From berke@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From berke@MIT.EDU Wed Sep 7 22:06:12 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 j8826CaJ026104 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:12 -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 j8826AaH023833 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:11 -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 j88267sD020661 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 7 Sep 2005 22:06:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j882678B016678; Wed, 7 Sep 2005 22:06:07 -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 ; Wed, 7 Sep 2005 22:06:07 -0400 Message-ID: <20050907220607.r9aa25eyjz4g4gg4@webmail.mit.edu> Date: Wed, 7 Sep 2005 22:06:07 -0400 From: Allison Berke To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Questions 1 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 X-IMAPbase: 1126748715 174 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk Status: RO X-UID: 1 Content-Length: 973 X-Keywords: NonJunk NotJunk Hi, I found the shortcomings of set theory (pgs. 12, 13, and Russell appendix) to be the most interesting. I'd like to know what the "simple ways" in which sets can be built are, and how they were decided upon. I'd also like to know more about the Banach-Tarski paradox: I understand what it implies, and how it can say that it's true (every piece must be infinitely "jagged", to an extent that goes beyond possible atomic subdivisions) but it would seem that infinite jaggedness would then be entirely uninteresting. How did the paradox become so famous? That the Axiom of Choice has to be assumed probably has something to do with it--especially for detractors of the Axiom--but it seems that even assuming the possibility of infinite subdivisions lends an absurdity to any conclusions that stem from it. If we can assume infinite subdivision of mathematical matter, can we propose infinite subdivision of mathematical time? And what would the point be? -Allison Berke From sheldons@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From sheldons@MIT.EDU Sat Sep 10 15:24:33 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 j8AJOXaJ029416 for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:33 -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 j8AJOVDo000651 for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:32 -0400 (EDT) Received: from [10.0.0.2] (209-6-159-26.c3-0.smr-ubr3.sbo-smr.ma.cable.rcn.com [209.6.159.26]) (authenticated bits=0) (User authenticated as sheldons@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8AJOOmO007145 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Sat, 10 Sep 2005 15:24:25 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <1EC3B9B2-1A79-4E4E-9A99-D0AD840E87BC@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Sheldon Chan Subject: Required Reading Comments 1 Date: Sat, 10 Sep 2005 15:24:23 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 2 Content-Length: 326 X-Keywords: NonJunk NotJunk 1.4 Proof by Cases (page 3) I found that the example of proof by cases to be most difficult to comprehend. I'm not exactly sure why Case 1.2 falls logically under Case 1, since, it basically implies that at least 2 people did meet x and not 3. Why aren't the two cases 1. Two have met x, and 2. Three have not met x? From ksindi@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From ksindi@MIT.EDU Sat Sep 10 17:39:32 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 j8ALdWaJ011814 for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:32 -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 j8ALdUhu026648 for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:31 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-SIX-O-NINE.MIT.EDU [18.244.7.98]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ALdLJg020131 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Sat, 10 Sep 2005 17:39:24 -0400 (EDT) Message-Id: <6.2.3.4.2.20050910170801.03b61eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Sat, 10 Sep 2005 17:39:07 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kamil Sindi Subject: Reading 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-UID: 98 Content-Length: 473 X-Keywords: NonJunk NotJunk " 2. Predicates (p. 4) A predicate is a proposition whose truth depends on the value of one or more variables." I found this passage surprising for its over-emphasis on mathematical terminology. However, I appreciated the 'simplicity by convention', which differentiates mathematics from many other disciplines. The example with P(n)= "n is a perfect square" made the concept of the predicate a little confusing, but reading on cleared all the misconceptions I had. From aeon@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From aeon@MIT.EDU Sun Sep 11 22:37:10 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 j8C2bAaJ000510 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:10 -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 j8C2b92D016621 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:09 -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 j8C2b7pU029283 for <6042-probs@theory.csail.MIT.EDU>; Sun, 11 Sep 2005 22:37:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8C2b7lk010499; Sun, 11 Sep 2005 22:37:07 -0400 Received: from NEW-FOUR-NINETY-THREE.MIT.EDU (NEW-FOUR-NINETY-THREE.MIT.EDU [18.241.6.238]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Sun, 11 Sep 2005 20:37:07 -0600 Message-ID: <20050911203707.c93buqpcs7400osk@webmail.mit.edu> Date: Sun, 11 Sep 2005 20:37:07 -0600 From: John Marrero To: 6042-probs@theory.csail.MIT.EDU Subject: LN2 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-UID: 99 Content-Length: 632 X-Keywords: NonJunk NotJunk Page 3, section "1.4 Proof by Cases" This was quite interesting, particularly the example given of the strangers and clubs. It seems like a useful method, but could it be used as easily for complicated expressions? If there's an example that addresses that issue, I'd very much like to see it. Page 10, section "3.3 Sequences" I understood what the idea behind the product operation was, but wasn't totally grasping it through the examples provided. Perhaps a step-by-step demonstration of it would help (on an example simple enough to actually be able to do this without tedium)? -- John Marrero MIT Class of '08 (305) 812.7042 From xiaoranz@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From xiaoranz@MIT.EDU Mon Sep 12 18:50:28 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 j8CMoSaJ012134 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:28 -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 j8CMoRVr019180 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:27 -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 j8CMoKu9025543 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 12 Sep 2005 18:50:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8CMoKFW031481; Mon, 12 Sep 2005 18:50:20 -0400 Received: from MCCORMICK-THREE-FORTY-TWO.MIT.EDU (MCCORMICK-THREE-FORTY-TWO.MIT.EDU [18.240.6.87]) (User authenticated as xiaoranz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 12 Sep 2005 18:50:19 -0400 Message-ID: <20050912185019.7e396ewb940kkkkg@webmail.mit.edu> Date: Mon, 12 Sep 2005 18:50:19 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comment 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-UID: 100 Content-Length: 740 X-Keywords: NonJunk NotJunk Passage: 2.6 Validity, Page 7 of week 2 Reading. I thought the prove of validity of qualifier is difficult, because it isn't quite mathematically obvious to me and I would like to have it discussed more in depth in the lecture. Maybe an explanation about how to approach these problems would be best. I had some difficulties with the qualifier validity question on the tutor problems also. Xiaoran (Sharon) Zhang ************************************************** Xiaoran (Sharon) Zhang Class of 2008 Department of Biology & Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology 320 Memorial Drive, Cambridge, MA 02139 E-mail: xiaoranz@mit.edu ************************************************** From lye@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lye@MIT.EDU Mon Sep 12 20:58:23 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 j8D0wNaJ029248 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:23 -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 j8D0wMX6012301 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:22 -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 j8D0wKMF019463 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 20:58:20 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8D0wKfl023507; Mon, 12 Sep 2005 20:58:20 -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 ; Mon, 12 Sep 2005 20:58:20 -0400 Message-ID: <20050912205820.tom5ow5cbrgo8400@webmail.mit.edu> Date: Mon, 12 Sep 2005 20:58:20 -0400 From: Linda Ye To: 6042-probs@theory.csail.MIT.EDU Subject: required email 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-UID: 101 Content-Length: 511 X-Keywords: NonJunk NotJunk >From Course Notes, Week 2: Predicates & Sets Section 1.4, pg. 3: "We first argue that at least one of these two cases must hold. We?ll prove this by contradiction. Namely, suppose neither case holds. This means that at most 2 people in the group met x and at most 2 did not meet x. This leaves at least 1 of the remaining 5 people unaccounted for." You could explain the same idea with the pigeonhole principle: out of 5 people, at least 3 of them 1) all met x, or 2) all did not meet x. Linda (Lunduo) Ye From adnaan@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From adnaan@MIT.EDU Mon Sep 12 21:09:22 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 j8D19MaJ032222 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:22 -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 j8D19KX6019582 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:20 -0400 (EDT) Received: from AdnaanJiwaji (NEW-THIRTY-NINE.MIT.EDU [18.241.5.39]) (authenticated bits=0) (User authenticated as adnaan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D19IZ0021655 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:09:18 -0400 (EDT) Message-ID: <003401c5b7ff$b4da0e20$2705f112@AdnaanJiwaji> From: "Adnaan Jiwaji" To: <6042-probs@theory.csail.mit.edu> Subject: Week 2 Comments Date: Mon, 12 Sep 2005 20:08:52 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0031_01C5B7D5.CBB17950" 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: -1.71 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 102 Content-Length: 1424 X-Keywords: NonJunk NotJunk Paragraph 2.6 Validity page 7 The difference between a propositional formula and a predicate formula is not clear from the paragraph and would be good to have it clarified in class.  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0031_01C5B7D5.CBB17950 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Paragraph 2.6 Validity page 7 The difference between a propositional formula and a predicate formula = is not clear from the paragraph and would be good to have it clarified = in class. ------=_NextPart_000_0031_01C5B7D5.CBB17950 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Paragraph 2.6 Validity page = 7
 
The difference between a propositional = formula and=20 a predicate formula is not clear from the paragraph and would be good to = have it=20 clarified in class.
 
------=_NextPart_000_0031_01C5B7D5.CBB17950-- From cwong08@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From cwong08@MIT.EDU Mon Sep 12 21:42:24 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 j8D1gOaJ005280 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:24 -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 j8D1gNAG007493 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:24 -0400 (EDT) Received: from [127.0.0.1] (PKT-NINETY-SIX.MIT.EDU [18.216.1.96]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8D1g93b025775 for <6042-probs@theory.lcs.mit.edu>; Mon, 12 Sep 2005 21:42:19 -0400 (EDT) Message-ID: <43262E65.7010401@mit.edu> Date: Mon, 12 Sep 2005 21:41:57 -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.lcs.mit.edu Subject: week 2 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-UID: 103 Content-Length: 137 X-Keywords: NonJunk NotJunk The stuff about breaking math by disproving ZFC axioms (page 12) was very intersting, and I would like to hear it elaborated in class. From mdmurray@MIT.EDU Wed Sep 14 21:47:26 2005 X-Coding-System: undecided-unix Mail-from: From mdmurray@MIT.EDU Mon Sep 12 21:50:42 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 j8D1ogaJ006044 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:42 -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 j8D1ofAG007713 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:41 -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 j8D1oc0v004705 for <6042-probs@theory.csail.mit.edu>; Mon, 12 Sep 2005 21:50:38 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: <27302626-1266-40D7-85F1-599792A4DDD1@mit.edu> Content-Type: multipart/alternative; boundary=Apple-Mail-3-92665577 From: Michael Murray Subject: Week 2 Comments Date: Mon, 12 Sep 2005 21:50:37 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.988 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 104 Content-Length: 9899 X-Keywords: NonJunk NotJunk 3.5 Functions A function assigns an element of one set, called the domain, to elements of another set, called the codomain. The notation f : A ! B indicates that f is a function with domain, A, and codomain, B. The familiar notation f(a) = b indicates that f assigns to a the element b. A function f : A ! B is: total if every element of A is assigned to some element of B; otherwise, f is called a partial function. surjective if every element of B is mapped to at least once, that is, 8b 2 B9a 2 A. f(a) = b. injective if every element of B is mapped to at most once. bijective if f is total, surjective, and injective. In particular, each element of B is mapped to exactly once. The names surjective and injective are hopelessly unmemorable and nondescriptive. Some authors us the term onto for surjective and one-to-one for injective, which are shorter but arguably no more memorable. Here are a couple examples: page 11 I found this section to be quite difficult to understand at first, it took me a while to get what was going on with f(A) and B. Now that I understand it though (thanks to another student in the class who helped me), I think it is a nice topic.  -=- MIME -=-  --Apple-Mail-3-92665577 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=WINDOWS-1252; delsp=yes; format=flowed 3.5 Functions A function assigns an element of one set, called the domain, to =20 elements of another set, called the codomain. The notation f : A ! B indicates that f is a function with domain, A, and codomain, B. The =20 familiar notation =93f(a) =3D b=94 indicates that f assigns to a the element b. A function f : A ! B is: =95 total if every element of A is assigned to some element of B; =20 otherwise, f is called a partial function. =95 surjective if every element of B is mapped to at least once, that =20= is, 8b 2 B9a 2 A. f(a) =3D b. =95 injective if every element of B is mapped to at most once. =95 bijective if f is total, surjective, and injective. In particular, =20= each element of B is mapped to exactly once. The names =93surjective=94 and =93injective=94 are hopelessly = unmemorable and =20 nondescriptive. Some authors us the term onto for surjective and one-to-one for injective, =20= which are shorter but arguably no more memorable. Here are a couple examples: page 11 I found this section to be quite difficult to understand at first, it =20= took me a while to get what was going on with f(A) and B. Now that I =20 understand it though (thanks to another student in the class who =20 helped me), I think it is a nice topic.= --Apple-Mail-3-92665577 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=WINDOWS-1252
3.5 = Functions
A = function assigns an element of one set, called the domain, to elements of another set, called = the
codomain. = The notation
f : A ! B
indicates that f = is a function with domain, A, and codomain, B. The familiar notation = =93f(a) =3D b=94
indicates that f assigns to a the element = b.
A = function f : A ! B is:
=95 total if every element of A is assigned to some = element of B; otherwise, f is called a partial
function.
=95 surjective if every element of B is mapped to at least = once, that is, 8b 2 B9a 2 A. f(a) =3D b.
=95 = injective = if every element = of B is mapped to at most once.
=95 = bijective = if f is total, = surjective, and injective. In particular, each element of B is mapped = to
exactly once.
The names =93surjective=94 and =93injective=94 = are hopelessly unmemorable and nondescriptive. = Some
authors us the term onto for surjective and one-to-one for injective, = which are shorter but arguably
no more memorable. = Here are a couple examples:

