From jeffhoff@MIT.EDU Mon Sep 26 21:22:38 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 j8R1McaJ012361 for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:38 -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 j8R1Mamx019819 for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:36 -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 j8R1MW6W009583 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Mon, 26 Sep 2005 21:22:33 -0400 (EDT) Message-Id: <5.2.1.1.2.20050926211914.05405898@hesiod> X-Sender: jeffhoff@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Mon, 26 Sep 2005 21:22:29 -0400 To: 6042-probs@theory.lcs.mit.edu From: "Jeffrey D. Hoff" Subject: [Hanson] Week 4 Comments Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.701 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 339 Content-Length: 192 X-IMAPbase: 1126748715 465 NonJunk $Label4 $Label1 $Label2 $Label3 $Label5 Junk $MDNSent $Forwarded NotJunk X-Keywords: NonJunk Section: 1.3 (Surjective and the like) Is Surjective and total the same? also, what is the difference between RB = A and BR = A ??? i had a problem with that on the online tutor questions From hectorb@MIT.EDU Mon Sep 26 21:52:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8R1qraJ017701 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 21:52:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8R1qomx006603; Mon, 26 Sep 2005 21:52:51 -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 j8R1qnWu015391; Mon, 26 Sep 2005 21:52:49 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8R1qnQR000481; Mon, 26 Sep 2005 21:52:49 -0400 Received: from BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU (BEXLEY-SIX-THIRTY-EIGHT.MIT.EDU [18.246.7.127]) (User authenticated as hectorb@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Mon, 26 Sep 2005 21:52:49 -0400 Message-ID: <20050926215249.yrdmzari4l4w4k4c@webmail.mit.edu> Date: Mon, 26 Sep 2005 21:52:49 -0400 From: Hector Beltran To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 4 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: 0.714 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 340 Content-Length: 477 X-Keywords: NonJunk "Definition 4.6. A topological sort of a partial order, , on a set, A, is a total ordering, @, on A such that a  b implies a @ b." pg. 13 (the mathematical symbols do not show) I might be wrong, but I think I remember from 6.034 that there's an easy, systematic way to find the topological sort of a DAG. Something about writing all the relations as a pair and them crossing them out or something like that. If I'm not mistaken can you please show this procedure. -Hector From lye@MIT.EDU Mon Sep 26 22:09:03 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 j8R293aJ018733 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:03 -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 j8R292mx015906 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:02 -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 j8R28xZq019216 for <6042-probs@theory.lcs.MIT.EDU>; Mon, 26 Sep 2005 22:09:00 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8R28xuQ002975; Mon, 26 Sep 2005 22:08:59 -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, 26 Sep 2005 22:08:59 -0400 Message-ID: <20050926220859.7fxq62sdjf480k8k@webmail.mit.edu> Date: Mon, 26 Sep 2005 22:08:59 -0400 From: Linda Ye To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Week 4 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: 341 Content-Length: 304 X-Keywords: NonJunk Page 3, Equivalence Relations, Def. 2.1 I don't understand what that symbol means in: "Given any total function, f, with domain A, define the binary relation f on A by the rule..." If the "=f" the relation itself? Is it a relation between A and the function f in the definition? Lunduo (Linda) Ye From ksindi@MIT.EDU Mon Sep 26 22:18:03 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 j8R2I3aJ019365 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:03 -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 j8R2I2mx021038 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:02 -0400 (EDT) Received: from KAMIL.mit.edu (SENIOR-FOUR-NINETY-THREE.MIT.EDU [18.244.6.238]) (authenticated bits=0) (User authenticated as ksindi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R2Hxot021157 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 22:18:00 -0400 (EDT) Message-Id: <6.2.3.4.2.20050926215317.039b3eb0@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Mon, 26 Sep 2005 22:17:57 -0400 To: 6042-probs@theory.csail.mit.edu From: Kamil Sindi Subject: [David] Week 4 Comments Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="=====================_724570327==.ALT" X-Spam-Score: 1.637 X-Spam-Level: * (1.637) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 342 Content-Length: 3485 X-Keywords: NonJunk --=====================_724570327==.ALT Content-Type: text/plain; charset="us-ascii"; format=flowed 6042- Reading Comments 4 Student: Kamil Sindi Tutor: David "3 Partial Orders Partial orders are another class of binary relations that are particularly important in Computer Science, with applications that include task scheduling, database concurrency control, and proving that computations terminate. A general example of a partial order is the subset relation, , on sets. In fact, we will define partial orders via the subset relation in much the same way we defined equivalence relations. Namely, for any element, a, we think of a function, g, such that g(a) is the set of properties that a has. Then we relate different elements according to how their properties compare. All partial order will arise in this way." I found the material in this reading, on he whole to be pretty straight forward. I'd like to see some applications and examples of partial orders and digraphs. Also if the fact that for total relations AR=B is not true is emphasized because I thought that it was both AR=B and RB=A (I learnt my lesson from the TPs). --=====================_724570327==.ALT Content-Type: text/html; charset="us-ascii" 6042- Reading Comments 4
Student: Kamil Sindi
Tutor: David

"3 Partial Orders
Partial orders are another class of binary relations that are particularly important in Computer Science,
with applications that include task scheduling, database concurrency control, and proving
that computations terminate.
A general example of a partial order is the subset relation, , on sets. In fact, we will define partial
orders via the subset relation in much the same way we defined equivalence relations. Namely,
for any element, a , we think of a function, g , such that g ( a ) is the set of properties that a has. Then
we relate different elements according to how their properties compare. All partial order will arise
in this way."

