19:55 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHJtaJ001122 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHJsid025136 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHFlHR025650; Fri, 23 Sep 2005 13:15:47 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHFi3a000127; Fri, 23 Sep 2005 13:15:44 -0400 (EDT) Message-ID: <43343824.4060900@mit.edu> Date: Fri, 23 Sep 2005 13:15:16 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz Subject: Re: slightly unclear References: <200509231432.j8NEWEi0025629@outgoing.mit.edu> In-Reply-To: <200509231432.j8NEWEi0025629@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------000209080207090105030401" X-Spam-Score: -1.406 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 4899 X-UID: 120 X-Keywords: This is a multi-part message in MIME format. --------------000209080207090105030401 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you. The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2. So, what we've proved is: P(1) P(3) -> P(4) P(4) -> P(5) P(5) -> P(6) P(6) -> P(7) ... Do you see that this does not imply P(n) for all n? DS Anton Katz wrote: > Hi, > > > > Two topics are still slightly unclear for me: > > Well-ordering and how is it possible to transfer way of proof to > another way. Is it done by math or is there a template to be used in > the transaction? Does this mean that everything that can be proved > using well-ordering can be proved using induction? > > > > Second topic is the false proof, I understand that the problem with > the horses proof was that we did not take into account different > horses at P(1) both in the tutorial problems and in the proof in the > reading I don't fully understand where exactly the proof becomes false. > > > > Thank you in advance, > > > > Anton. > > > --------------000209080207090105030401 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you.

The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2.  So, what we've proved is:

P(1)
P(3) -> P(4)
P(4) -> P(5)
P(5) -> P(6)
P(6) -> P(7)
...

Do you see that this does not imply P(n) for all n?

DS

Anton Katz wrote:

Hi,

 

Two topics are still slightly unclear for me:

Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction?

 

Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don’t fully understand where exactly the proof becomes false.

 

Thank you in advance,

 

Anton.

 