page = 11

I found this section to be quite difficult to = understand at first, it took me a while to get what was going on with = f(A) and B. Now that I understand it though (thanks to another student = in the class who helped me), I think it is a nice = topic.
= --Apple-Mail-3-92665577-- From ridell@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From ridell@MIT.EDU Mon Sep 12 22:51:16 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 j8D2pGaJ013616 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:16 -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 j8D2pFX6005702 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:15 -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 j8D2pCXC014129 for <6042-probs@theory.csail.MIT.EDU>; Mon, 12 Sep 2005 22:51:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8D2pChT031088; Mon, 12 Sep 2005 22:51:12 -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 ; Mon, 12 Sep 2005 22:51:12 -0400 Message-ID: <20050912225112.gdmcwte32pkwkws8@webmail.mit.edu> Date: Mon, 12 Sep 2005 22:51:12 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: Rebecca Idell 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: 0.226 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 105 Content-Length: 499 X-Keywords: NonJunk NotJunk 6.042 Comments On page 6, section 2.4 Order of Quantifiers, there is an example "every American has a dream." It is then translated into quantifier notation, but I do not understand the notation entirely, with lower and uppercase letters, and then creating some H(a,d), so I hope it will be further discussed in lecture. -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From shauni@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From shauni@MIT.EDU Tue Sep 13 00:08:40 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 j8D48eaJ020413 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:40 -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 j8D48cFB002382 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:38 -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 j8D48W8R001609 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:08:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8D48WCv017071; Tue, 13 Sep 2005 00:08:32 -0400 Received: from SIMMONS-FORTY-NINE.MIT.EDU (SIMMONS-FORTY-NINE.MIT.EDU [18.96.5.49]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 00:08:32 -0400 Message-ID: <20050913000832.ffo5y8fch7484cc4@webmail.mit.edu> Date: Tue, 13 Sep 2005 00:08:32 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.csail.MIT.EDU Subject: Week 2 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 X-UID: 106 Content-Length: 173 X-Keywords: NonJunk NotJunk I would like to have the functions section (p.11) discussed more fully in lecture. I especially would like to go over the terms total, surjective, injective, and bijective. From dangut@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dangut@MIT.EDU Tue Sep 13 00:24:05 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 j8D4O5aJ022133 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:24:05 -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 j8D4O4FB013139 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:24:04 -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 j8D4Nvt6004383 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 00:23:57 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8D4Nvct011692; Tue, 13 Sep 2005 00:23:57 -0400 Received: from MACGREGOR-THREE-EIGHTY-TWO.MIT.EDU (MACGREGOR-THREE-EIGHTY-TWO.MIT.EDU [18.239.6.127]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 00:23:57 -0400 Message-ID: <20050913002357.6dd1o18l5668sso4@webmail.mit.edu> Date: Tue, 13 Sep 2005 00:23:57 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Week 2 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-UID: 107 Content-Length: 361 X-Keywords: NonJunk NotJunk I think the most difficult part for me is the part about sets (page 9) since I haven't seen too many before. A question I have about hte copmlement of a set, they say the complement of all real positive numbers is all negative numbers plus zero, what keeps the complement from having imaginary and fraction values? Since these are not part of the set anyways. From hkhall@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From hkhall@MIT.EDU Tue Sep 13 00:42:13 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 j8D4gDaJ025558 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 00:42:13 -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 j8D4gBFB025529; Tue, 13 Sep 2005 00:42:11 -0400 (EDT) Received: from [18.96.6.222] (SIMMONS-FOUR-SEVENTY-SEVEN.MIT.EDU [18.96.6.222]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8D4g9hg007229 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Tue, 13 Sep 2005 00:42:10 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <80b4f15dbaa219a418cb1720b383f136@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Comments on LN2 Date: Tue, 13 Sep 2005 00:44:01 -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-UID: 108 Content-Length: 421 X-Keywords: NonJunk NotJunk I found section 2.1 of quantifying a predicate beginning on page 5 the most difficult concept in the reading. In my opinion this is due largely to the ambiguity inherent to the english language in its conversion to the language of mathematics and vice versa. While I don't want an english to math dictionary, more guidance in the conversion process would go a long way in understanding the method. -Harrison Hall From rnjacobs@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From rnjacobs@MIT.EDU Tue Sep 13 13:21:07 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 j8DHL7aJ030067 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:07 -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 j8DHL6uU027574 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:06 -0400 (EDT) Received: from m4-167-2.mit.edu (M4-167-2.MIT.EDU [18.53.0.86]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DHL4K9018842 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 13:21:04 -0400 (EDT) Received: (from rnjacobs@localhost) by m4-167-2.mit.edu (8.12.9) id j8DHL4Nt005486; Tue, 13 Sep 2005 13:21:04 -0400 (EDT) Message-Id: <200509131721.j8DHL4Nt005486@m4-167-2.mit.edu> To: 6042-probs@theory.lcs.MIT.EDU Subject: Comments for Reading due 14 September Date: Tue, 13 Sep 2005 13:21:04 -0400 From: r n jacobs X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 109 Content-Length: 387 X-Keywords: NonJunk NotJunk "2.5 Negating Quantifiers ... (not all)x. P(x) is equivalent to (exists)x. (not) P(x)." (page 7) I find myself wondering if this is related to DeMorgan's law. Also, "1.4 Proof by Cases" (page 3) I found the phrasing of the "in a group of 6, at least 3 have met or 3 have not met" bewildering, although now that I've reread it I understand. I may have been tired. - Robert Jacobs From bilodeau@MIT.EDU Wed Sep 14 21:47:29 2005 X-Coding-System: undecided-unix Mail-from: From bilodeau@MIT.EDU Tue Sep 13 15:28:23 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 j8DJSNaJ030895 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:23 -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 j8DJSMt4022947 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:22 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THREE-SEVENTY-SEVEN.MIT.EDU [18.239.6.122]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DJSI4M018839 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 15:28:18 -0400 (EDT) Message-Id: <200509131928.j8DJSI4M018839@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: Date: Tue, 13 Sep 2005 15:28:11 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5B877.C0DE2490" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcW4mUdEpMXj9GHERfCQVlzjeOLRMw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 3.757 X-Spam-Level: *** (3.757) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 110 Content-Length: 14404 X-Keywords: NonJunk NotJunk “• The union of sets X and Y (denoted X [ Y ) contains all elements appearing in X or Y or both. Thus, X [ Y = {1, 2, 3, 4}. • The intersection of X and Y (denoted X \ Y ) consists of all elements that appear in both X and Y . So X \ Y = {2, 3}. • The difference of X and Y (denoted X −Y ) consists of all elements that are in X, but not in Y . Therefore, X − Y = {1} and Y − X = {4}.” The main thing that I find difficult about the reading is the large number of symbols. The quote above is an example of the introduction of more such symbols. Having not had set theory or logical analysis before, most of this is difficult and confusing to me; I think I’ll make myself a table with all of the operators and symbols for easy reference.  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5B877.C0DE2490 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable =20 =E2=80=9C=E2=80=A2 The union of sets X and Y (denoted X [ Y ) contains = all elements appearing in X or Y or both. Thus, X [ Y =3D {1, 2, 3, 4}. =E2=80=A2 The intersection of X and Y (denoted X \ Y ) consists of all = elements that appear in both X and Y . So X \ Y =3D {2, 3}. =E2=80=A2 The difference of X and Y (denoted X =E2=88=92Y ) consists of = all elements that are in X, but not in Y . Therefore, X =E2=88=92 Y =3D {1} and Y =E2=88=92 X =3D {4}.=E2=80=9D =20 The main thing that I find difficult about the reading is the large = number of symbols. The quote above is an example of the introduction of = more such symbols. Having not had set theory or logical analysis = before, most of this is difficult and confusing to me; I think = I=E2=80=99ll make myself a table with all of the operators and symbols = for easy reference. ------=_NextPart_000_0000_01C5B877.C0DE2490 Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: quoted-printable

 

=E2=80=9C=E2=80=A2= The union of sets X and = Y (denoted X [ Y = ) contains all elements appearing in X or Y or

both. Thus, X [ Y = =3D {1, 2, 3, 4}.

=E2=80=A2 The intersection of X = and Y (denoted = X \ Y ) consists of all elements that appear in both X

and Y . So = X \ Y =3D {2, 3}.

=E2=80=A2 The difference of X = and Y (denoted = X =E2=88=92Y ) consists of all elements that are in X, but not in

Y . Therefore, X =E2=88=92 = Y = =3D {1} and Y =E2=88=92 = X = =3D {4}.=E2=80=9D

 

The main thing that I find difficult about the = reading is the large number of symbols.=C2=A0 The quote above is an example of the = introduction of more such symbols.=C2=A0 Having not had set theory or logical = analysis before, most of this is difficult and confusing to me; I think I=E2=80=99ll make = myself a table with all of the operators and symbols for easy = reference.