I found the material in this reading, on he whole to be pretty straight forward. I'd like to see some applications and examples of partial orders and digraphs. Also if the fact that for total relations AR=B is not true is emphasized because I thought that it was both AR=B and RB=A (I learnt my lesson from the TPs).
 --=====================_724570327==.ALT-- From a_lopez@MIT.EDU Mon Sep 26 23:09:02 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 j8R392aJ027248 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:09:02 -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 j8R391mx021463 for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:09:01 -0400 (EDT) Received: from [18.234.1.10] (DP-TEN.MIT.EDU [18.234.1.10]) (authenticated bits=0) (User authenticated as a_lopez@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8R38lu2002213 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Mon, 26 Sep 2005 23:08:53 -0400 (EDT) Message-ID: <4338B7C2.3080906@mit.edu> Date: Mon, 26 Sep 2005 23:08:50 -0400 From: Adriana Lopez User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [Jelani] Week 4 Readings Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.811 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 343 Content-Length: 240 X-Keywords: NonJunk The part of the readings that I found most confusing was Dilworth's Lemma and the Corollaries that followed it (pgs. 16-17). I still don't understand example 4.19: R is never defined. I'll have to wait for the in-class demo. Adriana From mdmurray@MIT.EDU Tue Sep 27 00:24:16 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 j8R4OFaJ007314 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:16 -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 j8R4OEF7028819 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:14 -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 j8R4OEOW023472 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:14 -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 j8R4OB0v018175 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 00:24:11 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <19A13BDF-BC24-467B-AD72-833AE3C2EBC5@mit.edu> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.csail.mit.edu From: Michael Murray Subject: [Hanson] E-mail comment on notes Date: Tue, 27 Sep 2005 00:23:55 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.671 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 344 Content-Length: 292 X-Keywords: Hi, Overall I felt that the notes were very dense and difficult to read in many areas. In particular, i thought the section on partial orders was difficult to understand. Definition 3.1 is unnecessarily complicated and hard to understand despite several reads. Thanks, Michael Murray From iyzhang@MIT.EDU Tue Sep 27 01:35:47 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 j8R5ZlaJ015354 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:47 -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 j8R5ZkZw007181 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:46 -0400 (EDT) Received: from [18.239.7.4] (MACGREGOR-FIVE-FIFTEEN.MIT.EDU [18.239.7.4]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8R5Zh0v020327 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 01:35:43 -0400 (EDT) Message-ID: <4338DA2B.5050003@mit.edu> Date: Tue, 27 Sep 2005 01:35:39 -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: [Sayan] Week 4 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: 345 Content-Length: 99 X-Keywords: I would be interested in hearing the graph theories about determining isomorphism. Thanks, Irene From hmzhou@blackbird.csail.mit.edu Tue Sep 27 12:00:09 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8RG09aJ021657; Tue, 27 Sep 2005 12:00:09 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8RG09rd019857; Tue, 27 Sep 2005 12:00:09 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8RG09VM019854; Tue, 27 Sep 2005 12:00:09 -0400 Date: Tue, 27 Sep 2005 12:00:09 -0400 (EDT) From: Hanson Zhou To: "Jeffrey D. Hoff" cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 4 Comments In-Reply-To: <5.2.1.1.2.20050926211914.05405898@hesiod> Message-ID: References: <5.2.1.1.2.20050926211914.05405898@hesiod> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 346 Content-Length: 586 X-Keywords: NonJunk Section 1.2 clarifies the notation that you're asking about. Surjective and total are NOT the same. For example, a relation that maps everything in A to one element in B would be total, but clearly not surjective. Again section 1.2 will help you to understand section 1.3. Go to office hours if this is still not clear. -Hanson On Mon, 26 Sep 2005, Jeffrey D. Hoff wrote: > > Section: 1.3 (Surjective and the like) > > Is Surjective and total the same? > also, what is the difference between > RB = A and BR = A ??? > > i had a problem with that on the online tutor questions > > From rnjacobs@MIT.EDU Tue Sep 27 14:20:27 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 j8RIKRaJ019820 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:27 -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 j8RIKPB2008091 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:26 -0400 (EDT) Received: from m2-225-9.mit.edu (M2-225-9.MIT.EDU [18.21.0.91]) (authenticated bits=56) (User authenticated as rnjacobs@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8RIKLDj023330 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:20:22 -0400 (EDT) Received: (from rnjacobs@localhost) by m2-225-9.mit.edu (8.12.9) id j8RIKNmn016435; Tue, 27 Sep 2005 14:20:23 -0400 Message-Id: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> To: 6042-probs@theory.csail.MIT.EDU Subject: [Jelani] reading response Date: Tue, 27 Sep 2005 14:20:23 -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: 347 Content-Length: 243 X-Keywords: NonJunk I found all the discussion on graphs familiar and entertaining. I think I've reviewed the paper enough by now that, in the process of trying to figure out something to comment on, I understand what I was fuzzy on at first. - Robert Jacobs From dangut@MIT.EDU Tue Sep 27 14:24:17 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 j8RIOHaJ020175 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:17 -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 j8RIOFB2011539 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:15 -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 j8RIODnG025422 for <6042-probs@theory.csail.MIT.EDU>; Tue, 27 Sep 2005 14:24:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8RIODkp004499; Tue, 27 Sep 2005 14:24:13 -0400 Received: from MACLAURIN-TWO-SEVENTY-EIGHT.MIT.EDU (MACLAURIN-TWO-SEVENTY-EIGHT.MIT.EDU [18.90.6.23]) (User authenticated as dangut@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 27 Sep 2005 14:24:13 -0400 Message-ID: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> Date: Tue, 27 Sep 2005 14:24:13 -0400 From: Daniel A Gutierrez To: 6042-probs@theory.csail.MIT.EDU Subject: Reading assignment #4 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: 0.666 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 348 Content-Length: 221 X-Keywords: I didn't really have much of a background in set relations, but I feel fairly confident in them. They're pretty natural feeling once I made the short step in my head and thinking of them like I do arrays in programming. From dshin@MIT.EDU Tue Sep 27 14:54:16 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 j8RIsGaJ028307 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 14:54:16 -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 j8RIsEBl003570; Tue, 27 Sep 2005 14:54:14 -0400 (EDT) Received: from [18.54.6.60] (NONE-THREE-FIFTEEN.MIT.EDU [18.54.6.60]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8RIrQ0v029657; Tue, 27 Sep 2005 14:53:26 -0400 (EDT) Message-ID: <43399522.70804@mit.edu> Date: Tue, 27 Sep 2005 14:53:22 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Daniel A Gutierrez CC: 6042-probs@theory.csail.mit.edu Subject: Re: Reading assignment #4 References: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> In-Reply-To: <20050927142413.aa9nkz9v9elckk0g@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 349 Content-Length: 441 X-Keywords: If this helps you to understand sets better, that's fine, but just be aware of one difference between sets and arrays: An array can have the same element twice. A set cannot. DS Daniel A Gutierrez wrote: >I didn't really have much of a background in set relations, but I feel fairly >confident in them. They're pretty natural feeling once I made the short step >in my head and thinking of them like I do arrays in programming. > > From kevin08@MIT.EDU Tue Sep 27 14:56:07 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 j8RIu6aJ028845 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:07 -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 j8RIu5B2012304 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:05 -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 j8RIu0sF013553 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 14:56:01 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8RIu0OS009664; Tue, 27 Sep 2005 14:56:00 -0400 Received: from SCI-12.MIT.EDU (SCI-12.MIT.EDU [18.51.0.197]) (User authenticated as kevin08@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Tue, 27 Sep 2005 14:56:00 -0400 Message-ID: <20050927145600.exez20leipkw8oks@webmail.mit.edu> Date: Tue, 27 Sep 2005 14:56:00 -0400 From: Kevin Y Wang To: 6042-probs@theory.lcs.MIT.EDU Subject: Friday 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: 350 Content-Length: 335 X-Keywords: NonJunk Dear Prof. Meyer and Rubinfeld, I found section 5.2 (Pg. 20-21) on this weeks reading to be slightly confusing, especially in terms of a bijection. If you could provide one or two more examples in the notes and explicitly state the bijection functions along with how you arrived at them, that would be very helpful. Thank you, Kevin From jehan@MIT.EDU Tue Sep 27 19:41:36 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 j8RNfaaJ030314 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:36 -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 j8RNfZxw008762 for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:35 -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 j8RNfRMc022808 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Tue, 27 Sep 2005 19:41:28 -0400 (EDT) Message-Id: <5.2.1.1.2.20050927193417.00befb90@po14.mit.edu> X-Sender: jehan@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Tue, 27 Sep 2005 19:39:26 -0400 To: 6042-probs@theory.csail.mit.edu From: Jehan deFonseka Subject: [Sayan] Week 4 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: 351 Content-Length: 108 X-Keywords: NonJunk I'm confused as to how to formally show a partial order, in terms of the process and the notation. Jehan From dowgun@MIT.EDU Tue Sep 27 23:45:58 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8S3jwaJ009057 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:58 -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 j8S3jv2l007973 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:57 -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 j8S3jsjX010016 for <6042-probs@theory.lcs.MIT.EDU>; Tue, 27 Sep 2005 23:45:55 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8S3js8m026304; Tue, 27 Sep 2005 23:45:54 -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, 27 Sep 2005 23:45:54 -0400 Message-ID: <20050927234554.dta9x43ao1kw8gg0@webmail.mit.edu> Date: Tue, 27 Sep 2005 23:45:54 -0400 From: Neil M Dowgun To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] ln4 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: -0.023 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 352 Content-Length: 576 X-Keywords: NonJunk After reading this week's notes, I would greatly appreciated it if we were to be given a chart of all the symbol that we use in this course. Now on top of 'less than' and 'subset of' we have 'less then in partial order' and 'less than in total order' (see page 13). And those are just the comparison ones! Anyway, I found Dilworth's Lemma the most interesting part of teh notes, because it is not an immediately obvious fact about partially ordered sets, and it could potentially used among laymen to make absurd sounding predictions that mathematically have to be true. Neil From shauni@MIT.EDU Wed Sep 28 04:08:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8S884aJ004172 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8S882uO008535 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:02 -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 j8S881to003949 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 04:08:01 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8S8814b013422; Wed, 28 Sep 2005 04:08:01 -0400 Received: from SIMMONS-ONE-SEVENTY-FIVE.MIT.EDU (SIMMONS-ONE-SEVENTY-FIVE.MIT.EDU [18.96.5.175]) (User authenticated as shauni@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 28 Sep 2005 04:08:01 -0400 Message-ID: <20050928040801.shdcyqt45pogcwow@webmail.mit.edu> Date: Wed, 28 Sep 2005 04:08:01 -0400 From: shauni@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] 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: 0.683 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 353 Content-Length: 361 X-Keywords: NonJunk (Digraphs p.11) I found it interesting that digraphs are formally the same as binary relations, since I seemed to better understand digraphs, while the concept of binary relations is still a bit confusing to me. I think the visual nature of the topic makes it easier to grasp. I think it would be helpful to use more visual examples for other concepts as well. From lmccart@MIT.EDU Wed Sep 28 08:57:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8SCvVaJ007717 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8SCvUaP018053 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:30 -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 j8SCvRj7000198 for <6042-probs@theory.lcs.MIT.EDU>; Wed, 28 Sep 2005 08:57:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8SCvR40029972; Wed, 28 Sep 2005 08:57:27 -0400 Received: from AP-EIGHTY-FIVE.MIT.EDU (AP-EIGHTY-FIVE.MIT.EDU [18.153.1.85]) (User authenticated as lmccart@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Wed, 28 Sep 2005 08:57:27 -0400 Message-ID: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> Date: Wed, 28 Sep 2005 08:57:27 -0400 From: Lauren McCarthy To: 6042-probs@theory.lcs.MIT.EDU Subject: [david] week 4 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: 0.618 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 354 Content-Length: 236 X-Keywords: NonJunk I was confused about the digraphs and the question on the tutor problems. The one that asked how many bijections there were of the star graph. I understand the things in the reading but I was unsure how to solve that problem. -lauren From minilek@theory.csail.mit.edu Wed Sep 28 09:14:03 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 j8SDE3w0009940 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:14:03 -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 j8SDE1aP029949; Wed, 28 Sep 2005 09:14:01 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SDDku2005432 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 28 Sep 2005 09:13:54 -0400 (EDT) Message-ID: <433A9724.1040201@theory.csail.mit.edu> Date: Wed, 28 Sep 2005 09:14:12 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Adriana Lopez Subject: Re: [Jelani] Week 4 Readings References: <4338B7C2.3080906@mit.edu> In-Reply-To: <4338B7C2.3080906@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 355 Content-Length: 544 X-Keywords: NonJunk Adriana, In example 4.19, aRb iff b is both older and taller than a (note that this relation is not total since if b is older but not taller than a, then neither aRb nor bRa hold). If you are still confused after today's lecture, please ask more questions. -Jelani Adriana Lopez wrote: > The part of the readings that I found most confusing was Dilworth's > Lemma and the Corollaries that followed it (pgs. 16-17). I still > don't understand example 4.19: R is never defined. I'll have to wait > for the in-class demo. > Adriana From minilek@theory.csail.mit.edu Wed Sep 28 09:15:56 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 j8SDFuw0010055 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:15:56 -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 j8SDFsaP001422; Wed, 28 Sep 2005 09:15:54 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8SDFhOl006155 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Wed, 28 Sep 2005 09:15:48 -0400 (EDT) Message-ID: <433A9798.9040207@theory.csail.mit.edu> Date: Wed, 28 Sep 2005 09:16:08 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: r n jacobs Subject: Re: [Jelani] reading response References: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> In-Reply-To: <200509271820.j8RIKNmn016435@m2-225-9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.413 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 356 Content-Length: 504 X-Keywords: NonJunk Were you able to solve the "Not So Quick Exercise" on page 17 of the notes? "Devise an efficient procedure for finding the longest increasing and the longest decreasing subsequence in any given sequence of integers." -Jelani r n jacobs wrote: >I found all the discussion on graphs familiar and entertaining. > >I think I've reviewed the paper enough by now that, in the process of >trying to figure out something to comment on, I understand what I was >fuzzy on at first. > > - Robert Jacobs > > From ajshafer@MIT.EDU Wed Sep 28 09:31:13 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 j8SDVDw0011175 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:13 -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 j8SDVBaP014982 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:12 -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 j8SDV5Jj012033 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 09:31:06 -0400 (EDT) Message-ID: <000201c5c430$e100fda0$6d04f712@ajshafer> From: "Andrew Shafer" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 4 Comments Date: Wed, 28 Sep 2005 09:31:05 -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: 357 Content-Length: 368 X-Keywords: NonJunk Page 16, Dilworth's Lemma. I'm not sure I understand what a chain or anti-chain are yet. Can you clarify those? A graph may be helpful to visualize it. Thanks, Andrew ---------------------------- Illegitmitatum Non Carborundum Est Andrew Shafer, MIT Blog http://shaferandrew.blogspot.com Si hoc legere scis numium eruditionis habes. ---------------------------- From cwong08@MIT.EDU Wed Sep 28 19:14:42 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 j8SNEgw0021188 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:42 -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 j8SNEfmi014956 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:41 -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 j8SNEeUM013791 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:40 -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 j8SNET0w018916 for <6042-probs@theory.csail.mit.edu>; Wed, 28 Sep 2005 19:14:37 -0400 (EDT) Message-ID: <433B23C5.4000306@mit.edu> Date: Wed, 28 Sep 2005 19:14:13 -0400 From: Chris Wong User-Agent: Mozilla Thunderbird 1.0 (Windows/20041206) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: Week 4 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: 358 Content-Length: 120 X-Keywords: NonJunk I found the explanation of isomorphism on pages 20-21 to be unclear. I'm still confused about what it actually means. From vixen@MIT.EDU Wed Sep 28 20:20:07 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 j8T0K7w0030769 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:07 -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 j8T0K6fc019460 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:06 -0400 (EDT) Received: from [18.243.2.26] (WALLFLOWER.MIT.EDU [18.243.2.26]) (authenticated bits=0) (User authenticated as vixen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T0K2um012391 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 20:20:03 -0400 (EDT) Mime-Version: 1.0 X-Sender: vixen@hesiod Message-Id: Date: Wed, 28 Sep 2005 20:19:05 -0400 To: 6042-probs@theory.lcs.mit.edu From: Amanda Seybold Subject: [Jelani] LN4 Content-Type: text/plain; charset="us-ascii" ; format="flowed" X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 359 Content-Length: 239 X-Keywords: NonJunk I was confused while reading the notes until pictures started showing up! Pictures--even little cirlces + arrows, with captions like, "The graph on the left is partial order, and the graph on the right is total order" would be useful. From benlu@MIT.EDU Wed Sep 28 23:14:16 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 j8T3EGw0002792 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:16 -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 j8T3EEk4002575 for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:14 -0400 (EDT) Received: from [18.244.3.46] (CAPSAICIN.MIT.EDU [18.244.3.46]) (authenticated bits=0) (User authenticated as benlu@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8T3EBHW013342 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Wed, 28 Sep 2005 23:14:11 -0400 (EDT) Message-ID: <433B5C07.7020905@mit.edu> Date: Wed, 28 Sep 2005 23:14:15 -0400 From: Benjamin Lu Organization: MIT User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [jelani][6.042] comments on reading 4 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.871 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 360 Content-Length: 277 X-Keywords: Gah! Strange symbols! What's up with that union looking thingy on the bottom of page 3? And what's the deal with the bent < signs? Are those supposed to be a new symbol used to denote partial order, binary relationships, or just a fun thing to throw around in Latex? ~Ben From hejing85@MIT.EDU Thu Sep 29 01:10:01 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 j8T5A1w0015428 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:10:01 -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 j8T5A0k4006894 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:10:00 -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 j8T59rsD001158 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 01:09:53 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8T59rT4007413; Thu, 29 Sep 2005 01:09:53 -0400 Received: from NEW-THREE-FORTY-NINE.MIT.EDU (NEW-THREE-FORTY-NINE.MIT.EDU [18.241.6.94]) (User authenticated as hejing85@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 01:09:53 -0400 Message-ID: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Date: Thu, 29 Sep 2005 01:09:53 -0400 From: Jing He To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] relations and graph theory 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: 361 Content-Length: 355 X-Status: A X-Keywords: NonJunk The most difficult part of this section for me to understand was what a total order was, since it wasn't given the formal definitions that the partial order came with in the reading. I also didn't understand the reasoning for the "proof" of Corollary 4.17 on page 16. The last section on graphs (and applications to computing) was the most interesting. From dnreshef@MIT.EDU Thu Sep 29 01:28:47 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T5Slw0016444 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8T5Skk4016361 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:46 -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 j8T5SguX003000 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 01:28:43 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8T5SgFB018830; Thu, 29 Sep 2005 01:28:42 -0400 Received: from FOURTHFRONT.MIT.EDU (FOURTHFRONT.MIT.EDU [18.206.0.66]) (User authenticated as dnreshef@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 01:28:42 -0400 Message-ID: <20050929012842.vwjqga46c3wo0ko0@webmail.mit.edu> Date: Thu, 29 Sep 2005 01:28:42 -0400 From: David N Reshef To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] READING ASSIGNMENT FOR 9/30 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: 362 Content-Length: 236 X-Keywords: NonJunk I thought that this week's reading was easier than the last few have been. I did however find the sections on relations and images (especially about properties of relations) on pages 1 and 2 the most confusing. Hope that helps. -Dave From icharny@mit.edu Thu Sep 29 03:28:14 2005 Return-Path: Received: from auathena (EASTCAMPUS-SIX-NINETY-EIGHT.MIT.EDU [18.238.5.187]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8T7SEw0028321 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 03:28:14 -0400 Received: (from icharny@localhost) by auathena (8.12.9) id j8T7SCC7021740; Thu, 29 Sep 2005 03:28:13 -0400 Subject: Week 4 Comments [Sayan] From: Isaac E Charny To: 6042-probs@theory.csail.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Thu, 29 Sep 2005 03:28:05 -0400 Message-Id: <1127978886.12799.1.camel@auathena> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 Status: RO X-UID: 363 Content-Length: 141 X-Keywords: NonJunk I'm still very hazy about the idea of partial order (section 3, page 7). Please add more real-life examples of partial orders. Isaac Charny From minilek@theory.csail.mit.edu Thu Sep 29 09:39:44 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 j8TDdiw0019001 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 09:39:44 -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 j8TDdgRw014988; Thu, 29 Sep 2005 09:39:42 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TDdSat023690 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 29 Sep 2005 09:39:32 -0400 (EDT) Message-ID: <433BEEA7.1020607@theory.csail.mit.edu> Date: Thu, 29 Sep 2005 09:39:51 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [jelani][6.042] comments on reading 4 References: <433B5C07.7020905@mit.edu> In-Reply-To: <433B5C07.7020905@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 364 Content-Length: 697 X-Keywords: NonJunk >> What's up with that union looking thingy on the bottom of page 3? It is a union symbol. It's saying A can be written as the union of elements of curvy-A (curvy-A is a set whose elements are sets). For the curvy-<, I'm not sure whether it's standard for all binary relations or just partial orders, or if it is even standard at all. I'll ask and get back to you. -Jelani Benjamin Lu wrote: > Gah! Strange symbols! > > What's up with that union looking thingy on the bottom of page 3? > > And what's the deal with the bent < signs? Are those supposed to be a > new symbol used to denote partial order, binary relationships, or just > a fun thing to throw around in Latex? > > ~Ben From minilek@theory.csail.mit.edu Thu Sep 29 09:48:21 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 j8TDmLw0019756 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 09:48:21 -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 j8TDmIRw023312; Thu, 29 Sep 2005 09:48:18 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TDmC5F027533 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Thu, 29 Sep 2005 09:48:15 -0400 (EDT) Message-ID: <433BF0B9.3010308@theory.csail.mit.edu> Date: Thu, 29 Sep 2005 09:48:41 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Benjamin Lu Subject: Re: [jelani][6.042] comments on reading 4 References: <433B5C07.7020905@mit.edu> In-Reply-To: <433B5C07.7020905@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 365 Content-Length: 538 X-Keywords: NonJunk It seems that the curvy-< (or \prec in LaTeX), is for partial orders. A previous year's version of the 6.042 lecture notes makes this more explicit: http://theory.lcs.mit.edu/classes/6.042/fall04/relations.pdf. -Jelani Benjamin Lu wrote: > Gah! Strange symbols! > > What's up with that union looking thingy on the bottom of page 3? > > And what's the deal with the bent < signs? Are those supposed to be a > new symbol used to denote partial order, binary relationships, or just > a fun thing to throw around in Latex? > > ~Ben From meyer@csail.mit.edu Thu Sep 29 12:02:31 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TG2Uw0018858 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 12:02:31 -0400 Received: (qmail 37751 invoked from network); 29 Sep 2005 16:02:30 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 16:02:29 -0000 Message-ID: <433C1019.6070903@csail.mit.edu> Date: Thu, 29 Sep 2005 12:02:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jing He CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] relations and graph theory reading comments References: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> In-Reply-To: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Content-Type: text/plain; charset=GB2312 Content-Transfer-Encoding: 7bit Status: RO X-UID: 366 Content-Length: 692 X-Keywords: 4.16 says there is a chain of size t, or an antichain of size n/t, for ANY t. So let t=sqrt(n) , and you get a chain of size sqrt(n) or an antichain of size n/sqrt(n), which also equals sqrt(n). So you 're sure to have a sqrt(n) size chain or antichain, which is what 4.17 says. Does that help? regards, A. Jing He wrote: >The most difficult part of this section for me to understand was what a total >order was, since it wasn't given the formal definitions that the partial order >came with in the reading. I also didn't understand the reasoning for the >"proof" of Corollary 4.17 on page 16. The last section on graphs (and >applications to computing) was the most interesting. > > From aeon@MIT.EDU Thu Sep 29 14:29:09 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 j8TIT9w0011586 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:09 -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 j8TIT7fP013780 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:07 -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 j8TIT5JO024338 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 14:29:05 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8TIT5N9001636; Thu, 29 Sep 2005 14:29:05 -0400 Received: from NEW-SIX-O-FOUR.MIT.EDU (NEW-SIX-O-FOUR.MIT.EDU [18.241.7.93]) (User authenticated as aeon@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 12:29:05 -0600 Message-ID: <20050929122905.o3evhbb71k0gwg00@webmail.mit.edu> Date: Thu, 29 Sep 2005 12:29:05 -0600 From: John Marrero To: 6042-probs@theory.lcs.MIT.EDU Subject: Reading Comments LN4 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: 0.704 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 367 Content-Length: 624 X-Keywords: NonJunk Page 17, "Undirected Graphs" This piqued my interest, because it relates to things such as data network topology or pathfinding, both important topics in computer science. It was surprising to find out that it could be expressed mathematically almost exactly as it is expressed in program code. Just a personal note. =) Page 20, "Isomorphism" The bit about isophomorphism existing iff there is a bijective function that relates G1 and G2 seems very important, but there wasn't really an example to illustrate that point.. I think it'd be great to use one (or several simple ones) in lecture, if possible. - John Marrero From yangc@MIT.EDU Thu Sep 29 14:55:51 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 j8TItpw0015337 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:51 -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 j8TItpBA028085 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:51 -0400 (EDT) Received: from CHRIS (MACGREGOR-FOUR-FIFTY-FOUR.MIT.EDU [18.239.6.199]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8TIth0v011087 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 14:55:43 -0400 (EDT) Message-Id: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> From: "Chris Yang" To: <6042-probs@theory.csail.mit.edu> Subject: [David] Week 4 Comments Date: Thu, 29 Sep 2005 14:55:31 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0036_01C5C505.D691B9B0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXFJ11GjcYVM07RT0GcIoKDDwuBqw== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1409 X-Spam-Score: 0.419 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 368 Content-Length: 2207 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0036_01C5C505.D691B9B0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit Is the problem of a general procedure for finding isomorphic graph related to P/N-P computability theory? It's pretty interesting how isomorphism is related to internet security - I look forward to hearing about it tomorrow. Chris Yang ------=_NextPart_000_0036_01C5C505.D691B9B0 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