--------------000209080207090105030401-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 12:44:05 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NGi5aJ027825; Fri, 23 Sep 2005 12:44:05 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j8NGi5rN027008; Fri, 23 Sep 2005 12:44:05 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j8NGi57E027007; Fri, 23 Sep 2005 12:44:05 -0400 Date: Fri, 23 Sep 2005 12:44:05 -0400 From: Sayan Mitra Message-Id: <200509231644.j8NGi57E027007@drake.csail.mit.edu> To: nedzel@MIT.EDU Subject: Re: Week 3 Comments Status: RO Content-Length: 898 X-UID: 121 X-Keywords: If a set W is well ordered then you can induct on its elements to prove the property in question. This may sound a little vague now but it will become clearer as we do more examples in class and in problem sets. Drop by in the office hours if you still have questions. -Sayan >From nedzel@MIT.EDU Fri Sep 23 09:45:25 2005 From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 3 Comments Date: Fri, 23 Sep 2005 09:45:11 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW4zeo7ykLa/WaNRiGOpegAnm9CXA== X-Spam-Score: -0.502 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 In section 4, I understand the Well Ordering Principle, but what is its significance to induction? - David From meyer@imap.theory.csail.mit.edu Fri Sep 23 12:25:48 2005 Return-Path: Received: from drake.csail.mit.edu (drake.csail.mit.edu [128.30.51.97]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NGPmaJ026211; Fri, 23 Sep 2005 12:25:48 -0400 Received: from drake.csail.mit.edu (localhost.localdomain [127.0.0.1]) by drake.csail.mit.edu (8.13.1/8.13.1) with ESMTP id j8NGPmoJ026871; Fri, 23 Sep 2005 12:25:48 -0400 Received: (from mitras@localhost) by drake.csail.mit.edu (8.13.1/8.13.1/Submit) id j8NGPmVi026870; Fri, 23 Sep 2005 12:25:48 -0400 Date: Fri, 23 Sep 2005 12:25:48 -0400 From: Sayan Mitra Message-Id: <200509231625.j8NGPmVi026870@drake.csail.mit.edu> To: lana@mit.edu Subject: Re: TP2 Status: RO Content-Length: 83 X-UID: 122 X-Keywords: you'll see one very nice application of the well ordering principle today. -sayan From meyer@imap.theory.csail.mit.edu Fri Sep 23 11:46:06 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.197]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NFk6aJ017921 for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 11:46:06 -0400 Received: by xproxy.gmail.com with SMTP id h28so824259wxd for <6042-staff@theory.csail.mit.edu>; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=XAk9zsQMl4HG2+TKdZixpVth6wDUlhQ3HM+T5Of7XSepP9tUsjaEY3ei9aOtK/BgMESnkU7lKY39WaG45Ro/B1CZpLf55zqa48cjdkQJMVkP/ilpFvzyIjTbD6sgy+UEaivqhSjf8w6R6llK/AOi8klG83vVARo14a9DQEc9CAs= Received: by 10.70.21.12 with SMTP id 12mr1128551wxu; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) Received: by 10.70.52.3 with HTTP; Fri, 23 Sep 2005 08:46:01 -0700 (PDT) Message-ID: Date: Fri, 23 Sep 2005 11:46:01 -0400 From: Jelani Nelson Reply-To: Jelani Nelson Sender: minilek@gmail.com To: r n jacobs Subject: Re: reading questions due sep 23 In-Reply-To: <200509231541.j8NFftZf007838@white-meteo.mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline References: <4333E709.5010506@theory.csail.mit.edu> <200509231541.j8NFftZf007838@white-meteo.mit.edu> Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8NFk6aJ017921 Status: RO Content-Length: 976 X-UID: 123 X-Keywords: Ok. If you find anything unclear later, feel free to ask. -Jelani On 9/23/05, r n jacobs wrote: > > The one on the tutor questions. I identified what was wrong fairly > fast (It wasn't proven for n=1), but it took me /forever/ to figure > out which line technically had the flaw. > > (I also still find the "every number is a product of primes" proof > (3.2, pg10) confusing. But I /think/ I understand strong induction > now.) > > - robert > > > Which fibonacci misproof? If you are talking about the fibonacci > > identity proven in the lecture notes, the identity is correct, and it is > > correctly proven using plain old induction (not strong induction). > > > > -Jelani > > > > r n jacobs wrote: > > > > >I found myself confused by the explanation of Strong > > >Induction. Although maybe the fibbonaci proof helped that. (The > > >definition made sense, but it took far to long for me to figure out > > >the answer to the fibbonaci misproof) > From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:34:50 2005 Message-ID: <43348308.5040000@csail.mit.edu> Date: Fri, 23 Sep 2005 18:34:48 -0400 From: "Prof. Albert R. Meyer" Reply-To: 6042-probs@theory.csail.mit.edu Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Karena Tyan CC: 6042-staff@theory.csail.mit.edu Subject: Re: Reading 3: Induction References: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> In-Reply-To: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1351 X-UID: 124 X-Keywords: I hope today's class clarified things some, but here's one way to understand why all the extra P's can be assumed: think of the falling dominoes explanation of induction: if you knock over the 1st domino, and knocking over the nth domino will cause the next (n+1st) one to get knocked over, then you can conclude by simple induction that they all will get knocked over. But when the cascade of falling dominoes gets to the n+1st, all the previous ones will have been knocked over already, so it's perfectly ok to use that fact in reasoning about how the n+1st one falls. If you're still feeling shaky about this, I'll second Hanson's suggestion that you take it up with a TA is office hours. I'd also be happy to meet you after classs or make some other appt with you to talk about this. regards, A. Karena Tyan wrote: >I forgot to include in my last e-mail: I refer to the passage that defines >Strong Induction at the top of page 10. Specifically, the part that confuses >me is: > >"The only