------=_NextPart_000_0000_01C5B877.C0DE2490-- From iyzhang@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From iyzhang@MIT.EDU Tue Sep 13 16:04:37 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 j8DK4baJ006668 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:37 -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 j8DK4aau014081 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:36 -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 j8DK4a4M028799 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:36 -0400 (EDT) Received: from [18.239.6.65] (MACGREGOR-THREE-TWENTY.MIT.EDU [18.239.6.65]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8DK4S0v022239 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 16:04:28 -0400 (EDT) Message-ID: <432730C7.2030807@mit.edu> Date: Tue, 13 Sep 2005 16:04:23 -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 2 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-UID: 111 Content-Length: 209 X-Keywords: NonJunk NotJunk I would like to discuss or see the Banach-Tarski theorem in the next class. Thanks, Irene p.s. in the function f : A -> B, A is A the codomain and B the domain? I think there may be an error in the reading From jstritar@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From jstritar@MIT.EDU Tue Sep 13 16:21:50 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 j8DKLoaJ009396 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:50 -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 j8DKLnt4021662 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:49 -0400 (EDT) Received: from [18.245.6.203] (BAKER-FOUR-FIFTY-EIGHT.MIT.EDU [18.245.6.203]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DKLcgm013329 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 16:21:42 -0400 (EDT) Message-ID: <432734D0.8020005@mit.edu> Date: Tue, 13 Sep 2005 16:21:36 -0400 From: Jon Stritar 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: -2.092 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 112 Content-Length: 523 X-Keywords: NonJunk NotJunk Lemma (Mapping Rule) on page 12. I thought the part pertaining to functions, and specifically to the sizes of the domain and codomains, was the most difficult part to understand. I guess I don't usually think of functions with respect to sets, and combined with a whole lot of new terminology, it takes a little while to get used to it. I guess comparing the sizes of the domains is like applying the function to every member of the starting set at the same time and comparing the results to the input. Jon Stritar From kevin08@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From kevin08@MIT.EDU Tue Sep 13 17:08:16 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 j8DL8FaJ018070 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:16 -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 j8DL8Ebf009964 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:14 -0400 (EDT) Received: from kevlar.mit.edu (NEXT-FOUR-THIRTY-NINE.MIT.EDU [18.242.6.184]) (authenticated bits=0) (User authenticated as kevin08@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DL8C23003951 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:08:12 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913170501.021becf8@hesiod> X-Sender: kevin08@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 17:08:09 -0400 To: 6042-probs@theory.lcs.mit.edu From: Kevin Wang Subject: 6.042 Reading Feedback Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.052 X-Spam-Level: * (1.052) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 113 Content-Length: 555 X-Keywords: NonJunk NotJunk I thought that the theorem in Section 1.4 (Pages 3-4) regarding the clubs/groups of 3 people was interesting, and evoked pictures of the pigeonhole principle. If you could talk about this proof more thoroughly (with diagrams!!) in class, that would be very useful. Also, if you could introduce the pigeonhole principle and talk about whether it relates to this problem or not, that would be great. Thanks, Kevin Wang Massachusetts Institute of Technology - Class of 2008 Electrical Engineering and Computer Science http://web.mit.edu/kevin08/www From ereid@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From ereid@MIT.EDU Tue Sep 13 17:15:27 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 j8DLFRaJ019591 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:27 -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 j8DLFQbf017327 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:26 -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 j8DLFITY006925 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 17:15:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <8AE1BD52-4336-4C28-8297-410A72A8F212@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: reading response Date: Tue, 13 Sep 2005 17:15:11 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -0.596 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 114 Content-Length: 199 X-Keywords: NonJunk NotJunk I'm finding it difficult to understand the product of sequences. I don't follow how the elements are chosen for the sequences, and what the rules are on which sequence to take the elements from. From dowgun@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dowgun@MIT.EDU Tue Sep 13 17:49:23 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 j8DLnNaJ031041 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:23 -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 j8DLnMbf018932 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:22 -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 j8DLnFAL019383 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 17:49:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8DLnFO8030598; Tue, 13 Sep 2005 17:49:15 -0400 Received: from PLP-ONE-FORTY-THREE.MIT.EDU (PLP-ONE-FORTY-THREE.MIT.EDU [18.218.1.143]) (User authenticated as dowgun@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 17:49:15 -0400 Message-ID: <20050913174915.4z7un1c55tmsgosw@webmail.mit.edu> Date: Tue, 13 Sep 2005 17:49:15 -0400 From: Neil M Dowgun To: 6042-probs@theory.csail.MIT.EDU Subject: response for 9/14 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-UID: 115 Content-Length: 951 X-Keywords: NonJunk NotJunk I feel that we should take some time in lecture to cover "functions" as applied to sets, which was written about in section 3.5 (pages 11-12) of the reading. It occurs to me that for a finite domain (and codomain), the only way to completely describe a set function is to specifically lay out which element of the codomain the domain is mapped too. I supposed that someone might have described a "sequential" function so that the first element of the domain maps to the first element of the codomain, but sets have no specific order, so this is impossible. Also, describing a function without knowing the codomain it is going to be mapped to seems impossible to me, unless you create the codomain FROM the application of the function to the domain. Is this allowed? Also, I don't see the difference between [ (upside-down A)x(backwards E)y(P(x,y)) ] and [ (backwards E)y(upside-down A)x(P(x,y)) ]. Does that change in order EVER matter? Neil Dowgun From hectorb@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From hectorb@MIT.EDU Tue Sep 13 17:56:16 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 j8DLuGaJ032132 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:56:16 -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 j8DLuEbf024816; Tue, 13 Sep 2005 17:56:14 -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 j8DLu7xk021725; Tue, 13 Sep 2005 17:56:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8DLu7nS016371; Tue, 13 Sep 2005 17:56:07 -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 ; Tue, 13 Sep 2005 17:56:07 -0400 Message-ID: <20050913175607.6p7cy2su6qe8gg84@webmail.mit.edu> Date: Tue, 13 Sep 2005 17:56:07 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 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 X-UID: 116 Content-Length: 326 X-Keywords: NonJunk NotJunk "When all variables in a formula are understood to take values from the same nonempty set D, it's conventional to omit mention of D. For example....It's easy to arrange for all variables to range over one domain." -pg 7 I did not find this explanation of variables over one domain so easy to understand. -Hector Beltran From lkini@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From lkini@MIT.EDU Tue Sep 13 17:57:05 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 j8DLv5aJ032183 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:05 -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 j8DLv4bf025414 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:04 -0400 (EDT) Received: from quickstation-macgregor.mit.edu (QUICKSTATION-MACGREGOR.MIT.EDU [18.239.0.167]) (authenticated bits=56) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DLuxFn022016 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 17:57:01 -0400 (EDT) Received: (from lkini@localhost) by quickstation-macgregor.mit.edu (8.12.9) id j8DLuwL9025952; Tue, 13 Sep 2005 17:56:58 -0400 (EDT) Subject: 09/14/05 Reading Assignment From: Lohith G Kini To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Tue, 13 Sep 2005 17:56:56 -0400 Message-Id: <1126648616.25942.3.camel@quickstation-macgregor.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 X-UID: 117 Content-Length: 413 X-Keywords: NonJunk NotJunk Hi, The passage that I found most difficult to comprehend and would like to have explained in detail during lecture is the proof by cases of the theorem regarding a collection of 6 people which includes a club of 3 people or a group of 3 strangers [page 3 of lecture notes 2]. One strategy that might help explain [me] this point is to show other examples of proofs that use proof by cases. Thanks, Lohith Kini From icharny@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From icharny@MIT.EDU Tue Sep 13 18:42:04 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 j8DMg4aJ005806 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:42:04 -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 j8DMg3UR017103 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:42:03 -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 j8DMft0x029377 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 18:41:59 -0400 (EDT) Message-ID: <432755B1.50201@mit.edu> Date: Tue, 13 Sep 2005 18:41:53 -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: 6.042 lecture notes 2 comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.486 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 118 Content-Length: 167 X-Keywords: NonJunk NotJunk Pages 11-12 of the lecture notes discuss funtions and mappings (section 3.5). Please discuss this topic a little more thoroughly at the next lecture. Isaac Charny From avalys@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From avalys@MIT.EDU Tue Sep 13 18:43:39 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 j8DMhdaJ005913 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:39 -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 j8DMhbbf001356 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:37 -0400 (EDT) Received: from [18.96.7.30] (SIMMONS-FIVE-FOURTY-ONE.MIT.EDU [18.96.7.30]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DMhISA005124 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 18:43:28 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <257831C0-3C54-4111-ABE3-8D4D18EAE1A6@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Alex Valys Subject: Week 2 Comments Date: Tue, 13 Sep 2005 18:43:19 -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-UID: 119 Content-Length: 381 X-Keywords: NonJunk NotJunk Unhappily, I don't understand the specifics of how functions work, and what it actually means when the notes say: "a function _assigns_ elements of one set..." Also, on the tutor problems, I don't understand how the definition of a function relating A and B has any affect on A and B themselves - specifically, what the difference is between |A| and |f(a)|. Alex Valys From adamreed@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: undecided-unix Mail-from: From adamreed@MIT.EDU Tue Sep 13 19:38:16 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 j8DNcFaJ013041 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:15 -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 j8DNcEbf008737 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:14 -0400 (EDT) Received: from [18.228.0.23] (BOLIVAR.MIT.EDU [18.228.0.23]) (authenticated bits=0) (User authenticated as adamreed@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8DNcCHq016048 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 19:38:12 -0400 (EDT) Message-ID: <4327632A.9000805@mit.edu> Date: Tue, 13 Sep 2005 19:39:22 -0400 From: Adam Reed User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050729) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 2 Reading 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-UID: 120 Content-Length: 331 X-Keywords: NonJunk NotJunk I found the following passage on the Order of Quantifiers the most difficult part of the reading to understand: "Swapping quantifiers in Goldbach's Conjecture creates a patently false statement." p. 6 The example "Every American has a dream" was easy to understand, but I couldn't figure out the Goldbach example. -Adam Reed From rshroff@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From rshroff@MIT.EDU Tue Sep 13 20:21:51 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 j8E0LpaJ019690 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:51 -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 j8E0LoEd006339 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:50 -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 j8E0LhWU024332 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 20:21:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E0LgbS008878; Tue, 13 Sep 2005 20:21:42 -0400 Received: from NEXT-FOUR-EIGHTY.MIT.EDU (NEXT-FOUR-EIGHTY.MIT.EDU [18.242.6.225]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 20:21:42 -0400 Message-ID: <20050913202142.77mf8mn1guiscc4w@webmail.mit.edu> Date: Tue, 13 Sep 2005 20:21:42 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: E-mail Comments for assigned 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: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 121 Content-Length: 324 X-Keywords: NonJunk NotJunk Reference: Section 1.4 Proof by cases, Pg 3 I never really understood the concept of 'Proof by Cases' before I read the article. I found the example extremely interesting and it greatly helped me to understand better. However I would like it if we are exposed to some more examples on proofs by this method. -Rahul Shroff From yangc@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: iso-8859-1-unix Mail-from: From yangc@MIT.EDU Tue Sep 13 20:24:27 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 j8E0OQaJ020218 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:27 -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 j8E0OPEd008114 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:25 -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 j8E0OMna024924 for <6042-probs@theory.csail.MIT.EDU>; Tue, 13 Sep 2005 20:24:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E0OM3h024139; Tue, 13 Sep 2005 20:24:22 -0400 Received: from MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) (User authenticated as yangc@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 20:24:22 -0400 Message-ID: <20050913202422.idtv2em8k288g0w8@webmail.mit.edu> Date: Tue, 13 Sep 2005 20:24:22 -0400 From: Christopher M Yang To: 6042-probs@theory.csail.MIT.EDU Subject: E-Mail 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: 0.907 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 122 Content-Length: 284 X-Keywords: NonJunk NotJunk Page 12, Part 4 (Does all this really work?) Are we comfortable living in a universe where volume is plastic and fluid? I've heard a lot about Godel's Incompleteness Theorem and the ZFC axioms - I'd like to hear more about that, even if it is in the most cursory sense. Chris Yang From juang@MIT.EDU Wed Sep 14 21:47:30 2005 X-Coding-System: raw-text-unix Mail-from: From juang@MIT.EDU Tue Sep 13 21:04:01 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 j8E141aJ023132 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:04:01 -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 j8E140Ed004813 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:04:00 -0400 (EDT) Received: from [18.238.6.150] (EASTCAMPUS-NINE-SEVENTEEN.MIT.EDU [18.238.6.150]) (authenticated bits=0) (User authenticated as juang@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E13wmb002604 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 21:03:59 -0400 (EDT) Message-ID: <432776ED.2070402@mit.edu> Date: Tue, 13 Sep 2005 21:03:41 -0400 From: Jason Juang User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.5) Gecko/20041201 Thunderbird/1.0RC1 Mnenhy/0.6.0.104 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=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 123 Content-Length: 671 X-Keywords: NonJunk NotJunk On page 12 of the reading: "No one knows whether the ZFC axioms are logically consistent; there is some possibility that one person might prove a proposition P and another might prove the proposition P. Then Math would be broken. This sounds like a crazy situation, but it has happened before. At the beginning of the 20th century, the logician Gotlob Frege made an initial attempt to axiomatize set theory using a few very plausible axioms. Several mathematicians most famously Bertrand Russell discovered that Freges axioms actually were self-contradictory!" I would like to see a further discussion of Russell's paradox in the next lecture. Jason Juang. From medrano@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From medrano@MIT.EDU Tue Sep 13 21:15:11 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 j8E1FBaJ024636 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:11 -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 j8E1FAEd012648 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:10 -0400 (EDT) Received: from newhouse-3.mit.edu (NEWHOUSE-3.MIT.EDU [18.241.2.250]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1F7Fk004798 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:15:08 -0400 (EDT) Received: (from medrano@localhost) by newhouse-3.mit.edu (8.12.9) id j8E1F717008017; Tue, 13 Sep 2005 21:15:07 -0400 Date: Tue, 13 Sep 2005 21:15:07 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: required comments for reading Message-ID: 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-UID: 124 Content-Length: 330 X-Keywords: NonJunk NotJunk This weeks reading on proof by contradiction is very straight forward; however, it would be nice to work with more deductive proofs. I would like to also take more time dealing with multiple and mixed quantifiers in Logic. The set theory was also straight forward. Please clearly explain Russell's paradox. Jesus Medrano From jehan@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From jehan@MIT.EDU Tue Sep 13 21:31:51 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 j8E1VpaJ027701 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:51 -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 j8E1VoEd024879 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:50 -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 j8E1VkNo007928 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:31:47 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913212552.00bb6808@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 21:31:48 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: Week 2 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-UID: 125 Content-Length: 211 X-Keywords: NonJunk NotJunk 3.5 Function on page 11 Could you explain them a bit better in class. They seemed to be just tagged on the reading without much of an explanation and I am still a bit confused as to their use. Thanks Jehan From dnreshef@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From dnreshef@MIT.EDU Tue Sep 13 21:43:21 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 j8E1hLaJ029619 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:21 -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 j8E1hJEd002762 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:20 -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 j8E1hDbA010216 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:43:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E1hDp8020185; Tue, 13 Sep 2005 21:43:13 -0400 Received: from MACGREGOR-FOUR-FIFTY-SIX.MIT.EDU (MACGREGOR-FOUR-FIFTY-SIX.MIT.EDU [18.239.6.201]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 21:43:13 -0400 Message-ID: <20050913214313.fx3vugr6f9a8ow0k@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:43:13 -0400 From: David N Reshef 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: 1.206 X-Spam-Level: * (1.206) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 126 Content-Length: 287 X-Keywords: NonJunk NotJunk READING COMMENTS: One thing that I found a little confusing was the difference between universal and existential quantification (pgs 5 and 6). Another confusing point was the business about Lemma (mapping rules) on page 12. I would appreciat disscusing this topic in lecture. -dave From pjs@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From pjs@MIT.EDU Tue Sep 13 21:44:35 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 j8E1iZaJ029668 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:35 -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 j8E1iXEd003761 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:34 -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 j8E1iQnN010466 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:44:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8E1iQIg020425; Tue, 13 Sep 2005 21:44:26 -0400 Received: from W20-575-76.MIT.EDU (W20-575-76.MIT.EDU [18.187.0.95]) (User authenticated as pjs@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 21:44:26 -0400 Message-ID: <20050913214426.jx0fhy62too4s0o0@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:44:26 -0400 From: Paul J Steiner To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Assignment 1 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-UID: 127 Content-Length: 411 X-Keywords: NonJunk NotJunk I found the wrap up section "Does All This Really Work?" on pages 12 and 13 the most interesting part of the reading. Having done some cursory reading on some of the interesting consequences of accepting the Axiom of Choice, I wasn't surprised by the Banach-Tarski theorem, but it still remains tremendously fascinating! It's also a bit of a chuckle that mathematics as is cannot prove itself. ;) PJ Steiner From nedzel@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From nedzel@MIT.EDU Tue Sep 13 21:45:14 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 j8E1jEaJ029720 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:14 -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 j8E1jDEd004241 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:13 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SIX-NINETY-FOUR.MIT.EDU [18.96.7.183]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E1j9MZ010607 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 21:45:10 -0400 (EDT) Message-Id: <200509140145.j8E1j9MZ010607@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: Week 2 Comments Date: Tue, 13 Sep 2005 21:44:59 -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: -0.02 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 128 Content-Length: 161 X-Keywords: NonJunk NotJunk In section 2.4, I don't understand the Goldbach's conjecture example. It seems to me that the both orderings of the quantifiers have the same meaning. - David From lana@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lana@MIT.EDU Tue Sep 13 21:49:18 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 j8E1nIaJ029899 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:18 -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 j8E1nHEd007036 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:17 -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 j8E1nAbW011387 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 21:49:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E1nAmt003121; Tue, 13 Sep 2005 21:49:10 -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 ; Tue, 13 Sep 2005 21:49:10 -0400 Message-ID: <20050913214910.urnbjqms7jco40wg@webmail.mit.edu> Date: Tue, 13 Sep 2005 21:49:10 -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: 3.14 X-Spam-Level: *** (3.14) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 129 Content-Length: 383 X-Keywords: NonJunk NotJunk The most interesting section for me was the paragraph on Page 14 explaining Russel's paradox. I was very surprising to see that a seemingly simple and intuitive notion of a set cannot be assumed in any situation. I would love to see some known attempts of establishing rules in order to determine when the use of the concepts of sets is appropriate in lecture. Svetlana Godlenberg From moscicki@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From moscicki@MIT.EDU Tue Sep 13 22:30:36 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 j8E2UaaJ007807 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:36 -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 j8E2UZEd006633 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:35 -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 j8E2UVFx019733 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 22:30:32 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E2UVD9032190; Tue, 13 Sep 2005 22:30:31 -0400 Received: from NEW-TWENTY-SIX.MIT.EDU (NEW-TWENTY-SIX.MIT.EDU [18.241.5.26]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 22:30:31 -0400 Message-ID: <20050913223031.8yjyyn9kjokgg448@webmail.mit.edu> Date: Tue, 13 Sep 2005 22:30:31 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 Reading Comments MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_4ytbw99wiav4" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.235 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 130 Content-Length: 4044 X-Keywords: NonJunk NotJunk Hope I did it right... Angelique Moscicki Week 2 Reading Comments *need to remember* to get contrapositive from positive, need to both negate and switch the order! Proof by cases procedure: 1. make some rules that have to be true concering the truth values of the statements in order for them to be consistent. 2. for a given value of a variable, try true or false to see if the statements are in fact consistent. The number of cases will be 2^n where n is the number of INDEPENDENT variables. Thus step 1 can make life easier. Predicates: A predicate is just like a Scheme predicate. Quantifiers: These are not so terribly new to me. Don't really have too many comments, only that they are always trickier when you actually use them than when they are explained in notes. Notes seem simple, useage somewhat harder. noting that the order is much more important that i had previously considered. Math data types: Used these before. Familiar with "element of" sign. Subsets(9): These are relatively new! Need to pay attention to these. Union: inclusive Intersection: exclusive Difference: (in the first) ^~ (in the second) Complement: domain - the set Power Set: familiar with it. All the subsets + the set Product of sets: set of the sequences where 1 element from each set All these things are not hard in themselves, just difficult to remember and keep straight, and it seems a little practice using them is required even though it doesn't at first appear to be the case. Set builder: just like filter from Scheme Functions: from earlier math, domain=domain, range=codomain. Total -> uses the whole domain Surjective -> all codomain used at least once Injective -> all codomain used at most once (this is a graphical function) Bijective -> one to one function; both of the above and therefore total.  -=- MIME -=-  This message is in MIME format. --=_4ytbw99wiav4 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit Hope I did it right... --=_4ytbw99wiav4 Content-Type: text/plain; charset=ISO-8859-1; name="week2comment.txt" Content-Disposition: attachment; filename="week2comment.txt" Content-Transfer-Encoding: 7bit Angelique Moscicki Week 2 Reading Comments *need to remember* to get contrapositive from positive, need to both negate and switch the order! Proof by cases procedure: 1. make some rules that have to be true concering the truth values of the statements in order for them to be consistent. 2. for a given value of a variable, try true or false to see if the statements are in fact consistent. The number of cases will be 2^n where n is the number of INDEPENDENT variables. Thus step 1 can make life easier. Predicates: A predicate is just like a Scheme predicate. Quantifiers: These are not so terribly new to me. Don't really have too many comments, only that they are always trickier when you actually use them than when they are explained in notes. Notes seem simple, useage somewhat harder. noting that the order is much more important that i had previously considered. Math data types: Used these before. Familiar with "element of" sign. Subsets(9): These are relatively new! Need to pay attention to these. Union: inclusive Intersection: exclusive Difference: (in the first) ^~ (in the second) Complement: domain - the set Power Set: familiar with it. All the subsets + the set Product of sets: set of the sequences where 1 element from each set All these things are not hard in themselves, just difficult to remember and keep straight, and it seems a little practice using them is required even though it doesn't at first appear to be the case. Set builder: just like filter from Scheme Functions: from earlier math, domain=domain, range=codomain. Total -> uses the whole domain Surjective -> all codomain used at least once Injective -> all codomain used at most once (this is a graphical function) Bijective -> one to one function; both of the above and therefore total. --=_4ytbw99wiav4-- From sergiob@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From sergiob@MIT.EDU Tue Sep 13 22:31:18 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 j8E2VIaJ007869 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:18 -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 j8E2VGEd007151 for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:17 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-FIVE-THIRTY-THREE.MIT.EDU [18.244.7.22]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E2V50T019881 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 13 Sep 2005 22:31:09 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913202016.034af478@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 20:30:49 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: Week 2 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 j8E2VIaJ007869 Status: RO X-UID: 131 Content-Length: 523 X-Keywords: NonJunk NotJunk I found the most shocking part of the reading assignment to be the comment on the Banach-Tarski Theorem on page 13. Its amazing that the same set of axioms that induces familiar arithmetic propositions like 3+3=6 also induces a theorem that states that a sphere can be split into 6 parts which can be rearranged to form 2 spheres of the same volume of the original. It is mind-boggling that math, as we know it, is based on a system of axioms with such obscure consequences. Sergio Bacallado MIT ID# 955.518.187 From ozcan@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From ozcan@MIT.EDU Tue Sep 13 22:56:40 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 j8E2ueaJ012903 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:40 -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 j8E2udQ2024508 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:39 -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 j8E2uciB009959 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:38 -0400 (EDT) Received: from ebrum.mit.edu (NEW-ONE-SIXTY-NINE.MIT.EDU [18.241.5.169]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E2uV3b002374 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225322.01fca0c8@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 22:56:26 -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: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 132 Content-Length: 190 X-Keywords: NonJunk NotJunk Hi, Can we go over the proof of the theorem on page 3: Every collection of 6 people includes a club of 3 people or a group of 3 strangers. It was kind of confusing. Thanks Yasin Ozcan From tonyng@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From tonyng@MIT.EDU Tue Sep 13 22:56:55 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 j8E2usaJ013181 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:54 -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 j8E2ufEd025305 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:41 -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 j8E2uYf8025268 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 22:56:35 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225217.01f02390@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 22:56:48 -0400 To: 6042-probs@theory.lcs.mit.edu From: Tony Ng Subject: Reading 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-UID: 133 Content-Length: 506 X-Keywords: NonJunk NotJunk I found Proof by Cases (Page 3, Section 1.4) particularly interesting and surprising. I never tried proving something by starting with a statement that I am not sure of, then analyzing what happens if it is true and what happens if it is false. Intuitively, I always thought that if I have a statement that I am not sure of, I cannot get anything from it, but apparently something can be concluded if the statement leads to the same thing when the statement is true and when the statement is false. From aston@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From aston@MIT.EDU Tue Sep 13 23:04:52 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 j8E34qaJ014932 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:04:52 -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 j8E34iuK029697 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:04:51 -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 j8E32trL026519 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:02:56 -0400 (EDT) Message-Id: <200509140302.j8E32trL026519@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Response Date: Tue, 13 Sep 2005 23:02:54 -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: AcW42MziUpWksdORSru3lDnRuw1dNQ== X-Spam-Score: 2.224 X-Spam-Level: ** (2.224) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 134 Content-Length: 348 X-Keywords: NonJunk NotJunk The swap of the quantifiers in Goldbach's Conjecture (page 6) slowed me down for a second in my reading as I tried to clarify what, exactly, was weird about it (why it was false), until I figured out that, as stated, we would need magical numbers p and q that sum to every even number. I'm not sure this was well clarified in the notes. - Aston From hzhou@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From hzhou@MIT.EDU Tue Sep 13 23:06:26 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 j8E36QaJ015001 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:26 -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 j8E36Ou8000475 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:25 -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 j8E368gH027089 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:06:09 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913225747.02a5c170@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 23:06:11 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: Interesting 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-UID: 135 Content-Length: 77 X-Keywords: NonJunk NotJunk I would like to see more discussion about the set notations. - Steven Zhou From jjmonzon@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From jjmonzon@MIT.EDU Tue Sep 13 23:22:44 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 j8E3MiaJ018030 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:44 -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 j8E3Mhu8012407 for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:43 -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 j8E3MZAk000172 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Tue, 13 Sep 2005 23:22:36 -0400 (EDT) Message-Id: <5.2.1.1.2.20050913220929.0272e7d8@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 13 Sep 2005 23:22:43 -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: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 136 Content-Length: 737 X-Keywords: NonJunk NotJunk I found it very confusing to understand section 2.4 and 2.5 (page 7-8) regarding validity of propositions. I find it hard to translate English propositions to math symbols and letters and vice versa. I also find negating quantifiers to be confusing if we put it in front of the predicate or in front of the "for all", "there exist" symbols. I think that a few more examples presented in class would help rather using elegant statements using math symbols. I know the symbols make a statement more elegant concise but people who are not really used to it will find it even more confusing. Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From kromer@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From kromer@MIT.EDU Tue Sep 13 23:23:37 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 j8E3NbaJ018281 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:37 -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 j8E3NZu8012994 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:36 -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 j8E3NOPa000352 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:25 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E3NOSc016989; Tue, 13 Sep 2005 23:23:24 -0400 Received: from BURTON-FIVE-SIXTY-SIX.MIT.EDU (BURTON-FIVE-SIXTY-SIX.MIT.EDU [18.247.7.55]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:23:24 -0400 Message-ID: <20050913232324.p5zmdou9s14oc8kc@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:23:24 -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-UID: 137 Content-Length: 396 X-Keywords: NonJunk NotJunk I found it surprising that "we must reject the axiom that every mathematically well-defined collection of objects is a set" (Course Notes, week 2, p. 14). The ZFC axioms are so general that I expected them to apply to every well-defined collection of objects. It seems strange that people used the concept of sets for years without being able to define precisely what a set it. -Katherine Romer From shreyes@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From shreyes@MIT.EDU Tue Sep 13 23:23:39 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 j8E3NdaJ018286 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:39 -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 j8E3Ncu8013031 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:38 -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 j8E3NaGN000385 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:23:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E3Na3o007265; Tue, 13 Sep 2005 23:23:36 -0400 Received: from NEXT-FOUR-FORTY-ONE.MIT.EDU (NEXT-FOUR-FORTY-ONE.MIT.EDU [18.242.6.186]) (User authenticated as shreyes@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:23:36 -0400 Message-ID: <20050913232336.oftclmyihby8ocw8@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:23:36 -0400 From: Shreyes Seshasai To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments Week 2 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-UID: 138 Content-Length: 864 X-Keywords: NonJunk $Label4 NotJunk Hi, This week's reading covered several topics which I was unfamiliar with, including sequences and surjective/injective functions. The part that I found most difficult (and would benefit from by seeing it in lecture) is the mapping rule. It is still a little unclear to me the relation between A and B if we know the type of the function. I was also confused about the following passage (page 12). "Everything about a function is captured by three sets: it domain, its codomain, and the set {(a, b) | f(a) = b} which is called the graph of f. Notice that the graph of a function does not determine by itself whether a function has any of the mapping properties above." This was a little confusing because looking at a graph seems to be the easiest way to quickly determine if the function is injective and/or surjective. See you in class, Shreyes Seshasai From crowell@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: iso-8859-1-unix Mail-from: From crowell@MIT.EDU Tue Sep 13 23:42:17 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 j8E3gHaJ019644 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:17 -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 j8E3gGu8025155 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:16 -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 j8E3g8k6004020 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 13 Sep 2005 23:42:08 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8E3g8uj009879; Tue, 13 Sep 2005 23:42:08 -0400 Received: from EASTCAMPUS-SEVEN-SIXTY.MIT.EDU (EASTCAMPUS-SEVEN-SIXTY.MIT.EDU [18.238.5.249]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 13 Sep 2005 23:42:08 -0400 Message-ID: <20050913234208.h24nzpn7eqf4wg00@webmail.mit.edu> Date: Tue, 13 Sep 2005 23:42:08 -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 X-UID: 139 Content-Length: 312 X-Keywords: NonJunk NotJunk I was most confused by section 7.1.2 on page 12 on "Implies." I have seen the other logical operations before (OR, AND, etc) but was unfamiliar with IMPLIES. At first the fact that a false P (in the relation P=>Q) made a true implication was troubling as well, as it seemed counter-intuitive. -Rob Crowell From brevzin@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From brevzin@MIT.EDU Wed Sep 14 00:00:03 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 j8E403aJ023592 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:03 -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 j8E402u8007148 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:02 -0400 (EDT) Received: from TP306.mit.edu (PLP-ONE-TWENTY-FOUR.MIT.EDU [18.218.1.124]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E3xvDN007092 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 00:00:00 -0400 (EDT) Message-Id: <6.1.2.0.1.20050913234422.01a7abb0@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Tue, 13 Sep 2005 23:59:56 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: Reading Assignment 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-UID: 140 Content-Length: 669 X-Keywords: NonJunk $Label4 NotJunk So it's pretty interesting that switching the orders of the definitions (such as the example with Goldbach's conjecture) provides such a VAST difference in meaning... what with "for each even integer n, there exist p,q primes such that p + q = n" and "there exist p,q primes such that for each even integer n, p + q = n". Sort of goes against common intuition that definitions are just definitions, sort of how one would expect all multiplication to be commutative. But you would be wrong. Also, in analysis, the set of natural numbers does refer to {1,2,3,...