Is the problem of a general procedure for finding = isomorphic graph related to P/N-P computability theory?  It’s pretty interesting how isomorphism is related to internet security – I = look forward to hearing about it tomorrow.

 

Chris Yang

------=_NextPart_000_0036_01C5C505.D691B9B0-- From cbossard@MIT.EDU Thu Sep 29 17:03:26 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 j8TL3Qw0007235 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:26 -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 j8TL3Pek023883 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:25 -0400 (EDT) Received: from m12-182-21.mit.edu (M12-182-21.MIT.EDU [18.19.0.52]) (authenticated bits=56) (User authenticated as cbossard@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TL3IqU029138 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 17:03:18 -0400 (EDT) Received: (from cbossard@localhost) by m12-182-21.mit.edu (8.12.9) id j8TL3HEO014430; Thu, 29 Sep 2005 17:03:17 -0400 (EDT) Subject: [David] Week 4 Comments From: Cynthia C Bossard To: 6042-probs@theory.lcs.MIT.EDU Content-Type: multipart/alternative; boundary="=-557AlRUN6Crp9U/9OLU5" Date: Thu, 29 Sep 2005 17:03:17 -0400 Message-Id: <1128027797.14379.3.camel@m12-182-21.mit.edu> Mime-Version: 1.0 X-Mailer: Evolution 2.0.4 X-Spam-Score: 1.077 X-Spam-Level: * (1.077) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 369 Content-Length: 1094 X-Keywords: NonJunk --=-557AlRUN6Crp9U/9OLU5 Content-Type: text/plain Content-Transfer-Encoding: 7bit Passage: mainly passage 2 Pages: 2-6 I believe that I have a general handle on most of the concepts, but I find that when trying to apply say partial order to something I do it improperly. My main mistakes are in testing for the equivalence relations. Can there be some extra real life type of problems given out on these? Cynthia --=-557AlRUN6Crp9U/9OLU5 Content-Type: text/html; charset=utf-8 Passage: mainly passage 2
Pages: 2-6
I believe that I have a general handle on most of the concepts, but I find that when trying to apply say partial order to something I do it improperly.  My main mistakes are in testing for the equivalence relations.  Can there be some extra real life type of problems given out on these?
Cynthia --=-557AlRUN6Crp9U/9OLU5-- From jstritar@MIT.EDU Thu Sep 29 17:13:01 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 j8TLD0w0008098 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:13:00 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TLCrek003260 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:12:54 -0400 (EDT) Received: from JSTRITAR.MIT.EDU (JSTRITAR.MIT.EDU [18.245.4.29]) (authenticated bits=0) (User authenticated as jstritar@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TLCk19002123 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:12:48 -0400 (EDT) Subject: [Sayan] Reading comments From: Jon Stritar To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Thu, 29 Sep 2005 17:12:47 -0400 Message-Id: <1128028367.7329.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 370 Content-Length: 198 X-Keywords: NonJunk 4.4 Parallel Task Scheduling I thought it was really cool when we formalized this concept. It seems like it could have some pretty interesting applications if it were written into a program. Jon From mukkala@MIT.EDU Thu Sep 29 17:46:41 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 j8TLkfw0012762 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:41 -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 j8TLkeek002675 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:40 -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 j8TLkWN5012048 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 17:46:33 -0400 (EDT) Message-Id: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Thu, 29 Sep 2005 17:46:34 -0400 To: 6042-probs@theory.lcs.mit.edu From: Praveen Pamidimukkala Subject: [Sayan] Week 4 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: 371 Content-Length: 505 X-Status: A X-Keywords: NonJunk In Section 3.2, the last 2 sentences read, "So < and < are total orders on [the set of all Real Numbers]. On the other hand, the subset relation is generally not total: any two finite sets of the same size will be incomparable under [the contained-in-or-equal subset relation]." Don't they mean just the [proper subset relation]? If the two finite sets contain the exact same elements, then aren't they comparable under [the contain-in-or-equal subset relation]? --Praveen Pamidimukkala Team 7 From avalys@MIT.EDU Thu Sep 29 18:10:50 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 j8TMAow0016866 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:50 -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 j8TMAmek019487 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:48 -0400 (EDT) Received: from [18.96.5.102] (SIMMONS-ONE-O-TWO.MIT.EDU [18.96.5.102]) (authenticated bits=0) (User authenticated as avalys@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMAYxG017937 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:42 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Alex Valys Subject: 6.042: [Hanson] Required reading comments Date: Thu, 29 Sep 2005 18:10:26 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: 0.087 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 372 Content-Length: 301 X-Keywords: NonJunk I was "surprised" to read in section 5.0 (page 18) about the varied applications of ugraphs: solving routing problems, scheduling, etc. I'm curious about how such applications work. Meaning, I understand digraphs and ugraphs in theory, but what are we supposed to do with them? Alex Valys From cvnguyen@MIT.EDU Thu Sep 29 18:10:54 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TMAsw0016879 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:54 -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 j8TMAqek019557 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:53 -0400 (EDT) Received: from MULTIVAC (EASTCAMPUS-ONE-O-FOUR-HUNDRED-TWENTY.MIT.EDU [18.238.7.153]) (authenticated bits=0) (User authenticated as cvnguyen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMAiX2017995 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:10:46 -0400 (EDT) Message-ID: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> From: "Chieu Nguyen" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] Week 4 comments Date: Thu, 29 Sep 2005 18:10:47 -0400 MIME-Version: 1.0 Content-Type: text/plain; format=flowed; charset="iso-8859-1"; reply-type=original Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 373 Content-Length: 40 X-Status: A X-Keywords: (Pages 11-21) Graphs are cool. --Chieu From aka_kame@MIT.EDU Thu Sep 29 18:12:20 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 j8TMCKw0016952 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:20 -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 j8TMCIek020758 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:19 -0400 (EDT) Received: from [18.234.1.130] (DP-ONE-THIRTY.MIT.EDU [18.234.1.130]) (authenticated bits=0) (User authenticated as aka_kame@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TMC9xY018330 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 18:12:10 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.csail.mit.edu Message-Id: Content-Type: multipart/alternative; boundary=Apple-Mail-6--599128064 From: Akari Kameyama Subject: Week 4 Comments Date: Thu, 29 Sep 2005 18:12:07 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -1.819 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 374 Content-Length: 8217 X-Keywords: NonJunk --Apple-Mail-6--599128064 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=UTF-8; delsp=yes; format=flowed In more formal terms, if G1 is a graph with vertices, V1 , and edges, =20= E1 , and likewise for G2 , then G1 is isomorphic to G2 iff there exists a bijective function f : V1 =20 =E2=86=92 V2 such that for every pair of vertices u, v =E2=88=88 V1 : u=E2=80=94v =E2=88=88 E1 iff f (u)=E2=80=94f = (v) =E2=88=88 E2 . I understand the overall gist of what this implies but I am finding =20 it difficult to interpret the terminology used above. So, as in =20 TP4.5 I can see how what is relevant is the number of edges to =20 vertices but in what specific way are they to be related?= --Apple-Mail-6--599128064 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=UTF-8
In more formal terms, if G1 is a graph with = vertices, V1 = , and = edges, E1 = , and likewise = for G2 = , = then=C2=A0
G1 is = isomorphic to G2 = iff there exists = a bijective function f : = V1=C2=A0=E2=86=92 V2 = such that for = every pair of=C2=A0
vertices u, v=C2=A0=E2=88=88 V1 :=C2=A0u=E2=80=94v=C2=A0=E2=88=88 E1 = iff = f = (u)=E2=80=94f (v) =E2=88=88 E2 = .=C2=A0