change from the ordinary induction principle is that strong induction >allows you to assume more stuff in the inductive step of your proof! ...In a >strong induction argument, you may assume that P(0), P(1), ..., and P(n) are >all true when you go to prove P(n + 1)." > >Why are you allowed to assume all these cases? > >- Karena > > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:23:07 2005 BCC: Ronitt Rubinfeld Message-ID: <43348049.9070206@csail.mit.edu> Date: Fri, 23 Sep 2005 18:23:05 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Hanson Zhou Subject: Re: Reading 3: Induction References: <20050923104849.gdiofdwamw0kwco8@webmail.mit.edu> In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 1028 X-UID: 125 X-Keywords: I think your answer to this student is insulting: she has done the reading and is asking about it -- and your advice is to do the reading (again?). I'll respond to her further. regards, A. Hanson Zhou wrote: >You are essentially asking what strong induction is and you should read >about it. If it remains confusing, go to office hours to get this cleared >up. > >-Hanson > >On Fri, 23 Sep 2005, Karena Tyan wrote: > > > >>I forgot to include in my last e-mail: I refer to the passage that defines >>Strong Induction at the top of page 10. Specifically, the part that confuses >>me is: >> >>"The only change from the ordinary induction principle is that strong induction >>allows you to assume more stuff in the inductive step of your proof! ...In a >>strong induction argument, you may assume that P(0), P(1), ..., and P(n) are >>all true when you go to prove P(n + 1)." >> >>Why are you allowed to assume all these cases? >> >>- Karena >> >>-- >>410 Memorial Drive >>Cambridge, MA 02139 >>(585)957-5923 >> >> >> From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:25:09 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHP7aJ002589 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:07 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6id003556 for <6042-staff@theory.lcs.MIT.EDU>; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHP6HR026006; Fri, 23 Sep 2005 13:25:06 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHP20v027960; Fri, 23 Sep 2005 13:25:03 -0400 (EDT) Message-ID: <43343A52.6060506@mit.edu> Date: Fri, 23 Sep 2005 13:24:34 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jason K Juang Subject: Re: Reading comments References: <1127487825.24672.1.camel@m56-129-2.mit.edu> In-Reply-To: <1127487825.24672.1.camel@m56-129-2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.9 required=3.7 tests=AWL,BAYES_00 autolearn=ham version=2.63 Status: RO Content-Length: 747 X-UID: 126 X-Keywords: My simple response would be that when you present a proof, you shouldn't state extraneous assumptions. There's nothing wrong with it logically, but it's not good style. For example, "a=b, therefore, a^2 = b^2." is good, but "a=b, and b=c, therefore, a^2 = b^2." is not really ideal. Distinguishing between strong and simple induction allows us to follow this "minimal-assumption" convention. Let me know if you're still confused. DS Jason K Juang wrote: >On page 10: "The only change from the ordinary induction principle is >that strong induction allows you to assume more stuff in the inductive >step of your proof!" > >So then why do we bother making a distinction between simple induction >and strong induction? > >Jason. > > From meyer@imap.theory.csail.mit.edu Fri Sep 23 13:19:57 2005 Return-Path: Received: from south-station-annex.mit.edu (SOUTH-STATION-ANNEX.MIT.EDU [18.72.1.2]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8NHJtaJ001122 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHJsid025136 for <6042-staff@theory.lcs.mit.edu>; Fri, 23 Sep 2005 13:19:55 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8NHFlHR025650; Fri, 23 Sep 2005 13:15:47 -0400 (EDT) Received: from [192.168.1.4] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8NHFi3a000127; Fri, 23 Sep 2005 13:15:44 -0400 (EDT) Message-ID: <43343824.4060900@mit.edu> Date: Fri, 23 Sep 2005 13:15:16 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz Subject: Re: slightly unclear References: <200509231432.j8NEWEi0025629@outgoing.mit.edu> In-Reply-To: <200509231432.j8NEWEi0025629@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------000209080207090105030401" X-Spam-Score: -1.406 X-Scanned-By: MIMEDefang 2.42 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on theory.csail.mit.edu X-Spam-Level: X-Spam-Status: No, hits=-4.8 required=3.7 tests=AWL,BAYES_00,HTML_50_60, HTML_MESSAGE autolearn=no version=2.63 Status: RO Content-Length: 4899 X-UID: 127 X-Keywords: This is a multi-part message in MIME format. --------------000209080207090105030401 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you. The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2. So, what we've proved is: P(1) P(3) -> P(4) P(4) -> P(5) P(5) -> P(6) P(6) -> P(7) ... Do you see that this does not imply P(n) for all n? DS Anton Katz wrote: > Hi, > > > > Two topics are still slightly unclear for me: > > Well-ordering and how is it possible to transfer way of proof to > another way. Is it done by math or is there a template to be used in > the transaction? Does this mean that everything that can be proved > using well-ordering can be proved using induction? > > > > Second topic is the false proof, I understand that the problem with > the horses proof was that we did not take into account different > horses at P(1) both in the tutorial problems and in the proof in the > reading I don't fully understand where exactly the proof becomes false. > > > > Thank you in advance, > > > > Anton. > > > --------------000209080207090105030401 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You will see a good example of the WOP today in class - let us know if it still confuses you.