}. But there IS a footnote which states that in some systems they start at 0. (Boris) Barry Revzin From bakster@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From bakster@MIT.EDU Wed Sep 14 00:23:37 2005 Return-Path: Received: from bexxxley.mit.edu (BEXXXLEY.MIT.EDU [18.246.1.238]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8E4NaaJ027310 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:23:36 -0400 Received: (from bakster@localhost) by bexxxley.mit.edu (8.12.9) id j8E4NaQv000708; Wed, 14 Sep 2005 00:23:36 -0400 Date: Wed, 14 Sep 2005 00:23:36 -0400 (EDT) From: Alexander G Bakst X-X-Sender: bakster@bexxxley To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Reading Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 141 Content-Length: 188 X-Keywords: NonJunk NotJunk I found Russel's Paradox very confusing - perhaps we could go over that in lecture? I would also like to go over section 3.4, set builder notation in class, if possible. Alexander Bakst From hsoumare@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From hsoumare@MIT.EDU Wed Sep 14 00:30:29 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 j8E4UTaJ027981 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:29 -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 j8E4USpm025007 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:29 -0400 (EDT) Received: from [18.241.5.187] (NEW-ONE-EIGHTY-SEVEN.MIT.EDU [18.241.5.187]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E4UL0v013001 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:30:22 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v733) 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: Hamidou Soumare Subject: Week 2 Required Reading Comments Date: Wed, 14 Sep 2005 00:38:33 -0400 X-Mailer: Apple Mail (2.733) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 142 Content-Length: 260 X-Keywords: NonJunk NotJunk I would like the proof of the theorem about strangers and clubs on Page 3 to be explained in class because I did not understand it at all. I dont see how the cases discussed account for everything and how they prove the theorem. Thanks. Hamidou Soumare From letrec@MIT.EDU Wed Sep 14 21:47:31 2005 X-Coding-System: undecided-unix Mail-from: From letrec@MIT.EDU Wed Sep 14 00:40:38 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 j8E4ecaJ030496 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:38 -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 j8E4ebu8002746 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:37 -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 j8E4eSpg013322 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 00:40:29 -0400 (EDT) Message-ID: <4327B6D3.8040906@mit.edu> Date: Wed, 14 Sep 2005 01:36:19 -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: Page 8, paragraph 2 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: -1.451 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 143 Content-Length: 627 X-Keywords: NonJunk NotJunk -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 It is clear that the implication doesn't work both ways for the generic case, However -- the interpretation of the converse of the statement initially felt a bit confusing as it used the domain of integers. At first the conclusion felt a bit trivial, but proves essential to constructing predicates across a single domain. -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.2 (GNU/Linux) Comment: Using GnuPG with Thunderbird - http://enigmail.mozdev.org iD4DBQFDJ7bR60r5GXIeeLsRAl0PAJ9EnuzaZaOUazy5lV9l3UH8pRIcTwCYqL7X be6ICpBORkWzzIw6+h3PZg== =bCLV -----END PGP SIGNATURE----- From sil_03@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From sil_03@MIT.EDU Wed Sep 14 00:58:10 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 j8E4wAaJ032515 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:10 -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 j8E4w9u8013335 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:09 -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 j8E4w3Wv015546 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 00:58:03 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8E4w3K4012386; Wed, 14 Sep 2005 00:58:03 -0400 Received: from NEW-TWO-SEVENTY-ONE.MIT.EDU (NEW-TWO-SEVENTY-ONE.MIT.EDU [18.241.6.16]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 00:58:03 -0400 Message-ID: <20050914005803.w6euvys4rqfk8s08@webmail.mit.edu> Date: Wed, 14 Sep 2005 00:58:03 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: week 2 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 X-UID: 144 Content-Length: 204 X-Keywords: NonJunk NotJunk Section 3.5(Functions) page 12: I did not understand the meaning of the graph of f which is {(a,b)|f(a)=b}. I'm not even sure how you would read it. A few examples in the next lecture might clarify it. From mracich@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mracich@MIT.EDU Wed Sep 14 01:08:57 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 j8E58vaJ001981 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:57 -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 j8E58uFj025698 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:56 -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 j8E58tiB012618 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:55 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY-FIVE.MIT.EDU (MACGREGOR-FOUR-EIGHTY-FIVE.MIT.EDU [18.239.6.230]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E58q0v014273 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:08:52 -0400 (EDT) Subject: Comments for Course Notes, Week 2 (Predicates & Sets) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 14 Sep 2005 01:08:49 -0400 Message-Id: <1126674529.8290.23.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.372 X-Spam-Level: * (1.372) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 145 Content-Length: 362 X-Keywords: NonJunk NotJunk I found all of section 3.5, Functions, (starting on page 11) to be very confusing. I found it difficult to grasp the concepts because I have never seen them before, and they come across as being very non-intuitive (and it doesn't help that the terms all sound alike). I would very much appreciate it if we went over this section in lecture. Moira Racich From pgroudas@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From pgroudas@MIT.EDU Wed Sep 14 01:13:48 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 j8E5DmaJ002331 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:48 -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 j8E5Dku8022586 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:47 -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 j8E5DdIN017530 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 01:13:40 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E5Ddqk029686; Wed, 14 Sep 2005 01:13:39 -0400 Received: from 18.233.0.185 ([18.233.0.185]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 01:13:39 -0400 Message-ID: <20050914011339.vdq8zinsl40k08gk@webmail.mit.edu> Date: Wed, 14 Sep 2005 01:13:39 -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: -1.351 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 146 Content-Length: 231 X-Keywords: NonJunk NotJunk I would like the examples of validity on page 8 to be explained more fully. I just feel that there should be an easy way to think about it that will click in my mind and the reading wasn't doing it for me. Thanks. -Paul Groudas From jeffhoff@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From jeffhoff@MIT.EDU Wed Sep 14 01:19:41 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 j8E5JfaJ003533 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:41 -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 j8E5Jdu8025871 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:40 -0400 (EDT) Received: from JERSEY.mit.edu (JERSEY.MIT.EDU [18.235.0.193]) (authenticated bits=0) (User authenticated as jeffhoff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E5JVNn018195 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:19:32 -0400 (EDT) Message-Id: <5.2.1.1.2.20050914010639.028e6d70@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Wed, 14 Sep 2005 01:19:30 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Jeffrey D. Hoff" Subject: Reading Comments 1 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.689 X-Spam-Level: * (1.689) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 147 Content-Length: 544 X-Keywords: NonJunk NotJunk Would like to have discussed more fully it in the next lecture. Page: 8 Under Mathematical Data Types Definition of Natural Numbers: In class and in the notes it has been said that the natural numbers include zero. However, a quick Google search [ http://www.google.com/search?hl=en&lr=&oi=defmore&q=define:Natural+Numbers ] shows that 9 out of 10 definitions are { 1,2,3... } (the one says could be either). Why is there multiple definitions and also why do we use the one that seems to be the odd definition? Thanks, Jeffrey D. Hoff From mpapi@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mpapi@MIT.EDU Wed Sep 14 01:27:00 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 j8E5R0aJ005406 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:27:00 -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 j8E5QxFj021691 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:27:00 -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 j8E5QxiB012985 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:26:59 -0400 (EDT) Received: from MACGREGOR-ONE-THIRTY.MIT.EDU (MACGREGOR-ONE-THIRTY.MIT.EDU [18.239.5.130]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8E5Qt3a007030 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 01:26:55 -0400 (EDT) Subject: Reading comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: multipart/signed; micalg=pgp-sha1; protocol="application/pgp-signature"; boundary="=-fW9rLwW2Ge/MIKv1vKtw" Date: Wed, 14 Sep 2005 01:26:54 -0400 Message-Id: <1126675615.27426.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 148 Content-Length: 1456 X-Keywords: NonJunk NotJunk Reading comments: On page 12, just before section 4, where the "graph" of a function is discussed: "Everything about a function is captured by..." and also "Notice that the graph of a function does not determine by itself...". I understand this concept, for the most part. However, I would like to have this discussed more fully in the next lecture, particularly with regard to why it's significant, and maybe to better understand it through some examples. -- Matt This is a digitally signed message part  -=- MIME -=-  --=-fW9rLwW2Ge/MIKv1vKtw Content-Type: text/plain Content-Transfer-Encoding: quoted-printable Reading comments: On page 12, just before section 4, where the "graph" of a function is discussed: "Everything about a function is captured by..." and also "Notice that the graph of a function does not determine by itself...". I understand this concept, for the most part. However, I would like to have this discussed more fully in the next lecture, particularly with regard to why it's significant, and maybe to better understand it through some examples. -- Matt --=-fW9rLwW2Ge/MIKv1vKtw Content-Type: application/pgp-signature; name=signature.asc Content-Description: This is a digitally signed message part -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.2.5 (GNU/Linux) iD8DBQBDJ7SeiNjR4eJ+VcIRAhK6AKCnEyYtKSvz7pc2I8yeRkP8fM7bBACghWQB ipdl4FDqap4Nq+HjKW5ocrk= =WJ2y -----END PGP SIGNATURE----- --=-fW9rLwW2Ge/MIKv1vKtw-- From mwangi@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mwangi@MIT.EDU Wed Sep 14 01:57:22 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 j8E5vMaJ013555 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:57:22 -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 j8E5v1u8017795 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:57:01 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-FOUR-SEVENTY-FIVE.MIT.EDU [18.241.6.220]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E5uwsX022355 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 01:56:58 -0400 (EDT) Message-Id: <6.2.1.2.2.20050914014400.03abde20@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 14 Sep 2005 01:56:54 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: Week 2 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-UID: 149 Content-Length: 381 X-Keywords: NonJunk NotJunk Passage: 3.3 sequences Page: 10 I found this passage most difficult because of the idea of multiplying two sets. Even after reading the passage it remained unclear what it means to multiply two sets. The example did little to help since it multiplied the set of natural numbers by itself and it was not possible to tell which numbers came from which set in the answer. From alisonc@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From alisonc@MIT.EDU Wed Sep 14 02:04:43 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 j8E64haJ014145 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:04:43 -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 j8E64e97021517 for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:04:41 -0400 (EDT) Received: from [18.238.6.66] (EASTCAMPUS-EIGHT-THIRTY-THREE.MIT.EDU [18.238.6.66]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E611KV022729 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 14 Sep 2005 02:01:02 -0400 (EDT) Message-ID: <4327BCA2.3020009@mit.edu> Date: Wed, 14 Sep 2005 02:01:06 -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.csail.mit.edu Subject: Reading comments -- Course Notes, Week 2 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-UID: 150 Content-Length: 383 X-Keywords: NonJunk NotJunk I found the section on quantifiers (pages 6&7) most difficult -- attempting to logically dissect pieces of English as naturally spoken (as opposed to carefully phrased verbose math-english) is really tough, so the examples took quite a bit of thought to really get them to sink in. (They were entertaining though -- I'd like to try skiing over magma sometime.) Alison Cichowlas From mrivas03@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From mrivas03@MIT.EDU Wed Sep 14 02:36:35 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 j8E6aYaJ017947 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 02:36:34 -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 j8E6aX95007730; Wed, 14 Sep 2005 02:36:33 -0400 (EDT) Received: from el-ternero.mit.edu (MRIVAS03.MIT.EDU [18.237.0.105]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E6aPxT025094 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 14 Sep 2005 02:36:27 -0400 (EDT) Message-Id: <6.2.3.4.2.20050914013211.02d8ec78@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Wed, 14 Sep 2005 02:36:05 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: Reading 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-UID: 151 Content-Length: 476 X-Keywords: NonJunk NotJunk In the reading page number 3 - Reading Number 2 {Predicates and Sets} - Passage Proof by Cases the split of cases into subcases became surprisingly difficult to follow. Explanation of Theorem and Proof would be helpful. The difference of the two assertions one which is valid and the other not valid on page 8 of Reading number 2 was not quite clear. The second proof showing why the second was not valid was clear but the first proof was not. Thanks, Manuel Rivas From ryan786@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From ryan786@MIT.EDU Wed Sep 14 03:02:41 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 j8E72faJ022222 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:41 -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 j8E72e95020231 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:40 -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 j8E72bol026364 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:02:37 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8E72bIx004010; Wed, 14 Sep 2005 03:02:37 -0400 Received: from PSK-TWENTY-EIGHT.MIT.EDU (PSK-TWENTY-EIGHT.MIT.EDU [18.217.1.28]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:02:37 -0400 Message-ID: <20050914030237.iqf7kklo2n8kwwwk@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:02:37 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: required 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-UID: 152 Content-Length: 378 X-Keywords: NonJunk NotJunk I found the segment about functions (3.5 FUNCTIONS, p.11/14) most difficult. I had never seen surjective, injective, or bijective before, so I had to read it over a few times. I still don't fully grasp the definitions provided, although the visual representation helped. When doing the tutorial problems for this section, I had to constantly re-look at the pdf. -Ryan Young From scholtz@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From scholtz@MIT.EDU Wed Sep 14 03:18:17 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 j8E7IHaJ024739 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:17 -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 j8E7IF95027753 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:15 -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 j8E7ICb8027043 for <6042-probs@theory.csail.MIT.EDU>; Wed, 14 Sep 2005 03:18:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8E7ICC9021498; Wed, 14 Sep 2005 03:18:12 -0400 Received: from c-24-34-20-46.hsd1.ma.comcast.net (c-24-34-20-46.hsd1.ma.comcast.net [24.34.20.46]) (User authenticated as scholtz@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:18:12 -0400 Message-ID: <20050914031812.6b9vnnk4s68g4440@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:18:12 -0400 From: Edward A Scholtz To: 6042-probs@theory.csail.MIT.EDU Subject: Reading Comments - LN2 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: 1.801 X-Spam-Level: * (1.801) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 153 Content-Length: 258 X-Keywords: NonJunk NotJunk I haven't had much experience with sets and it would help if you spent some time going over the notation. In section 3.2 on page 9, I understood the subset symbol but I was confused by the subset symbol without the bar underneath it (the two are not equal?) From rshearer@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From rshearer@MIT.EDU Wed Sep 14 03:21:06 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 j8E7L6aJ024943 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:06 -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 j8E7L595029104 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:05 -0400 (EDT) Received: from [18.240.7.234] (MCCORMICK-SEVEN-FORTY-FIVE.MIT.EDU [18.240.7.234]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7KxaJ027171 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:21:03 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) Content-Transfer-Encoding: 7bit Message-Id: <15b0f8a81bd1e0c484a4ad7b01c2b04f@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: Comments Date: Wed, 14 Sep 2005 03:20:58 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 154 Content-Length: 592 X-Keywords: NonJunk NotJunk "The product operation is one link between sets and sequences. A product of sets, S1 * S2 * S3 * ... SN, is a new set consisting of all sequences where the first component is drawn from S1, the second from S2, and so forth" I found the above passage, found on page 10 of the notes, to be confusing and I would like it to be discussed in the lecture. I think that if I saw a full example of the process one goes through to get the product of two sets I would be able to understand it. The way it is worded in the notes doesn't give me a good idea of the steps I should go through. From crowell@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From crowell@MIT.EDU Wed Sep 14 03:26:15 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 j8E7QFaJ026457 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:15 -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 j8E7QE95001602 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:14 -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 j8E7Q715027399 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 03:26:07 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8E7Q7Bt021796; Wed, 14 Sep 2005 03:26:07 -0400 Received: from EASTCAMPUS-SEVEN-SIXTY.MIT.EDU (EASTCAMPUS-SEVEN-SIXTY.MIT.EDU [18.238.5.249]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 03:26:07 -0400 Message-ID: <20050914032607.j658zx21e64g8wos@webmail.mit.edu> Date: Wed, 14 Sep 2005 03:26:07 -0400 From: Robert Crowell To: 6042-probs@theory.lcs.MIT.EDU Subject: Re: Reading comments References: <20050913234208.h24nzpn7eqf4wg00@webmail.mit.edu> <20050914032442.lp7423xe2x6sc8o0@webmail.mit.edu> In-Reply-To: <20050914032442.lp7423xe2x6sc8o0@webmail.mit.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.041 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 155 Content-Length: 684 X-Keywords: NonJunk NotJunk It seems I misunderstood and sent comments for the first week reading. Here's my comment for the second week. I was most surprised by the closing remarks of the notes (page 13-14). It is amazing that so much depends on a mathematics which has no axiomatic foundation. -Rob Crowell Quoting Robert Crowell : > >> I was most confused by section 7.1.2 on page 12 on "Implies." I >> have seen the >> other logical operations before (OR, AND, etc) but was unfamiliar >> with IMPLIES. >> At first the fact that a false P (in the relation P=>Q) made a true >> implication was troubling as well, as it seemed counter-intuitive. >> >> -Rob Crowell >> >> >> >> > > > From kjhollen@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From kjhollen@MIT.EDU Wed Sep 14 03:30:41 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 j8E7UfaJ026857 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:41 -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 j8E7Ud95003738 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:40 -0400 (EDT) Received: from MACGREGOR-ONE-O-NINE.MIT.EDU (MACGREGOR-ONE-O-NINE.MIT.EDU [18.239.5.109]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E7UZ2m027611 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 03:30:38 -0400 (EDT) Subject: reading for September 14 From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Wed, 14 Sep 2005 03:30:35 -0400 Message-Id: <1126683035.24797.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 156 Content-Length: 462 X-Keywords: NonJunk NotJunk Hello, I found the passage about Sequences (on page 10 of ln2) somewhat confusing. I understand what a sequence is, but the multiplication of sets and sequences is something that is difficult to understand and that I hope will be discussed more in lecture. Also, the terms for functions in section 3.5 (on page 11) were explained well but they are brand new to me and I would really appreciate learning them in more depth in lecture. Kate Hollenbach kjhollen From petek@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From petek@MIT.EDU Wed Sep 14 05:02:41 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 j8E92eaJ006335 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:40 -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 j8E92dDp016496 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:39 -0400 (EDT) Received: from [18.194.1.144] (SN-ONE-FORTY-FOUR.MIT.EDU [18.194.1.144]) (authenticated bits=0) (User authenticated as petek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8E92bdw000865 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 05:02:38 -0400 (EDT) Message-ID: <4327E734.9050908@mit.edu> Date: Wed, 14 Sep 2005 05:02:44 -0400 From: Pete Kruskall User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Comments Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -1.886 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 157 Content-Length: 885 X-Keywords: NonJunk NotJunk Latter half of section 2.6, on page 8:

The fact that mathematical symbols are not yet second-nature for me means that shuffling them around makes it difficult for me to follow what's being done.  Perhaps I need to take a second look later on, but it's hard for me to see why the last statement is not true save for the example given.


 
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From kkdb@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: undecided-unix Mail-from: From kkdb@MIT.EDU Wed Sep 14 08:02:59 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 j8EC2waJ032227 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:58 -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 j8EC2vko022772 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:57 -0400 (EDT) Received: from KKDB (ASHDOWN-FIVE-NINETY-FOUR.MIT.EDU [18.250.7.83]) (authenticated bits=0) (User authenticated as kkdb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EC2tux013786 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:02:55 -0400 (EDT) Message-ID: <001601c5b924$3dfb5590$5307fa12@KKDB> From: "Kaustuv DeBiswas" To: <6042-probs@theory.lcs.mit.edu> Subject: *Required* Comments for Reading Date: Wed, 14 Sep 2005 08:02:55 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0013_01C5B902.B661C0E0" 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: -1.431 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 158 Content-Length: 2469 X-Keywords: NonJunk NotJunk Passage 1.4 - Proof by Cases, Pg 3 I found the theorem surprising and would like the proof to be discussed in the next lecture. It was surprising because in a collection of 6 people, there might be a situation where nobody has met each other. The cases in the example need clarification. -------------------------------------------------------------------------------------------------------------------------------- Kaustuv DeBiswas | MIT - SMArchS - Computation | kkdb@mit.edu | +1 (617) 230 6471  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_0013_01C5B902.B661C0E0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Passage 1.4 - Proof by Cases, Pg 3 I found the theorem surprising and would like the proof to be discussed = in the next lecture. It was surprising because in a collection of 6 = people, there might be a situation where nobody has met each other. The = cases in the example need clarification. -------------------------------------------------------------------------= ------------------------------------------------------- Kaustuv DeBiswas | MIT - SMArchS - Computation | kkdb@mit.edu | +1 = (617) 230 6471 ------=_NextPart_000_0013_01C5B902.B661C0E0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Passage 1.4 - Proof by Cases, Pg=20 3
I found the theorem surprising and = would like the=20 proof to be discussed in the next lecture. It was surprising because in = a=20 collection of 6 people, there might be a situation where nobody has met = each=20 other. The cases in the example need clarification.
 