I understand = the overall gist of what this implies but I am finding it difficult to = interpret the terminology used above.=C2=A0 So, as in TP4.5 I can see = how what is relevant is the number of edges to vertices but in what = specific way are they to be related?
= --Apple-Mail-6--599128064-- From meyer@csail.mit.edu Thu Sep 29 18:14:33 2005 Return-Path: Received: from smtp106.mail.sc5.yahoo.com (smtp106.mail.sc5.yahoo.com [66.163.169.226]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TMEXw0018292 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:14:33 -0400 Received: (qmail 43163 invoked from network); 29 Sep 2005 22:14:32 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp106.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 22:14:31 -0000 Message-ID: <433C674B.7010806@csail.mit.edu> Date: Thu, 29 Sep 2005 18:14:35 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 4 Comments References: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 375 Content-Length: 788 X-Keywords: If they had the same elements, they'd be the same set, not two sets :-) But I grant you it would have been clearer to say "two distinct sets," and I'm editing the notes to say that right now. Thanks for the comment. Regards, A. Praveen Pamidimukkala wrote: > In Section 3.2, the last 2 sentences read, "So < and < are total > orders on [the set of all Real Numbers]. On the other hand, the subset > relation is generally not total: any two finite sets of the same size > will be incomparable under [the contained-in-or-equal subset > relation]." Don't they mean just the [proper subset relation]? If > the two finite sets contain the exact same elements, then aren't they > comparable under [the contain-in-or-equal subset relation]? > > --Praveen Pamidimukkala > Team 7 From meyer@csail.mit.edu Thu Sep 29 18:17:11 2005 Return-Path: Received: from smtp102.mail.sc5.yahoo.com (smtp102.mail.sc5.yahoo.com [216.136.174.140]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j8TMHBw0019156 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 18:17:11 -0400 Received: (qmail 57507 invoked from network); 29 Sep 2005 22:17:09 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp102.mail.sc5.yahoo.com with SMTP; 29 Sep 2005 22:17:09 -0000 Message-ID: <433C67E9.1050704@csail.mit.edu> Date: Thu, 29 Sep 2005 18:17:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] Week 4 comments References: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> In-Reply-To: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 376 Content-Length: 253 X-Keywords: Glad you think so, but we'd like more specific reactions to particular sections of the notes that caught your attention. I'll look forward to fuller comments next week. regards, A. Chieu Nguyen wrote: > (Pages 11-21) Graphs are cool. > > --Chieu From fluff@MIT.EDU Thu Sep 29 19:38:35 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 j8TNcZw0031657 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:35 -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 j8TNcXek017057 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:33 -0400 (EDT) Received: from [18.244.6.19] (SENIOR-TWO-SEVENTY-FOUR.MIT.EDU [18.244.6.19]) (authenticated bits=0) (User authenticated as fluff@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNcQEu004076 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:38:26 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Crystal Chao Subject: [Jelani] week 4 reading Date: Thu, 29 Sep 2005 19:38:00 -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: 377 Content-Length: 248 X-Keywords: NonJunk I pretty much sort of got everything except page 2, 1.2 and 1.3. You know, the notation "we'll get a lot of mileage out of" (why?). I didn't see it being used anywhere else. I have no idea what those sections are trying to tell me. ~Crystal From sil_03@MIT.EDU Thu Sep 29 19:49:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TNn4w0000760 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:49:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TNmpek023955 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:48:52 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNmjGD005640 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 19:48:45 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8TNmjNw030238; Thu, 29 Sep 2005 19:48:45 -0400 Received: from NEW-FOUR-O-NINE.MIT.EDU (NEW-FOUR-O-NINE.MIT.EDU [18.241.6.154]) (User authenticated as sil_03@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Thu, 29 Sep 2005 19:48:45 -0400 Message-ID: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> Date: Thu, 29 Sep 2005 19:48:45 -0400 From: "Silvia F. Baptista" To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] 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: 378 Content-Length: 261 X-Keywords: NonJunk Total orders: I dont understand the definition given on page 8 section 3.2. I also don't understand what comparable means. g(a) : I dont understand this function and the weird less than sign thing on page 7 section 3.1. How do you read it? Silvia Baptista From sriaz@MIT.EDU Thu Sep 29 19:56:46 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8TNukw0001998 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:56:46 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8TNtfPM027606 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:56:45 -0400 (EDT) Received: from [18.242.6.136] (NEXT-THREE-NINETY-ONE.MIT.EDU [18.242.6.136]) (authenticated bits=0) (User authenticated as sriaz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8TNsa59006512 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 19:54:36 -0400 (EDT) Message-ID: <433C7EBE.5090601@mit.edu> Date: Thu, 29 Sep 2005 19:54:38 -0400 From: Sameer Riaz User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Jelani] Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.638 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 379 Content-Length: 200 X-Keywords: NonJunk There was one thing that confused me about this week's reading. I would like more explanation about partial orders (Section 3, Page 7), as I did not fully understand the defintion Thanks Sameer From sheldons@MIT.EDU Thu Sep 29 20:07:50 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 j8U07ow0003304 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:50 -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 j8U07mNS004283 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:48 -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 j8U07eiu008469 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:07:41 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Sheldon Chan Subject: [David] Week 4 Comments Date: Thu, 29 Sep 2005 20:07:39 -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: 380 Content-Length: 232 X-Keywords: NonJunk I found the concepts of minimal and maximal the hardest to grasp, on page 14. I know we discussed it in class, but simply the idea of how specific elements within a set can be both minimal and maximal is interesting. Sheldon From hzhou@MIT.EDU Thu Sep 29 20:20:37 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 j8U0Kaw0004949 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:36 -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 j8U0KZNS012074 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:35 -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 j8U0KJLB009051 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:20:29 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> X-Sender: hzhou@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 20:11:17 -0400 To: 6042-probs@theory.lcs.mit.edu From: Steven Zhou Subject: comment Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 381 Content-Length: 267 X-Keywords: NonJunk I am still really confused with equivalence relation. Is the relation the function itself, or a description of how a set maps to the other? I was wondering if possible, in class, you show some more examples of how to write a equivalence relation. - Steve Zhou From tonyng@MIT.EDU Thu Sep 29 20:51:11 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 j8U0pBw0010840 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:11 -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 j8U0pANS003735 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:10 -0400 (EDT) Received: from TNG.mit.edu (BURTON-TWO-TWENTY-SIX.MIT.EDU [18.247.5.226]) (authenticated bits=0) (User authenticated as tonyng@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0p7JB013783 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:51:08 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> X-Sender: tonyng@hesiod X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 20:51:15 -0400 To: 6042-probs@theory.csail.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: 382 Content-Length: 353 X-Keywords: I am confused by Images and Reverse Images (Section 1.2 page 2). I don't understand why the answers are what they are on the tutorial problems and the tutorial problems refer me to section 1.4 of the lecture notes for the explanation but section 1.4 does not exist. I find the notation confusing and I don't get why a relation R is total iff RB=A. From bilodeau@MIT.EDU Thu Sep 29 20:54:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U0s4w0011120 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U0s3NS005608 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:03 -0400 (EDT) Received: from pbilodeau (MACGREGOR-THIRTY.MIT.EDU [18.239.5.30]) (authenticated bits=0) (User authenticated as bilodeau@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0s12M014170 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 20:54:01 -0400 (EDT) Message-Id: <200509300054.j8U0s12M014170@outgoing.mit.edu> From: "Peter Bilodeau" To: <6042-probs@theory.lcs.mit.edu> Subject: [Jelani] reading comments 9/30 Date: Thu, 29 Sep 2005 20:53:51 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C537.E5E488C0" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXFWWxqWceFAiHpQ7qkkgg+RRv+Kg== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.274 X-Spam-Level: * (1.274) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 383 Content-Length: 2129 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C537.E5E488C0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Peter Bilodeau The most difficult part of the reading was dilworth's dilemma and the ordered subset thing. While the demo with height and age in class was entertaining, the concept got lost somehow. ------=_NextPart_000_0000_01C5C537.E5E488C0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Peter Bilodeau

 

The most difficult part of the reading was = dilworth’s dilemma and the ordered subset thing.  While the demo with height and age = in class was entertaining, the concept got lost somehow.