The horse proof has a step P(n) -> P(n+1), but the proof of that step demanded n>2.  So, what we've proved is:

P(1)
P(3) -> P(4)
P(4) -> P(5)
P(5) -> P(6)
P(6) -> P(7)
...

Do you see that this does not imply P(n) for all n?

DS

Anton Katz wrote:

Hi,

 

Two topics are still slightly unclear for me:

Well-ordering and how is it possible to transfer way of proof to another way. Is it done by math or is there a template to be used in the transaction? Does this mean that everything that can be proved using well-ordering can be proved using induction?

 

Second topic is the false proof, I understand that the problem with the horses proof was that we did not take into account different horses at P(1) both in the tutorial problems and in the proof in the reading I don’t fully understand where exactly the proof becomes false.

 

Thank you in advance,

 

Anton.

 


--------------000209080207090105030401-- From meyer@imap.theory.csail.mit.edu Fri Sep 23 18:54:04 2005 Message-ID: <4334878A.6000409@csail.mit.edu> Date: Fri, 23 Sep 2005 18:54:02 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Angelique E Moscicki CC: 6042-staff@theory.csail.mit.edu Subject: Re: comment References: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> In-Reply-To: <20050923014649.s2van2mf2o000ckw@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 486 X-UID: 128 X-Keywords: It was a dense 10 or so pages of reading, and even if you knew and understood it all, I can't believe you found it all equally well :-) explained, with enough examples, etc. Also, do you see how Well Ordering proofs can be converted to Induction proofs, and vice-versa?-- that was tricky. I'll look forward to more responsive comments from you next week. regards, A. Angelique E Moscicki wrote: >Sorry there isn't as much this time, its just rather more familiar material. > From meyer@imap.theory.csail.mit.edu Sat Sep 24 02:41:44 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8O6fiaJ014234 for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:44 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8O6fgI1009021 for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:42 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8O6fcl9029104 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 24 Sep 2005 02:41:39 -0400 (EDT) Message-ID: <433565AD.3040501@mit.edu> Date: Sat, 24 Sep 2005 06:41:49 -0800 From: Benjamin Lu User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [6.042] comments on reading 3 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.449 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-Status: A Content-Length: 452 X-UID: 129 X-Keywords: NonJunk Hello, My apologies for the late comment, I completely forgot to send this earlier. "The Well Ordering Principle looks nothing like the induction axiom, and it may seem obvious but useless" -pg 12 I agreed with this sentiment, probably because using Well Ordering in proofs seems so backwards and counterintuitive. I think it would be cool to do a bunch of Well Ordering proofs to try and develop a little more intuition for them. Cheers, ~Ben From meyer@imap.theory.csail.mit.edu Sat Sep 24 17:00:51 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8OL0paJ016114 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 17:00:51 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8OL0QNT009414 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 17:00:50 -0400 (EDT) Received: from w92-130-webmail-5.mit.edu (W92-130-WEBMAIL-5.MIT.EDU [18.7.22.136]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8OKxJcW006806 for <6042-probs@theory.lcs.MIT.EDU>; Sat, 24 Sep 2005 16:59:19 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8OKxJKu003616; Sat, 24 Sep 2005 16:59:19 -0400 Received: from AP-ONE-TWENTY-NINE.MIT.EDU (AP-ONE-TWENTY-NINE.MIT.EDU [18.153.1.129]) (User authenticated as ridell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sat, 24 Sep 2005 16:59:19 -0400 Message-ID: <20050924165919.924oi2ic8e50ckgc@webmail.mit.edu> Date: Sat, 24 Sep 2005 16:59:19 -0400 From: Rebecca Idell To: 6042-probs@theory.lcs.MIT.EDU Subject: 042 reading MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.721 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO Content-Length: 383 X-UID: 130 X-Keywords: NonJunk $Forwarded On page 3-4 i nthe reading is the template for the induction proof. When we do proofs in clas, I don't completely understand which parts of the proof correspond to which parts of the template. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From meyer@imap.theory.csail.mit.edu Mon Sep 26 08:23:25 2005 Message-ID: <4337E83E.8020403@csail.mit.edu> Date: Mon, 26 Sep 2005 08:23:26 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [6.042] comments on reading 3 References: <433565AD.3040501@mit.edu> In-Reply-To: <433565AD.3040501@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 550 X-UID: 131 X-Keywords: Was Friday lecture cool enough? :-) Regards, A. Benjamin Lu wrote: > Hello, > > My apologies for the late comment, I completely forgot to send this > earlier. > > "The Well Ordering Principle looks nothing like the induction axiom, > and it may seem obvious but useless" -pg 12 > > I agreed with this sentiment, probably because using Well Ordering in > proofs seems so backwards and counterintuitive. I think it would be > cool to do a bunch of Well Ordering proofs to try and develop a little > more intuition for them. > > Cheers, > ~Ben