----------------------------------------------------------------= ----------------------------------------------------------------
Kaust= uv=20 DeBiswas | MIT - SMArchS - Computation |  kkdb@mit.edu  | +1 (617) 230=20 6471
 
------=_NextPart_000_0013_01C5B902.B661C0E0-- From kktyan@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From kktyan@MIT.EDU Wed Sep 14 08:21:29 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 j8ECLTaJ001524 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:29 -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 j8ECLRko006429 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:28 -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 j8ECLQxK018162 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:21:26 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8ECLQAj019872; Wed, 14 Sep 2005 08:21:26 -0400 Received: from BURTON-TWO-FORTY-ONE.MIT.EDU (BURTON-TWO-FORTY-ONE.MIT.EDU [18.247.5.241]) (User authenticated as kktyan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:21:26 -0400 Message-ID: <20050914082126.by2bkffais1mw444@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:21:26 -0400 From: Karena Tyan To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.032 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 j8ECLTaJ001524 Status: RO X-UID: 159 Content-Length: 1028 X-Keywords: NonJunk NotJunk Here are my comments on the first two readings for 6.042J: Good & bad proofs: In the reading for good and bad proofs, I found Theorem 5.1 (DeMorgan's Law for Sets) and its subsequent proof on page 8 most difficult to understand, primarily because I am unfamiliar with set notation and meaning. Overall, the entire reading and its concepts were not too difficult to understand; after looking up the meaning of the set notation online, I was able to understand the proof and theorem much more easily. Proofs by Contradiction and Cases/Predicate Logic: The most surprising part of this reading was the product operation on page 10: "The product operation is one link between sets and sequences. A product of sets, S1S2?Sn, is a new set consisting of all sequences where the first component is drawn from S1, the second from S2, and so forth." I had not known that one could take the product of two sets, and that the result was a set of sequences. - Karena Tyan -- 410 Memorial Drive Cambridge, MA 02139 (585)957-5923 From cbossard@MIT.EDU Wed Sep 14 21:47:32 2005 X-Coding-System: iso-8859-1-unix Mail-from: From cbossard@MIT.EDU Wed Sep 14 08:27:38 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 j8ECRcaJ002520 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:38 -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 j8ECRbko010714 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:37 -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 j8ECRVMR019448 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:27:31 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8ECRVim004877; Wed, 14 Sep 2005 08:27:31 -0400 Received: from NEXT-SEVEN-THIRTEEN.MIT.EDU (NEXT-SEVEN-THIRTEEN.MIT.EDU [18.242.7.202]) (User authenticated as cbossard@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:27:31 -0400 Message-ID: <20050914082731.93o0c5h9waec8ccw@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:27:31 -0400 Disposition-Notification-To: cbossard@MIT.EDU X-Confirm-Reading-To: cbossard@mit.edu X-PMRQC: 1 From: cbossard@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: Week 2 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: 0.548 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 160 Content-Length: 336 X-Keywords: NonJunk $MDNSent NotJunk passage: Section 3.5 Functions pgs. 11 - 12 I would like to see this dicussed more fully in class because I am having trouble with the injection, surjection, etc. concerning functions. It would be easier if I say some actual examples that had words or numbers assigned to a and b to help me make better sense of them. Cynthia Bossard From mmt@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From mmt@MIT.EDU Wed Sep 14 08:34:50 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 j8ECYoaJ004011 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:50 -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 j8ECYnko015999 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:49 -0400 (EDT) Received: from all-night-tool.mit.edu (ALL-NIGHT-TOOL.MIT.EDU [18.7.16.70]) (authenticated bits=56) (User authenticated as mmt@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8ECYlZ3021230 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:34:47 -0400 (EDT) Received: (from mmt@localhost) by all-night-tool.mit.edu (8.12.9) id j8ECYlrv013542; Wed, 14 Sep 2005 08:34:47 -0400 (EDT) Date: Wed, 14 Sep 2005 08:34:47 -0400 From: Mark M Tobenkin To: 6042-probs@theory.lcs.MIT.EDU Subject: Required Comments Message-ID: <20050914123447.GB13098@all-night-tool.mit.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.5.6i X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 161 Content-Length: 512 X-Keywords: NonJunk NotJunk The most interesting aspect of the ln2.pdf reading for me was the Power Set/Sequence section of the reading. Particularly, there seems to be a relationship between the "power set" P(A), which implies AxA and the multiplication operator for sequences (NxN) which is implicit in the reading. Can we make more explicit what the actual members of a power set are, and the origins of the sequence/set multiplication is? -- As for a picture, if it isn't worth a thousand words, the hell with it. - Ad Reinhardt From ajshafer@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From ajshafer@MIT.EDU Wed Sep 14 08:45:43 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 j8ECjhaJ006659 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:43 -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 j8ECjfko025575 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:41 -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 j8ECjXk2024285 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 08:45:34 -0400 (EDT) Message-ID: <000701c5b92a$30caa8c0$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.lcs.mit.edu> Subject: Set Functions: ln2.pdf Date: Wed, 14 Sep 2005 08:45:29 -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 X-UID: 162 Content-Length: 393 X-Keywords: NonJunk NotJunk On page 11 you describe functions that map one set to another. I'm still confused by the four types: total, surjective, injective, and bijective. Can we get an example of each of these? -Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From lmccart@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: iso-8859-1-unix Mail-from: From lmccart@MIT.EDU Wed Sep 14 08:59:13 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 j8ECxDaJ008551 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:13 -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 j8ECxBko007616 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:11 -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 j8ECxAer028412 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 08:59:10 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8ECxAO3012543; Wed, 14 Sep 2005 08:59:10 -0400 Received: from AP-FIFTY-ONE.MIT.EDU (AP-FIFTY-ONE.MIT.EDU [18.153.1.51]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 08:59:10 -0400 Message-ID: <20050914085910.oy8307ow1sgsooww@webmail.mit.edu> Date: Wed, 14 Sep 2005 08:59:10 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments 9/14 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-UID: 163 Content-Length: 344 X-Keywords: NonJunk NotJunk All of the material in this reading seemed pretty straight forward, but the passage I found most confusing was section 3.5: Functions. I understood the definitions for total, surjective, injective, and bijective, but I still found the tutor problem covering this information slightly confusing. I would therefore like this reviewed in class. From aka_kame@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: emacs-mule-unix Mail-from: From aka_kame@MIT.EDU Wed Sep 14 09:40:15 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 j8EDeFaJ019548 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:15 -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 j8EDeEko019666 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:14 -0400 (EDT) Received: from [18.234.1.72] (DP-SEVENTY-TWO.MIT.EDU [18.234.1.72]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EDe7cs015660 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:08 -0400 (EDT) Message-ID: <43282837.1090503@mit.edu> Date: Wed, 14 Sep 2005 09:40:07 -0400 From: Akari Kameyama User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 2 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 164 Content-Length: 502 X-Keywords: NonJunk NotJunk A function f : A ! B is: total if every element of A is assigned to some element of B; otherwise, f is called a partial function. surjective if every element of B is mapped to at least once, that is, 8b 2 B9a 2 A. f(a) = b. injective if every element of B is mapped to at most once. bijective if f is total, surjective, and injective. In particular, each element of B is mapped to exactly once. This is confusing, it would be helpful if there were more specific examples given in lecture. From veracarr@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From veracarr@MIT.EDU Wed Sep 14 09:40:24 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 j8EDeOaJ019582 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:24 -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 j8EDeORT006899 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:24 -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 j8EDeL0v000765 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 09:40:22 -0400 (EDT) Message-ID: <432844A7.2080708@mit.edu> Date: Wed, 14 Sep 2005 09:41:27 -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: Comments #1 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 165 Content-Length: 403 X-Keywords: NonJunk NotJunk Functions (pg 11) I found this concept interesting in how they denote the relative size of the function's domain and codomain. However, the part I found difficult was when I had to apply these concepts, for example, in the tutor problems. I didn't understand the cases where surjection and injection ended up being equal to the other set. E.g. If f is a surjection/injection then |f(A)| ____ |B|. From mukkala@MIT.EDU Wed Sep 14 21:47:33 2005 X-Coding-System: undecided-unix Mail-from: From mukkala@MIT.EDU Wed Sep 14 10:13:55 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 j8EEDsaJ025718 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:55 -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 j8EEDrko028258 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:53 -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 j8EEDjdA000919 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:13:47 -0400 (EDT) Message-Id: <6.2.1.2.2.20050914100717.01eb2288@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Wed, 14 Sep 2005 10:13:46 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: Reading Comments 9/14 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-UID: 166 Content-Length: 394 X-Keywords: NonJunk NotJunk In Section 2.4, entitled Order of Quantifiers, it explains how the order of different kinds of quantifiers changes the meaning of a proposition. I understand the example with the set of Americans and the set of Dreams. However, I don't understand how swapping quantifiers in Goldbach's Conjecture creates a "patently false statement"; it seems true to me. Thanks, Praveen Pamidimukkala From yaser@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From yaser@MIT.EDU Wed Sep 14 10:16:11 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 j8EEGBaJ026487 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:11 -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 j8EEG9ko001225 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:10 -0400 (EDT) Received: from Charizard (NEW-THREE-TWENTY-FIVE.MIT.EDU [18.241.6.70]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8EEG4DU002194 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:16:06 -0400 (EDT) Message-Id: <200509141416.j8EEG4DU002194@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: .042 email on readings Date: Wed, 14 Sep 2005 10:16:02 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000C_01C5B915.511612A0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcW5NtTAkvN6oBFZQ7W8eW1sDHHcVg== X-Spam-Score: -0.229 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 167 Content-Length: 13771 X-Keywords: NonJunk $Forwarded NotJunk *Required* Comments for Reading: send email to 6042-probs@theory.lcs.mit.edu citing a passage in the reading - including its page number - and explaining, in at most three sentences, why you found it 1. most difficult, or 2. most surprising, or 3. would like to have discussed more fully it in the next lecture. Passage: "2.5 Negating Quantifiers" (page 7) I found this slightly difficult (#1 above) to follow near the expression of the logic equivalence (line marked "(1)") because the inversion, simple as it is, is not making intuitive sense to me. Could you please discuss this more fully in the next lecture? Thanks! (Please note, my athena id is actually "ykhan" in case you're using that for scorekeeping). Also- on the tutor, question TP 2.1.1 number 3: I am confident that the answer is R>P (getting an A implies you got an A on the final). However, I tried entering that and it didnd't accept. I then tried all sorts of combinations and permutations (and spent a long time doing it). In the end I got frustrated, wrote some random answer (in my case P>R as that was the last of the 20 some combos), and hit submit, only to find that the expected answer was "R->P". So could I get my point back as my original answer would have been correct had the expected answer not had the "-" in it? (In your notation note at the top, we were told NOT to put in the "-" in our "implies" answer). Thank you very much, and have wonderful time! _Yaser  -=- MIME -=-  This is a multi-part message in MIME format. ------=_NextPart_000_000C_01C5B915.511612A0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit *Required* Comments for Reading: send email to 6042-probs@theory.lcs.mit.edu citing a passage in the reading - including its page number - and explaining, in at most three sentences, why you found it 1. most difficult, or 2. most surprising, or 3. would like to have discussed more fully it in the next lecture. Passage: "2.5 Negating Quantifiers" (page 7) I found this slightly difficult (#1 above) to follow near the expression of the logic equivalence (line marked "(1)") because the inversion, simple as it is, is not making intuitive sense to me. Could you please discuss this more fully in the next lecture? Thanks! (Please note, my athena id is actually "ykhan" in case you're using that for scorekeeping). Also- on the tutor, question TP 2.1.1 number 3: I am confident that the answer is R>P (getting an A implies you got an A on the final). However, I tried entering that and it didnd't accept. I then tried all sorts of combinations and permutations (and spent a long time doing it). In the end I got frustrated, wrote some random answer (in my case P>R as that was the last of the 20 some combos), and hit submit, only to find that the expected answer was "R->P". So could I get my point back as my original answer would have been correct had the expected answer not had the "-" in it? (In your notation note at the top, we were told NOT to put in the "-" in our "implies" answer). Thank you very much, and have wonderful time! _Yaser ------=_NextPart_000_000C_01C5B915.511612A0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

*Required* Comments for Reading: send email to =

6042-probs@theory.lcs.mit.edu

citing a passage in the reading = – including its page number – and explaining, in at most three = sentences, why you found it

  1. most difficult, or
  2. most surprising, or
  3. would like to have discussed more fully it in the next = lecture.