------=_NextPart_000_0000_01C5C537.E5E488C0-- From brevzin@MIT.EDU Thu Sep 29 20:58:21 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 j8U0wLw0011396 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:21 -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 j8U0wKNS008484 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:20 -0400 (EDT) Received: from BARRY.mit.edu ([18.218.1.202]) (authenticated bits=0) (User authenticated as brevzin@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U0w5ua014767 (version=TLSv1/SSLv3 cipher=DES-CBC3-SHA bits=168 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 20:58:11 -0400 (EDT) Message-Id: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> X-Sender: brevzin@hesiod X-Mailer: QUALCOMM Windows Eudora Version 6.1.2.0 Date: Thu, 29 Sep 2005 20:58:07 -0400 To: 6042-probs@theory.csail.mit.edu From: Barry Revzin Subject: [Hanson] 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: 384 Content-Length: 860 X-Keywords: So my question is actually from the in-class 3d: Describe a set with a unique minimal element and no mimumum. I don't think this set exists. So we have a set with n elemetns x1... xn, for every xi there exists a xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it would be minimal, so we can say that this set is arranged in some order, so that xi !> xi+1 for all i. But then, by transivity, it is easy to see that x1 < xi for all i, but then x1 would be a minimum! But there can't be one ? The only way this would work is if you have a group of elements related circularly with a disjoint element. i.e. Some element x, with three other elements a,b,c such that a < b and b < c and c < a. Thus, x is minimal but there is no minimum. But this is ridiculous because partial orders are transitive. So, Does 3d have a solution? Barry From jjmonzon@MIT.EDU Thu Sep 29 21:10:28 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 j8U1ASw0012907 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:28 -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 j8U1ANNS016568 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:24 -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 j8U1AGVi016527 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:10:17 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> X-Sender: jjmonzon@po9.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 21:10:30 -0400 To: 6042-probs@theory.csail.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: 385 Content-Length: 407 X-Keywords: I found the CR RC AR BR notation for relations quite confusing especially in the tutorial problems. I wish you could go over that part in class. When I answered the question, the tutorial also said to refer to part 1.4 of the notes which doesn't exist. Sincerely, Joshua Jen C. Monzon Massachusetts Institute of Technology Electrical Engineering with Computer Science jjmonzon@mit.edu 617-803-7497 From dshin@MIT.EDU Thu Sep 29 21:27:34 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 j8U1RYw0015049 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:27:34 -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 j8U1RWuE010765; Thu, 29 Sep 2005 21:27:33 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1RU3a004901; Thu, 29 Sep 2005 21:27:30 -0400 (EDT) Message-ID: <433C947C.80104@mit.edu> Date: Thu, 29 Sep 2005 21:27:24 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Joshua Jen C. Monzon" CC: 6042-probs@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929210803.0214fd68@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 386 Content-Length: 673 X-Keywords: Sorry, Joshua, the tutorial should have read part 1.3, not 1.4. Please take a look at that section, as well as section 1.2, where the notation is defined. Let me know if you still are confused after that. DS Joshua Jen C. Monzon wrote: > I found the CR RC AR BR notation for relations quite confusing > especially in the tutorial problems. I wish you could go over that > part in class. When I answered the question, the tutorial also said to > refer to part 1.4 of the notes which doesn't exist. > > Sincerely, > > > Joshua Jen C. Monzon > Massachusetts Institute of Technology > Electrical Engineering with Computer Science > jjmonzon@mit.edu 617-803-7497 > From dshin@MIT.EDU Thu Sep 29 21:32:25 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 j8U1WPw0015281 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 21:32:25 -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 j8U1WOuE010872; Thu, 29 Sep 2005 21:32:24 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1WK3a005084; Thu, 29 Sep 2005 21:32:21 -0400 (EDT) Message-ID: <433C959F.2010304@mit.edu> Date: Thu, 29 Sep 2005 21:32:15 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Tony Ng CC: 6042-probs@theory.csail.mit.edu Subject: Re: reading comments References: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929204720.02105a00@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 387 Content-Length: 768 X-Keywords: Sorry, the TP should read section 1.3, not section 1.4 (it has been modified accordingly). The set RB denotes the set of all a in A such that aRb for some b in B (definition given in section 1.2). A relation is total if domain element is related to some codomain element. In other words, R is total if for every a in A, aRb for some b in B. Do you see, then, why RB=A? DS Tony Ng wrote: > I am confused by Images and Reverse Images (Section 1.2 page 2). I > don't understand why the answers are what they are on the tutorial > problems and the tutorial problems refer me to section 1.4 of the > lecture notes for the explanation but section 1.4 does not exist. I > find the notation confusing and I don't get why a relation R is total > iff RB=A. > From ridell@MIT.EDU Thu Sep 29 21:52:21 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 j8U1qLw0016864 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:21 -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 j8U1qJNS016707 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:19 -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 j8U1qCYq023797 for <6042-probs@theory.csail.MIT.EDU>; Thu, 29 Sep 2005 21:52:12 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U1qCSx017231; Thu, 29 Sep 2005 21:52:12 -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 ; Thu, 29 Sep 2005 21:52:12 -0400 Message-ID: <20050929215212.8u1na8vqmsw8c4k8@webmail.mit.edu> Date: Thu, 29 Sep 2005 21:52:12 -0400 From: Rebecca Idell To: 6042-probs@theory.csail.MIT.EDU Subject: week 4 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: 388 Content-Length: 329 X-Keywords: NonJunk I am having trouble understanding partial orders. In class, it would help if we could do more problems similar to example 3.4 on page 8. -Rebecca Idell -- Rebecca Idell Massachusetts Institute of Technology Electrical Engineering and Computer Science, Class of 2007 479 Commonwealth Avenue Boston, MA 02215 (617) 875-0889 From dshin@MIT.EDU Thu Sep 29 21:56:13 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 j8U1uDw0017035 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 21:56:13 -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 j8U1uCe4008218 for <6042-probs@theory.lcs.MIT.EDU>; Thu, 29 Sep 2005 21:56:13 -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 j8U1uCB9029980; Thu, 29 Sep 2005 21:56:12 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U1uA3a005949; Thu, 29 Sep 2005 21:56:10 -0400 (EDT) Message-ID: <433C9B34.4020006@mit.edu> Date: Thu, 29 Sep 2005 21:56:04 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: "Silvia F. Baptista" CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] comments References: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> In-Reply-To: <20050929194845.vfu5rlutx0kk0ck8@webmail.mit.edu> Content-Type: multipart/alternative; boundary="------------040202050508030303070709" X-Spam-Score: -1.658 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 389 Content-Length: 4197 X-Keywords: This is a multi-part message in MIME format. --------------040202050508030303070709 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sorry, the definition on page 8 section 3.2 had a typo. We have corrected it. It now reads: "...given any two numbers, one will be bigger *than* the other..." (before, it read "that" instead of "than") Basically, an order is total if whenever you take two distinct elements x and y in the set, one of the following will be true: "x is bigger than y" or "y is bigger than x" This may seem obvious if the example you have in mind is something like the natural numbers or the real numbers. However, think about the divisors-relation (call it "div-rel") we used in the class problems, where x is less than y iff x divides y. In this setting, neither "4 is-less-than-with-respect-to-div-rel 5" nor "5 is-less-than-with-respect-to-div-rel 4" are true. So this order is not total. Since you can't tell me which of the two (4 or 5) are bigger, they're not comparable. Otherwise, they would be comparable. And, wouldn't it be convenient to have a simple symbol we could use instead of this cumbersome "is-less-than-with-respect-to-div-rel"? Well, that's why we have the "weird less than sign thing"! Just convenient notation, when it's clear from the contex what the order relation is. The g(a) function is a little harder to explain over email. I'd suggest trying to read over section 3.1 carefully - if you're still stuck, stop by office hours. Hope that helps. DS Silvia F. Baptista wrote: >Total orders: I dont understand the definition given on page 8 section 3.2. I >also don't understand what comparable means. > >g(a) : I dont understand this function and the weird less than sign thing on >page 7 section 3.1. How do you read it? > >Silvia Baptista > > --------------040202050508030303070709 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Sorry, the definition on page 8 section 3.2 had a typo.  We have corrected it.  It now reads:

"...given any two numbers, one will be bigger than the other..."

(before, it read "that" instead of "than")

Basically, an order is total if whenever you take two distinct elements x and y in the set, one of the following will be true:

"x is bigger than y"

or

"y is bigger than x"

This may seem obvious if the example you have in mind is something like the natural numbers or the real numbers.  However, think about the divisors-relation (call it "div-rel") we used in the class problems, where x is less than y iff x divides y.  In this setting, neither "4 is-less-than-with-respect-to-div-rel 5" nor "5 is-less-than-with-respect-to-div-rel 4" are true.  So this order is not total.

Since you can't tell me which of the two (4 or 5) are bigger, they're not comparable.  Otherwise, they would be comparable.

And, wouldn't it be convenient to have a simple symbol we could use instead of this cumbersome "is-less-than-with-respect-to-div-rel"?  Well, that's why we have the "weird less than sign thing"!  Just convenient notation, when it's clear from the contex what the order relation is.

The g(a) function is a little harder to explain over email.  I'd suggest trying to read over section 3.1 carefully - if you're still stuck, stop by office hours.

Hope that helps.

DS

Silvia F. Baptista wrote: 
Total orders: I dont understand the definition given on page 8 section 3.2.  I
also don't understand what comparable means.

g(a) : I dont understand this function and the weird less than sign thing on
page 7 section 3.1.   How do you read it?

Silvia Baptista

--------------040202050508030303070709-- From dshin@MIT.EDU Thu Sep 29 22:18:02 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 j8U2I1w0018334 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 22:18:02 -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 j8U2I0e4001604 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 22:18:00 -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 j8U2I0B9000478; Thu, 29 Sep 2005 22:18:00 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U2Hu0v002992; Thu, 29 Sep 2005 22:17:56 -0400 (EDT) Message-ID: <433CA04E.7020802@mit.edu> Date: Thu, 29 Sep 2005 22:17:50 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Steven Zhou CC: 6042-probs@theory.lcs.mit.edu Subject: Re: comment References: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> In-Reply-To: <5.2.1.1.2.20050929200541.02a7a2e0@po9.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.779 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 390 Content-Length: 785 X-Keywords: A (binary) relation R on a set S is a subset of SxS. We write "aRb" if (a,b) is in R. For example, if R is "less than" on the set N (natural numbers), R contains (0,1), (0,2), (0,3), ..., (1,2), (1,3), ... (a whole bunch of more pairs). We write "0R6" to mean "(0,6) is in R". In this particular example, we typically use the symbol '<' instead of 'R'. An equivalence relation is a type of relation that obeys reflexivity, symmetry, and transitivity. Hope that helps DS Steven Zhou wrote: > I am still really confused with equivalence relation. Is the relation > the function itself, or a description of how a set maps to the other? > I was wondering if possible, in class, you show some more examples of > how to write a equivalence relation. > > - Steve Zhou > From sergiob@MIT.EDU Thu Sep 29 23:28:04 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3S4w0032671 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:28:04 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3S2pT024036 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:28:02 -0400 (EDT) Received: from mit-kf7uwcnbdr4.mit.edu (SENIOR-THREE-FIFTY-EIGHT.MIT.EDU [18.244.6.103]) (authenticated bits=0) (User authenticated as sergiob@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U3Ro18009582 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:27:55 -0400 (EDT) Message-Id: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> X-Sender: sergiob@po14.mit.edu X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Thu, 29 Sep 2005 21:27:46 -0600 To: 6042-probs@theory.csail.mit.edu From: Sergio Bacallado Subject: [David] Week 4 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: 391 Content-Length: 292 X-Keywords: NonJunk I think the most interesting part of the reading was the section on proofs using Dilworth's lemma. You can prove very useful and non-trivial theorems using this lemma. I would like to know more about devising methods for search of fast paths of scheduling, etc. Sergio Bacallado Group 1 From aston@MIT.EDU Thu Sep 29 23:31:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U3VPw0000522 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U3VNpT026399 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:23 -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 j8U3VGQX010136 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:31:17 -0400 (EDT) Message-Id: <200509300331.j8U3VGQX010136@outgoing.mit.edu> From: "Aston Motes" To: <6042-probs@theory.csail.mit.edu> Subject: Week 4 Comments Date: Thu, 29 Sep 2005 23:33:15 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_006D_01C5C54E.2AB23450" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFb7E3iFSJ01CzRnixYzBsGk8qUg== X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 392 Content-Length: 2345 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_006D_01C5C54E.2AB23450 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I was extremely excited by the mention of the possible ramifications of proving/disproving the isomorphism between graphs (short of the brute force approach). I'm looking forward to hearing about it in the lecture tomorrow. - Aston ------=_NextPart_000_006D_01C5C54E.2AB23450 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I was extremely excited by the mention of the = possible ramifications of proving/disproving the isomorphism between graphs = (short of the brute force approach). I’m looking forward to hearing about it = in the lecture tomorrow.

         =   

         =    - Aston

------=_NextPart_000_006D_01C5C54E.2AB23450-- From dshin@MIT.EDU Thu Sep 29 23:40:48 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 j8U3emw0001592 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:40:48 -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 j8U3ekhP013864; Thu, 29 Sep 2005 23:40:46 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3ed3a009264; Thu, 29 Sep 2005 23:40:39 -0400 (EDT) Message-ID: <433CB3AF.5060405@mit.edu> Date: Thu, 29 Sep 2005 23:40:31 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Yang CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 4 Comments References: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> In-Reply-To: <200509291855.j8TIth0v011087@outgoing-legacy.mit.edu> 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: 393 Content-Length: 1028 X-Keywords: An excellent question! Graph isomorphism represents a huge gap in our current understanding of complexity theory. We don't know whether it is easy to solve (i.e., we don't know whether graph-isomorphism is in P), and we also don't have strong evidence that it is hard to solve (i.e., we don't know whether graph-isomorphism is NP-complete). There aren't many other computational problems like this - for all other problems, we usually either have proven that it's easy, or have provided strong evidence that it is hard. Sorry for being a bit vague, but we don't quite have the machinery built up to talk precisely about notions of "easiness" or "hardness" or "evidence" - I can point you to further resources if you find this interesting. DS Chris Yang wrote: > Is the problem of a general procedure for finding isomorphic graph > related to P/N-P computability theory? It’s pretty interesting how > isomorphism is related to internet security – I look forward to > hearing about it tomorrow. > > Chris Yang > From dshin@MIT.EDU Thu Sep 29 23:44:22 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 j8U3iLw0001697 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 23:44:21 -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 j8U3iKhP013932; Thu, 29 Sep 2005 23:44:20 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3iE0v005945; Thu, 29 Sep 2005 23:44:15 -0400 (EDT) Message-ID: <433CB486.3090302@mit.edu> Date: Thu, 29 Sep 2005 23:44:06 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Cynthia C Bossard CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 Comments References: <1128027797.14379.3.camel@m12-182-21.mit.edu> In-Reply-To: <1128027797.14379.3.camel@m12-182-21.mit.edu> Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.606 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 394 Content-Length: 516 X-Keywords: The next pset has a few problems on partial orders/equivalence relations. Hopefully, this will give you the practice you need. DS Cynthia C Bossard wrote: > Passage: mainly passage 2 > Pages: 2-6 > I believe that I have a general handle on most of the concepts, but I > find that when trying to apply say partial order to something I do it > improperly. My main mistakes are in testing for the equivalence > relations. Can there be some extra real life type of problems given > out on these? > Cynthia From dshin@MIT.EDU Thu Sep 29 23:55:05 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 j8U3t4w0002277 for <6042-probs@theory.csail.mit.edu>; Thu, 29 Sep 2005 23:55:04 -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 j8U3t2hP014218; Thu, 29 Sep 2005 23:55:03 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3ss0v006342; Thu, 29 Sep 2005 23:54:55 -0400 (EDT) Message-ID: <433CB706.7080909@mit.edu> Date: Thu, 29 Sep 2005 23:54:46 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Sergio Bacallado CC: 6042-probs@theory.csail.mit.edu Subject: Re: [David] Week 4 Comments References: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> In-Reply-To: <5.2.1.1.2.20050929212427.00c40e38@po14.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 395 Content-Length: 1066 X-Keywords: Scheduling is a fascinating problem. There are all sorts of variants on the scheduling problem (single processor/multiple processor, set-up times, penalties associated with uncompleted tasks, etc.). Some of these variants are easy to solve, but we have very strong evidence that others are hard to solve. Here's a sample of some of the variants for which we have strong evidence of their hardness: http://www.nada.kth.se/~viggo/wwwcompendium/node173.html (The concept of "easy", "hard", and "evidence" might seem subjective to you, but they are actually mathematically rigorous terms - to give their exact definitions is beyond the scope of this class). DS Sergio Bacallado wrote: > I think the most interesting part of the reading was the section on > proofs using Dilworth's lemma. You can prove very useful and > non-trivial theorems using this lemma. I would like to know more about > devising methods for search of fast paths of scheduling, etc. > > Sergio Bacallado > Group 1 > From dshin@MIT.EDU Thu Sep 29 23:56:38 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 j8U3ucw0003207 for <6042-probs@theory.lcs.mit.edu>; Thu, 29 Sep 2005 23:56:38 -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 j8U3uahP014246; Thu, 29 Sep 2005 23:56:36 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U3uS0v006411; Thu, 29 Sep 2005 23:56:28 -0400 (EDT) Message-ID: <433CB764.3070902@mit.edu> Date: Thu, 29 Sep 2005 23:56:20 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Lauren McCarthy CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [david] week 4 reading References: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> In-Reply-To: <20050928085727.svxcu9nbxwpw48w0@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 396 Content-Length: 351 X-Keywords: Did you submit an answer to check the rationale given in the TP solution? DS Lauren McCarthy wrote: >I was confused about the digraphs and the question on the tutor problems. The >one that asked how many bijections there were of the star graph. I understand >the things in the reading but I was unsure how to solve that problem. >-lauren > > From bens@MIT.EDU Fri Sep 30 00:07:35 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 j8U47Zw0006637 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:35 -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 j8U47YpT021239 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:34 -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 j8U47Rr4015592 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:07:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U47RkF018421; Fri, 30 Sep 2005 00:07:27 -0400 Received: from LSLATER.MIT.EDU (LSLATER.MIT.EDU [18.221.0.171]) (User authenticated as bens@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 00:07:27 -0400 Message-ID: <20050930000727.7g9rx2spwprksswo@webmail.mit.edu> Date: Fri, 30 Sep 2005 00:07:27 -0400 From: Benjamin M Schwartz To: 6042-probs@theory.lcs.MIT.EDU Subject: Relations II 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: 397 Content-Length: 135 X-Keywords: NonJunk P. 11: Directed graphs are equivalent to relations? That's so awesome! That totally explains how to store graphs in a data structure. From dshin@MIT.EDU Fri Sep 30 00:08:58 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 j8U48ww0006867 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 00:08:58 -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 j8U48uhP014538; Fri, 30 Sep 2005 00:08:56 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U48l0v006840; Fri, 30 Sep 2005 00:08:48 -0400 (EDT) Message-ID: <433CBA47.5010209@mit.edu> Date: Fri, 30 Sep 2005 00:08:39 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chris Wong CC: 6042-probs@theory.csail.mit.edu Subject: Re: Week 4 Comments References: <433B23C5.4000306@mit.edu> In-Reply-To: <433B23C5.4000306@mit.edu> Content-Type: multipart/alternative; boundary="------------090402000509070000020202" X-Spam-Score: -1.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 398 Content-Length: 3184 X-Keywords: This is a multi-part message in MIME format. --------------090402000509070000020202 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Formally, a graph G is a set of vertices V, and a set of edges E, which is a subset of VxV. G=(V,E) and H'=(V',E') are isomorphic if there exists a bijection f: V->V' such that (u,v) is in E iff (f(u),f(v)) is in E'. Informally... Think about the graph your team drew for the last in-class problem on Wednesday (involving 6.042 TA's trying to defeat Microsoft). Your team probably drew the graph differently from other teams around you - maybe your "path" went from left to right, while theirs went top to bottom. Maybe they had a pair of edges cross each other, while you arranged your nodes so that no edges needed to cross each other. Even despite all these differences, what would you say if a TA came by and said, "Nope, your graph is wrong - look at that team's graph - their graph is the correct one." You'd say, "No, but they're actually the same graph! They encode the same information! *We just drew them differently*!" Isomorphism tries to capture this notion, that the two graphs actually are equivalent in some sense. Hope that helps, DS Chris Wong wrote: > I found the explanation of isomorphism on pages 20-21 to be unclear. > I'm still confused about what it actually means. > --------------090402000509070000020202 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Formally, a graph G is a set of vertices V, and a set of edges E, which is a subset of VxV.  G=(V,E) and H'=(V',E') are isomorphic if there exists a bijection f: V->V' such that (u,v) is in E iff (f(u),f(v)) is in E'. 

Informally...

Think about the graph your team drew for the last in-class problem on Wednesday (involving 6.042 TA's trying to defeat Microsoft).  Your team probably drew the graph differently from other teams around you - maybe your "path" went from left to right, while theirs went top to bottom.  Maybe they had a pair of edges cross each other, while you arranged your nodes so that no edges needed to cross each other. 

Even despite all these differences, what would you say if a TA came by and said, "Nope, your graph is wrong - look at that team's graph - their graph is the correct one."  You'd say, "No, but they're actually the same graph!  They encode the same information!  We just drew them differently!"

Isomorphism tries to capture this notion, that the two graphs actually are equivalent in some sense. 

Hope that helps,

DS

Chris Wong wrote:
I found the explanation of isomorphism on pages 20-21 to be unclear. I'm still confused about what it actually means.


--------------090402000509070000020202-- From dshin@MIT.EDU Fri Sep 30 00:15:23 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 j8U4FNw0007264 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 00:15:23 -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 j8U4FLhP014667; Fri, 30 Sep 2005 00:15:21 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U4F70v007071; Fri, 30 Sep 2005 00:15:08 -0400 (EDT) Message-ID: <433CBBC3.4010707@mit.edu> Date: Fri, 30 Sep 2005 00:14:59 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Barry Revzin CC: 6042-probs@theory.csail.mit.edu Subject: Re: [Hanson] Comments References: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> In-Reply-To: <6.1.2.0.1.20050929205341.019c6650@po14.mit.edu> Content-Type: multipart/alternative; boundary="------------060500010706000409040001" X-Spam-Score: -1.872 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 399 Content-Length: 2935 X-Keywords: This is a multi-part message in MIME format. --------------060500010706000409040001 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit You have successfully proven that any *finite* set with a unique minimal element must have a minimum. Now, can you describe a set with a unique minimal element and no minimum? DS Barry Revzin wrote: > So my question is actually from the in-class 3d: Describe a set with a > unique minimal element and no mimumum. I don't think this set exists. > > So we have a set with n elemetns x1... xn, for every xi there exists a > xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it > would be minimal, so we can say that this set is arranged in some > order, so that xi !> xi+1 for all i. But then, by transivity, it is > easy to see that x1 < xi for all i, but then x1 would be a minimum! > But there can't be one ? > > The only way this would work is if you have a group of elements > related circularly with a disjoint element. i.e. Some element x, with > three other elements a,b,c such that a < b and b < c and c < a. Thus, > x is minimal but there is no minimum. But this is ridiculous because > partial orders are transitive. > > So, > > Does 3d have a solution? > > Barry > --------------060500010706000409040001 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit You have successfully proven that any finite set with a unique minimal element must have a minimum.

Now, can you describe a set with a unique minimal element and no minimum?

DS

Barry Revzin wrote:
So my question is actually from the in-class 3d: Describe a set with a unique minimal element and no mimumum. I don't think this set exists.

So we have a set with n elemetns x1... xn, for every xi there exists a xj such that xj < xi EXCEPT for x1. Now if any xi is unrelated, it would be minimal, so we can say that this set is arranged in some order, so that xi !> xi+1 for all i. But then, by transivity, it is easy to see that x1 < xi for all i, but then x1 would be a minimum! But there can't be one ?

The only way this would work is if you have a group of elements related circularly with a disjoint element. i.e. Some element x, with three other elements a,b,c such that a < b and b < c and c < a. Thus, x is minimal but there is no minimum. But this is ridiculous because partial orders are transitive.

So,

Does 3d have a solution?

Barry


--------------060500010706000409040001-- From ozcan@MIT.EDU Fri Sep 30 00:19:11 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 j8U4JAw0007464 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:10 -0400 Received: from central-city-carrier-station.mit.edu (CENTRAL-CITY-CARRIER-STATION.MIT.EDU [18.7.7.72]) by south-station-annex.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4J99x025940 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:10 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4J9MZ003306 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:09 -0400 (EDT) Received: from ebrum.mit.edu (OZCAN.MIT.EDU [18.241.3.106]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U4J13b010587 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 00:19:02 -0400 (EDT) Message-Id: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> X-Sender: ozcan@hesiod (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.1 Date: Fri, 30 Sep 2005 00:18:59 -0400 To: 6042-probs@theory.lcs.mit.edu From: Yasin Ozcan Subject: [hanson]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: 400 Content-Length: 254 X-Status: A X-Keywords: NonJunk Hi, The only comments I have about this week's readings is that it was too long. Also, I am a little bit confused by the notation in the True/False questions in the Tutor problems, i.e. Question 4.4. Can we go over it in the lecture thanks yasin From xiaoranz@MIT.EDU Fri Sep 30 00:34:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U4Ygw0008988 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U4YepT009373 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:41 -0400 (EDT) Received: from w92-130-webmail-6.mit.edu (W92-130-WEBMAIL-6.MIT.EDU [18.7.22.137]) ) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U4YY1w018862 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 00:34:34 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U4YYtg030629; Fri, 30 Sep 2005 00:34:34 -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 ; Fri, 30 Sep 2005 00:34:34 -0400 Message-ID: <20050930003434.peberlpwka0osskk@webmail.mit.edu> Date: Fri, 30 Sep 2005 00:34:34 -0400 From: "Xiaoran (Sharon) Zhang" To: 6042-probs@theory.lcs.MIT.EDU Subject: [Hanson] Week 4 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: 401 Content-Length: 593 X-Keywords: NonJunk I was confused about isomorphism for undirected graphs. Page 17, section 5. I am still unsure which characteristics to evaluate when determining whether two graphs are isomorphic or not. The correspondence between the vertex of the graphs are not clear to me. Sharon ************************************************** 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 lkini@MIT.EDU Fri Sep 30 01:05:30 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 j8U55Uw0010930 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:30 -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 j8U55TpT028874 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:29 -0400 (EDT) Received: from [18.239.5.240] (MACGREGOR-TWO-FORTY.MIT.EDU [18.239.5.240]) (authenticated bits=0) (User authenticated as lkini@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U55O64022100 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:05:26 -0400 (EDT) Message-ID: <433CC791.7090107@mit.edu> Date: Fri, 30 Sep 2005 01:05:21 -0400 From: Lohith Kini User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: [Hanson] Week 4 Reading Comments Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.164 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 402 Content-Length: 218 X-Keywords: NonJunk Hi, I was totally confused about parallel task scheduling and theorem 4.11. I didn't understand the corollaries either (because of this). I would like it if you could go over that a little bit more. Thanks, Lohith From clintonb@MIT.EDU Fri Sep 30 01:09:42 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U59gw0012656 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U59epT001589 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:41 -0400 (EDT) Received: from ccbibmx30 (NEW-EIGHTY-TWO.MIT.EDU [18.241.5.82]) (authenticated bits=0) (User authenticated as clintonb@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U59WYL022496 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:09:33 -0400 (EDT) Message-Id: <200509300509.j8U59WYL022496@outgoing.mit.edu> From: "Clinton Blackburn" To: <6042-probs@theory.lcs.mit.edu> Subject: Jelani: Reading #4 Date: Fri, 30 Sep 2005 01:08:26 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_000C_01C5C55B.771A3830" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFfP0id5mRPCReRxCm/RaUpus+WA== X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 403 Content-Length: 3146 X-Status: A X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_000C_01C5C55B.771A3830 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I'm just totally confused by this subject. I don't know. --- Clinton Blackburn DLP - Have you seen it? ------=_NextPart_000_000C_01C5C55B.771A3830 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I’m just totally confused by this subject. I = don’t know…

 

---

Clinton Blackburn

DLP – Have you seen = it? 