 

Passage: = “2.5 Negating Quantifiers” (page 7)

 

I found this = slightly difficult (#1 above) to follow near the expression of the logic = equivalence (line marked “(1)”) because the inversion, simple as it is, = is not making intuitive sense to me. Could you please discuss this more fully = in the next lecture?

 

Thanks!

 

(Please note, = my athena id is actually = “ykhan” in case you’re using that for scorekeeping)…

 

Also- on the = tutor, question TP 2.1.1 number 3:

I am = confident that the answer is R>P (getting an A implies you got an A on the final). = However, I tried entering that and it didnd’t = accept. I then tried all sorts of combinations and permutations (and spent a long = time doing it). In the end I got frustrated, wrote some random answer (in my = case P>R as that was the last of the 20 some combos), and hit submit, only = to find that the expected answer was “R->P”. =

 

So could I = get my point back as my original answer would have been correct had the = expected answer not had the “-“ in it? (In = your notation note at the top, we were told NOT to put in the “-“ in our “implies” = answer).

 

Thank you = very much, and have wonderful time!

 

_Yaser =

------=_NextPart_000_000C_01C5B915.511612A0-- From jacques@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: iso-8859-1-unix Mail-from: From jacques@MIT.EDU Wed Sep 14 10:23:40 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 j8EENdaJ029455 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:39 -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 j8EENcko010437 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:38 -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 j8EENVII006014 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:23:31 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8EENVLR026872; Wed, 14 Sep 2005 10:23:31 -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 ; Wed, 14 Sep 2005 10:23:31 -0400 Message-ID: <20050914102331.q6018mus1eck4884@webmail.mit.edu> Date: Wed, 14 Sep 2005 10:23:31 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: Comments for Reading: Week 2 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-UID: 168 Content-Length: 461 X-Keywords: NonJunk NotJunk On page 11 of Course Notes, Week 2: "surjective if every element of B is mapped to at least once, that is, ... injective if every element of B is mapped to at most once." I would like to see this covered in lecture, or more examples given, because I found the descriptions a little confusing (easy to mix up). As it says in the text, the names are "hopelessly unmemorable and nondescriptive," so anything to help me get a handle on which is which. From cvnguyen@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From cvnguyen@MIT.EDU Wed Sep 14 10:33:08 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 j8EEX8aJ032766 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:08 -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 j8EEX7ko022106 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:07 -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 j8EEX0Qv011056 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:33:01 -0400 (EDT) Message-ID: <001301c5b939$3777a6f0$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: Reading Comments Week 2 Date: Wed, 14 Sep 2005 10:33:03 -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: -0.596 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 169 Content-Length: 364 X-Keywords: NonJunk NotJunk I found the discussion of Godel's incompleteness theorem on page 13 of the notes particularly worthy of discussion in class. From what I understand of it, the theorem demonstrates that it is impossible to know everything about a system from just the underlying rules of the system themselves. This has interesting philosophical consequences. --Chieu Nguyen From harelw@mit.edu Wed Sep 14 21:47:34 2005 X-Coding-System: undecided-unix Mail-from: From harelw@gmail.com Wed Sep 14 10:36:06 2005 Return-Path: Received: from zproxy.gmail.com (zproxy.gmail.com [64.233.162.192]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EEa6aJ000650 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 10:36:06 -0400 Received: by zproxy.gmail.com with SMTP id o1so133297nzf for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 07:36:01 -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:mime-version:content-type; b=KAgeNdTclPl3aeL/GJz2ZPbcBKKyKuAwgnvGYx2DTjZPnKlcP6VX9bv7Zvdej8d9jgJlJupikIafN+EppdT8PJ6y6bbUwiV4Lx+35dQ/CA0TiWzDmw2ORCjIGBsdZxO5Ve+sH5SCqW+SaNd8HvVymZEqk5ZbU+rHjC1MxnEuoCw= Received: by 10.36.33.19 with SMTP id g19mr2103558nzg; Wed, 14 Sep 2005 07:35:53 -0700 (PDT) Received: by 10.36.109.9 with HTTP; Wed, 14 Sep 2005 07:35:52 -0700 (PDT) Message-ID: <8c5248a80509140735499af1ba@mail.gmail.com> Date: Wed, 14 Sep 2005 10:35:53 -0400 From: "Harel M. Williams" Reply-To: "Harel M. Williams" Sender: harelw@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Reading for 9/14 Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_20643_29364537.1126708553001" Status: RO X-UID: 170 Content-Length: 863 X-Keywords: NonJunk NotJunk I found functions to be most difficult because there weren't many examples in the reading. Also, there are a lot of definitions to know. ~Harel Williams  -=- MIME -=-  ------=_Part_20643_29364537.1126708553001 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found functions to be most difficult because there weren't many examples= =20 in the reading. Also, there are a lot of definitions to know. ~Harel Williams ------=_Part_20643_29364537.1126708553001 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found functions to be most difficult because there weren't many examples in the reading. Also, there are a lot of definitions to know.

~Harel Williams
------=_Part_20643_29364537.1126708553001-- From hejing85@MIT.EDU Wed Sep 14 21:47:34 2005 X-Coding-System: gb2312-unix Mail-from: From hejing85@MIT.EDU Wed Sep 14 10:43:20 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 j8EEhKaJ006159 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:20 -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 j8EEhJko004390 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:19 -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 j8EEhGC5016020 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 10:43:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8EEhGFB024704; Wed, 14 Sep 2005 10:43:16 -0400 Received: from NEW-ONE-FIFTEEN.MIT.EDU (NEW-ONE-FIFTEEN.MIT.EDU [18.241.5.115]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 10:43:16 -0400 Message-ID: <20050914104316.t28wssw0592s8c84@webmail.mit.edu> Date: Wed, 14 Sep 2005 10:43:16 -0400 From: Jing He To: 6042-probs@theory.lcs.MIT.EDU Subject: 6042 reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=GB2312 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-UID: 171 Content-Length: 238 X-Keywords: NonJunk NotJunk I found that the passage on page 11 about the functions surjective, injective, and bijective the most difficult to understand. This could be because I had never heard of them before and it's hard to visualize how A maps to B. --Jing He From rehughes@MIT.EDU Wed Sep 14 21:47:35 2005 X-Coding-System: undecided-unix Mail-from: From rehughes@MIT.EDU Wed Sep 14 11:10:58 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 j8EFAwaJ011829 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:58 -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 j8EFAvRJ000461 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:57 -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 j8EFAv93028305 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:57 -0400 (EDT) Received: from [18.186.6.142] (TANG-ELEVEN-SIXTY-FIVE.MIT.EDU [18.186.6.142]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8EFAo0v006311 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:10:50 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v622) To: 6042-probs@theory.lcs.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-1-227077757 From: Richard Hughes Subject: Required Comments for Reading Date: Wed, 14 Sep 2005 11:10:49 -0400 X-Mailer: Apple Mail (2.622) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 172 Content-Length: 782 X-Keywords: NonJunk NotJunk Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ -=- MIME -=-  --Apple-Mail-1-227077757 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; format=flowed Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ --Apple-Mail-1-227077757 Content-Transfer-Encoding: 7bit Content-Type: text/enriched; charset=US-ASCII Page 11: Functions. This was easily the hardest simply because injunctive, surjective, and so on are such opaque terms. What are their root words? /no idea/ --Apple-Mail-1-227077757-- From fgreen@MIT.EDU Wed Sep 14 21:47:35 2005 X-Coding-System: iso-8859-1-unix Mail-from: From fgreen@MIT.EDU Wed Sep 14 14:17:56 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 j8EIHuaJ020108 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:56 -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 j8EIHt0p026771 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:55 -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 j8EIHmrv017992 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 14 Sep 2005 14:17:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8EIHmKQ015788; Wed, 14 Sep 2005 14:17:48 -0400 Received: from GREENONE.MIT.EDU (GREENONE.MIT.EDU [18.238.2.183]) (User authenticated as fgreen@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 14 Sep 2005 14:17:48 -0400 Message-ID: <20050914141748.0wct7vu18ug4c0ck@webmail.mit.edu> Date: Wed, 14 Sep 2005 14:17:48 -0400 From: Forrest O Green 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 X-UID: 173 Content-Length: 108 X-Keywords: NonJunk NotJunk I found page 11 section 3.5 to be somewhat confusing primarily on account of the terminology. -Forrest From manosdefierro@gmail.com Wed Sep 14 21:47:35 2005 X-Coding-System: undecided-unix Mail-from: From manosdefierro@gmail.com Wed Sep 14 14:46:01 2005 Return-Path: Received: from mintaka.lcs.mit.edu (mintaka.lcs.mit.edu [18.26.0.36]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8EIk0aJ027513 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 14:46:00 -0400 Received: from wproxy.gmail.com (wproxy.gmail.com [64.233.184.197]) by mintaka.lcs.mit.edu (8.12.9/8.12.9) with ESMTP id j8EIk0SF046965 for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 14:46:00 -0400 (EDT) (envelope-from manosdefierro@gmail.com) Received: by wproxy.gmail.com with SMTP id i5so12897wra for <6042-probs@theory.lcs.mit.edu>; Wed, 14 Sep 2005 11:44:55 -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=jE8tapmEaHjrVEtj3yqP/8agrYaZUOqii86iTAjCp+/JhMydUFutglKGjZi4irDxMb6iTiTp2xrivxCmMj3mLm/Q4lqB6yMlmxzTQr3SMjPICe41dafjSnCJkT+Ow4RBD5vTFS/DPE0XxIK39/uFxaC+udDxD0ZIyR/Rpie1TPw= Received: by 10.54.44.11 with SMTP id r11mr1133758wrr; Wed, 14 Sep 2005 03:09:01 -0700 (PDT) Received: by 10.54.151.13 with HTTP; Wed, 14 Sep 2005 03:09:01 -0700 (PDT) Message-ID: <2bac7c5d050914030950528d75@mail.gmail.com> Date: Wed, 14 Sep 2005 06:09:01 -0400 From: Mario Marrufo Reply-To: joeyrufo@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: comments for reading Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_17744_4553015.1126692541798" Status: RO X-UID: 174 Content-Length: 883 X-Keywords: NonJunk NotJunk I found the lemma on page 12 somewhat confusing as I am unfamiliar with the terms that were used ("surjective," etc.). (But basically, everything made sense.)  -=- MIME -=-  ------=_Part_17744_4553015.1126692541798 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found the lemma on page 12 somewhat confusing as I am unfamiliar with the= =20 terms that were used ("surjective," etc.). (But basically, everything made= =20 sense.) ------=_Part_17744_4553015.1126692541798 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline I found the lemma on page 12 somewhat confusing as I am unfamiliar with the terms that were used ("surjective," etc.). (But basically, everything made sense.)
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