 

 

------=_NextPart_000_000C_01C5C55B.771A3830-- From zev@MIT.EDU Fri Sep 30 01:42:26 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 j8U5gQw0015480 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:26 -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 j8U5gLhP016462 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:21 -0400 (EDT) Received: from [18.243.2.60] (PRINTER-ON-FIRE.MIT.EDU [18.243.2.60]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U5gE0v010315 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 01:42:14 -0400 (EDT) Message-ID: <433CD035.4010005@mit.edu> Date: Fri, 30 Sep 2005 01:42:13 -0400 From: Zev Benjamin User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050726) X-Accept-Language: en-us, en MIME-Version: 1.0 To: 6042-probs@theory.csail.mit.edu Subject: [jelani] week 4 comments Content-Type: text/plain; charset=UTF-8; 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: 404 Content-Length: 370 X-Keywords: NonJunk The only sections I found confusing were 1.2 and 1.3 on page 2. The rest of the notes were really cool. It was a lot of information, but the connections drawn between them was interesting. The fact that set theory can be examined from so many angles is fascinating. In addition, I enjoyed the practical examples (here's how we can do parallel processing). Zev From ereid@MIT.EDU Fri Sep 30 01:55:49 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8U5tnw0016425 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:49 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U5tmuA029863 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:48 -0400 (EDT) Received: from [18.242.6.66] (NEXT-THREE-TWENTY-ONE.MIT.EDU [18.242.6.66]) (authenticated bits=0) (User authenticated as ereid@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U5tJPa026454 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:19 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) Content-Transfer-Encoding: 7bit Message-Id: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Elizabeth Reid Subject: [Hanson] Reading Comments Date: Fri, 30 Sep 2005 01:55:15 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.078 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 405 Content-Length: 195 X-Keywords: NonJunk If they could talk a bit about parallel task scheduling in lecture that would be nice. I was wondering if there was any mathematical way to maximize it, or if it's more of a guess and check? From mpapi@MIT.EDU Fri Sep 30 01:55:58 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 j8U5tww0016442 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:58 -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 j8U5tvgG027672 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:57 -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 j8U5tvGR005298 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:57 -0400 (EDT) Received: from MACGREGOR-THREE-FORTY-SIX.MIT.EDU (MACGREGOR-THREE-FORTY-SIX.MIT.EDU [18.239.6.91]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U5tl3a014044 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 01:55:47 -0400 (EDT) Subject: [Jelani] Week 4 Comments From: Matt Papi To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 01:55:45 -0400 Message-Id: <1128059745.12411.40.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.4.0 Content-Transfer-Encoding: 7bit X-Spam-Score: 1.041 X-Spam-Level: * (1.041) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 406 Content-Length: 390 X-Keywords: NonJunk The section on images, following "The set C R is called the image of C under R"; the notes mention that it's notation "that we'll get a lot of mileage out of". It might be helpful to see some examples or have the topic appear somehow in an in-class problem. While the concept itself isn't that hard to understand, I could imagine situations where it could be used in tricky ways. -- Matt From crowell@MIT.EDU Fri Sep 30 02:26:43 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 j8U6Qgw0020313 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8U6QeuA017073 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:40 -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 j8U6QcIw028433 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:26:38 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U6QckL029923; Fri, 30 Sep 2005 02:26:38 -0400 Received: from EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU (EASTCAMPUS-EIGHT-SIXTY-THREE.MIT.EDU [18.238.6.96]) (User authenticated as crowell@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 02:26:38 -0400 Message-ID: <20050930022638.fxnit9klgug0kk8k@webmail.mit.edu> Date: Fri, 30 Sep 2005 02:26:38 -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: 407 Content-Length: 180 X-Keywords: NonJunk I'm a little unclear on the section involving the AR=B notation found on page 2 in section 1.2 It's a bit confusing as to what is meant by having AR = something. -Rob Crowell From mracich@MIT.EDU Fri Sep 30 02:48:47 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 j8U6mlw0022002 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:47 -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 j8U6mkMf017924 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:46 -0400 (EDT) Received: from MACGREGOR-THREE-SIXTY-TWO.MIT.EDU (MACGREGOR-THREE-SIXTY-TWO.MIT.EDU [18.239.6.107]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8U6mh3a015432 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:48:44 -0400 (EDT) Subject: [Sayan] Comments for Course Notes, Week 4 (Binary Relations) From: Moira Racich To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 02:48:40 -0400 Message-Id: <1128062920.30191.6.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.0.2 Content-Transfer-Encoding: 7bit X-Spam-Score: 0.999 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 408 Content-Length: 181 X-Keywords: NonJunk I found section 3.5, Products and Restrictions of Relations, (located on page 10) kind of confusing. I would find it very helpful if this was reviewed in class. Moira Racich From ryan786@MIT.EDU Fri Sep 30 02:49:01 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 j8U6n1w0022046 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:49:01 -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 j8U6mxuA000082 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:48:59 -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 j8U6mwYu029662 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 02:48:58 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U6mwiR013101; Fri, 30 Sep 2005 02:48:58 -0400 Received: from PSK-SIX.MIT.EDU (PSK-SIX.MIT.EDU [18.217.1.6]) (User authenticated as ryan786@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 02:48:58 -0400 Message-ID: <20050930024858.plht40d3x3lcssg0@webmail.mit.edu> Date: Fri, 30 Sep 2005 02:48:58 -0400 From: Ryan Young Reply-to: ryoung@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [sayan] 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: -0.071 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 409 Content-Length: 209 X-Keywords: NonJunk This weeks reading was alright, since I have had some experience with graphs of this nature. Partial orders will probably give me the most trouble, but overall I think I understand the material. -Ryan Young From kjhollen@MIT.EDU Fri Sep 30 02:57:16 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 j8U6vGw0023955 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:57:16 -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 j8U6vFuA004912 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:57:15 -0400 (EDT) Received: from MACGREGOR-FOUR-EIGHTY.MIT.EDU (MACGREGOR-FOUR-EIGHTY.MIT.EDU [18.239.6.225]) (authenticated bits=0) (User authenticated as kjhollen@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U6utP6000037 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 02:56:59 -0400 (EDT) Subject: [Hanson] confused about isomorphism From: Kate Hollenbach To: 6042-probs@theory.lcs.mit.edu Content-Type: text/plain Date: Fri, 30 Sep 2005 02:56:55 -0400 Message-Id: <1128063415.13619.2.camel@localhost.localdomain> Mime-Version: 1.0 X-Mailer: Evolution 2.2.1.1 Content-Transfer-Encoding: 7bit X-Spam-Score: 3.227 X-Spam-Level: *** (3.227) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 410 Content-Length: 273 X-Keywords: NonJunk I think that I've almost got this concept (graphs and isomorphism) but the explanations in the reading (p. 20-22) didn't really give me a good feel for how to solve problems with isomorphism. It might be nice to have a few more examples of how to use/apply this property. From lana@MIT.EDU Fri Sep 30 03:04:36 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 j8U74aw0024273 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:36 -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 j8U74YuA008996 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:34 -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 j8U74RN8000360 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:04:27 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8U74RrL013710; Fri, 30 Sep 2005 03:04:27 -0400 Received: from NEXT-FOUR-TWENTY-THREE.MIT.EDU (NEXT-FOUR-TWENTY-THREE.MIT.EDU [18.242.6.168]) (User authenticated as lana@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:04:27 -0400 Message-ID: <20050930030427.q2wiidvufjls00o8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:04:27 -0400 From: Svetlana Goldenberg To: 6042-probs@theory.lcs.MIT.EDU Subject: TP4 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: 411 Content-Length: 258 X-Keywords: NonJunk The part of the reading that intrigued me the most was parallel task scheduling, particularly the question of existence of an efficient way of doing it. So I would like to know if there is a purely computational (i.e. non-intuitive) way of schceduling. Lana From adnaan@MIT.EDU Fri Sep 30 03:26:20 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 j8U7QKw0029048 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:20 -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 j8U7QJuA021127 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:19 -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 j8U7QFJA001313 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:26:16 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U7QFfA009788; Fri, 30 Sep 2005 03:26:15 -0400 Received: from NEW-FOUR-SIXTY-SEVEN.MIT.EDU (NEW-FOUR-SIXTY-SEVEN.MIT.EDU [18.241.6.212]) (User authenticated as adnaan@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:26:15 -0400 Message-ID: <20050930032615.lhnye9xspwoo4ck8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:26:15 -0400 From: Adnaan N Jiwaji To: 6042-probs@theory.lcs.MIT.EDU Subject: relation rwading - hanson 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: -0.043 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 412 Content-Length: 73 X-Keywords: NonJunk I found the distintion between partial and total order confusing Adnaan From juang@MIT.EDU Fri Sep 30 03:28:34 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 j8U7SYw0029158 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:34 -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 j8U7SXuA022312 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:33 -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 j8U7SLLg001364 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 03:28:26 -0400 (EDT) Message-ID: <433CE8F5.9030605@mit.edu> Date: Fri, 30 Sep 2005 03:27:49 -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.csail.mit.edu Subject: [David] Week 4 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: 413 Content-Length: 238 X-Keywords: NonJunk "So with an unlimited number of processors, the time to complete all the tasks is the size of the largest chain." Is there a general way to determine the fastest way of scheduling tasks if number of processors is finite? Jason Juang. From medrano@MIT.EDU Fri Sep 30 03:35:05 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 j8U7Z5w0029675 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:35:05 -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 j8U7Z4uA025933 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:35:04 -0400 (EDT) Received: from scrubbing-bubbles.mit.edu (SCRUBBING-BUBBLES.MIT.EDU [18.7.16.68]) (authenticated bits=56) (User authenticated as medrano@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U7YvDS001626 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:34:57 -0400 (EDT) Received: (from medrano@localhost) by scrubbing-bubbles.mit.edu (8.12.9) id j8U7Yvr8016362; Fri, 30 Sep 2005 03:34:57 -0400 (EDT) Date: Fri, 30 Sep 2005 03:34:56 -0400 (EDT) From: Jesus I Medrano To: 6042-probs@theory.lcs.MIT.EDU Subject: [Sayan] Isomorphism Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 414 Content-Length: 353 X-Keywords: NonJunk The examples in the text about isomorphism are simplistic. Is there something more complicated we can work with. Some isomorphism that will not be as obvious just so we can understand the concept a bit better. Changing the name of the vertices for A,B,C,D to 1,2,3,4 was not difficult to determine that those two graphs are isomorphic. Jesus Medrano From rshroff@MIT.EDU Fri Sep 30 03:58:32 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 j8U7wWw0030762 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:32 -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 j8U7wUuA009301 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:31 -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 j8U7wToG002494 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 03:58:29 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8U7wTDA001647; Fri, 30 Sep 2005 03:58:29 -0400 Received: from NEXT-SIXTY-SEVEN.MIT.EDU (NEXT-SIXTY-SEVEN.MIT.EDU [18.242.5.67]) (User authenticated as rshroff@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 03:58:29 -0400 Message-ID: <20050930035829.7jemghp8sj4s8sg8@webmail.mit.edu> Date: Fri, 30 Sep 2005 03:58:29 -0400 From: rshroff@MIT.EDU To: 6042-probs@theory.lcs.MIT.EDU Subject: [David]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: 0.694 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 415 Content-Length: 166 X-Keywords: NonJunk Reference: Theorem 4.11 I didn't really understand parallel task scheduling or its realted corollaries and would like it if we went over it in class. -Rahul Shroff From jacques@MIT.EDU Fri Sep 30 04:07:39 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 j8U87dw0031479 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:39 -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 j8U87cuA014246 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:38 -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 j8U87aMx002803 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 04:07:36 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8U87aT4002295; Fri, 30 Sep 2005 04:07:36 -0400 Received: from EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY.MIT.EDU (EASTCAMPUS-ONE-O-THREE-HUNDRED-FORTY.MIT.EDU [18.238.7.73]) (User authenticated as jacques@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 04:07:36 -0400 Message-ID: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> Date: Fri, 30 Sep 2005 04:07:36 -0400 From: jacques To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] Week 4 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: 0.917 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 416 Content-Length: 232 X-Keywords: NonJunk In the lecture notes, page 8, there is a very good definition of the property of antisymmetry. However, asymmetry is mentioned in the lecture slides but never really defined in the lecture notes. That's something I'd like to see. From scot@MIT.EDU Fri Sep 30 04:11:02 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 j8U8B2w0032679 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:11:02 -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 j8U8B1uA016199 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:11:01 -0400 (EDT) Received: from [18.227.1.135] (ZBT-ONE-THIRTY-FIVE.MIT.EDU [18.227.1.135]) (authenticated bits=0) (User authenticated as scot@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8U8Ar0Q002931 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 04:10:55 -0400 (EDT) Message-ID: <433CF299.5040004@mit.edu> Date: Fri, 30 Sep 2005 04:08:57 -0400 From: Scot Frank User-Agent: Thunderbird 1.4 (Windows/20050908) MIME-Version: 1.0 To: 6042-probs@theory.lcs.mit.edu Subject: Week 4 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: 417 Content-Length: 253 X-Keywords: NonJunk I found partial orders (from page 7) to be confusing during lecture. Perhaps it was gone over too quickly for me. Also, I had never heard of modulo before. I did not find it in the appendix or explained, so I wlll need to look it up. Thanks, Scot From knightfu@MIT.EDU Fri Sep 30 05:29:30 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 j8U9TTw0008718 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:29 -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 j8U9TSkI028204 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:28 -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 j8U9TMVY005553 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 05:29:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-6.mit.edu (8.12.4) id j8U9TLWS016208; Fri, 30 Sep 2005 05:29:21 -0400 Received: from EASTCAMPUS-ONE-FORTY-FOUR.MIT.EDU (EASTCAMPUS-ONE-FORTY-FOUR.MIT.EDU [18.248.5.144]) (User authenticated as knightfu@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 05:29:21 -0400 Message-ID: <20050930052921.ls56zys1yk8wsgkc@webmail.mit.edu> Date: Fri, 30 Sep 2005 05:29:21 -0400 From: Knight W Fu To: 6042-probs@theory.csail.MIT.EDU Subject: 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: 418 Content-Length: 1124 X-Keywords: NonJunk Definition 2.1. Given any total function, f, with domain A, define the binary relation :=f on A by the rule: a :=f b iff f(a) = f(b) (1) for all a, b 2 A. A binary relation is an equivalence relation iff it equals :=f for some f. (the above was taken from Relations II, page 3.) 2.3 Properties of Equivalence Relations Equivalence relations have some obvious properties that occur so frequently they merit names: Definition 2.7. A binary relation R on a set A is: ? reflexive iff for every a 2 A, aRa, Course Notes, Week 4: Binary Relations 5 ? symmetric iff for every a, a0 2 A, aRa0 implies a0 Ra, ? transitive iff for every a, b, c 2 A, [aRb and bRc] implies aRc. (the above was taken from Relations II, page 4 The excerpts were most interesting because in Algebra (18.701), equivalent relationships are defined to be a relationship that is reflexive, symmetric, and transitive. The definition offered here is actually a consequence of the fact that the inverse image of a function partitions the domain (and the partition defines an equivalent relationship under the inverse image and creates equivalent classes.) From shreyes19@gmail.com Fri Sep 30 06:07:42 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.193]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UA7gw0014053 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 06:07:42 -0400 Received: by xproxy.gmail.com with SMTP id t11so373023wxc for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 03:07:37 -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=IU5OwGbsTjTvFI3JXlP+viVoNI7szFgWq+xTNH3EJE88sxukfV5bhdG6nO/+w+k5Ln0g0QrsM6jt4QjfJmwz1xPj+rmoxstTjMD+uVJIhbFaOYR6mQdBc3eHk6BG1AFpIAGMjwshtZA73c8FFJEvQXPJl9uDjDlvIeYTdp0nC1Q= Received: by 10.70.105.11 with SMTP id d11mr840976wxc; Fri, 30 Sep 2005 03:07:37 -0700 (PDT) Received: by 10.70.123.14 with HTTP; Fri, 30 Sep 2005 03:07:37 -0700 (PDT) Message-ID: <3c33fe380509300307j2d1cb405ye12ab51ede8fb8e9@mail.gmail.com> Date: Fri, 30 Sep 2005 06:07:37 -0400 From: Shreyes Seshasai Reply-To: shreyes@mit.edu To: 6042-probs@theory.lcs.mit.edu Subject: [Sayan] Week 4 Reading Comments MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_1915_2104057.1128074857232" Status: RO X-UID: 419 Content-Length: 1974 X-Keywords: NonJunk ------=_Part_1915_2104057.1128074857232 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Good morning, Here are my comments on the week 4 reading: The part I found most difficult in this week's reading was the definitions on page 10. Specifically, the definition of restrictions. "Definition 3.13. Let R be a relation on a set, A, and let B be a subset of A. The restrictio= n of R to B is the relation on B whose graph is graph (R) \ (B =D7 B). " (pag= e 10). I understand the concept of a restriction, but the specific definition that's provided is a little confusing to me, especially the graph of R [int= ] (B x B) part. I like how it goes through the properties of relations for th= e restriction on a set, but perhaps a couple examples of how restrictions are used will be helping in getting past the technical definition. Thanks, Shreyes Seshasai Group 7 ------=_Part_1915_2104057.1128074857232 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Good morning,

Here are my comments on the week 4 reading:

The part I found most difficult in this week's reading was the definitions on page 10.  Specifically, the definition of restrictions.  "Definition 3.13. Let R be a relation on a set, A, and let B be a subset of A. The restriction of R to B is the relation on B whose graph is graph (R) \ (B =D7 B). " (page 10).  I understand the concept of a restriction, but the specific definition that's provided is a little confusing to me, especially the graph of R [int] (B x B) part.  I like how it goes through the properties of relations for the restriction on a set, but perhaps a couple examples of how restrictions are used will be helping in getting past the technical definition.

Thanks,
Shreyes Seshasai
Group 7
------=_Part_1915_2104057.1128074857232-- From miki_tnd@MIT.EDU Fri Sep 30 06:33:30 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 j8UAXUw0017034 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:30 -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 j8UAXTkI004709 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:29 -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 j8UAXMfR008073 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 06:33:23 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UAXMu1020980; Fri, 30 Sep 2005 06:33:22 -0400 Received: from PIKA-ONE-O-NINE.MIT.EDU (PIKA-ONE-O-NINE.MIT.EDU [18.214.1.109]) (User authenticated as miki_tnd@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 06:33:22 -0400 Message-ID: <20050930063322.bnc5qws54xz40gw8@webmail.mit.edu> Date: Fri, 30 Sep 2005 06:33:22 -0400 From: Thu Ngoc Duong To: 6042-probs@theory.lcs.MIT.EDU Subject: (Sayan) Reading Assignment 4 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: 420 Content-Length: 245 X-Keywords: NonJunk I found the section on Parellel Task Scheduling on pg. 14 pretty interesting. If there is time in class, it would be great if more strategies for parellel task scheduling (or related operations research applications) were covered. ~Thu Duong From natalia3@MIT.EDU Fri Sep 30 06:33:32 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 j8UAXWw0017043 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:32 -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 j8UAXUkI004716 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:31 -0400 (EDT) Received: from Drache (NEW-ONE-EIGHTY-NINE.MIT.EDU [18.241.5.189]) (authenticated bits=0) (User authenticated as natalia3@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UAXMSo008071 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 06:33:24 -0400 (EDT) Message-Id: <200509301033.j8UAXMSo008071@outgoing.mit.edu> From: "Natalia Chernenko" To: <6042-probs@theory.csail.mit.edu> Subject: Week 4 Comments Date: Fri, 30 Sep 2005 06:33:19 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0006_01C5C588.DA0286F0" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 Thread-Index: AcXFql+umw1OudKiQgSO9LCsteBOjA== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 1.15 X-Spam-Level: * (1.15) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 421 Content-Length: 1964 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0006_01C5C588.DA0286F0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit I found images and inverse images confusing (p. 2; 1-2). Could you please explain them more? Thank you. Natalia Chernenko ------=_NextPart_000_0006_01C5C588.DA0286F0 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

I found images and inverse images confusing (p. 2; = 1-2). Could you please explain them more? Thank = you.

 

Natalia Chernenko

------=_NextPart_000_0006_01C5C588.DA0286F0-- From pgroudas@MIT.EDU Fri Sep 30 07:02:16 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 j8UB2Gw0020442 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:16 -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 j8UB2EkI021065 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:14 -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 j8UB2Ds0009624 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 07:02:13 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UB2D5N022489; Fri, 30 Sep 2005 07:02:13 -0400 Received: from DKE-TWENTY-SEVEN.MIT.EDU (DKE-TWENTY-SEVEN.MIT.EDU [18.233.1.27]) (User authenticated as pgroudas@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 07:02:13 -0400 Message-ID: <20050930070213.8f5443mr1sz9c8ow@webmail.mit.edu> Date: Fri, 30 Sep 2005 07:02:13 -0400 From: Paul Groudas To: 6042-probs@theory.lcs.MIT.EDU Subject: reading comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 422 Content-Length: 223 X-Keywords: NonJunk I found the pages on isomorphic graphs were interesting (18-20). After doing the tutor problems, I thought that it would have been useful if it gave any strategys on determining the number of isomorphisms. -Paul Groudas From fgreen@MIT.EDU Fri Sep 30 07:15:38 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 j8UBFcw0022525 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:38 -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 j8UBFbkI029092 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:37 -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 j8UBFSAP010660 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 07:15:28 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UBFSln023216; Fri, 30 Sep 2005 07:15:28 -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 ; Fri, 30 Sep 2005 07:15:28 -0400 Message-ID: <20050930071528.6whrjgkfn0g0koos@webmail.mit.edu> Date: Fri, 30 Sep 2005 07:15:28 -0400 From: Forrest O Green To: 6042-probs@theory.csail.MIT.EDU Subject: Required Reading Comment MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 423 Content-Length: 139 X-Keywords: NonJunk I found the isomorphism test to be very interesting. I would be interested in knowing if it is NP complete or not. -Forrest Green From petek@MIT.EDU Fri Sep 30 07:54:55 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 j8UBstw0028935 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:55 -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 j8UBsnkI023074 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:49 -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 j8UBsjho014254 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:54:48 -0400 (EDT) Message-ID: <433D2792.5080900@mit.edu> Date: Fri, 30 Sep 2005 07:54:58 -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: [Jelani] Partial Orders ? Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Spam-Score: -1.892 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 424 Content-Length: 649 X-Keywords: NonJunk [3.1]

I think I've having most difficulty with the vocabulary for partial orders....


 
-- 
Pete Kruskall
28 The Fenway
Boston, MA 02215

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From zacharyozer@gmail.com Fri Sep 30 08:47:56 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.193]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UCltw0002444 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 08:47:56 -0400 Received: by xproxy.gmail.com with SMTP id h27so84558wxd for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 05:47:50 -0700 (PDT) DomainKey-Signature: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:sender:to:subject:mime-version:content-type:content-transfer-encoding:content-disposition; b=k2RP7SRfouHC6qRS6gg3Qopk7sn1eXIKO6tqA+r8gkzqyJt7Bv+q+TgkuynC7oCaUD6IGxe8JKjA6TJdTb1+KqBVGrjdbwPmbNEUyRBySqMzgPzxkKBgrdd8/W9zxYQ6lrYrLO8vcSbu6WEn5RNMe3lM0WG3nDcxfInonmGBd/8= Received: by 10.70.87.13 with SMTP id k13mr876067wxb; Fri, 30 Sep 2005 05:47:49 -0700 (PDT) Received: by 10.70.9.14 with HTTP; Fri, 30 Sep 2005 05:47:49 -0700 (PDT) Message-ID: <4f2613e50509300547k7cde16f7le8d435085afeb24f@mail.gmail.com> Date: Fri, 30 Sep 2005 08:47:49 -0400 From: Zachary Adam Ozer Reply-To: Zachary Adam Ozer Sender: zacharyozer@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: Email comments MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by theory.csail.mit.edu id j8UCltw0002444 Status: RO X-UID: 425 Content-Length: 592 X-Keywords: NonJunk While the reading does give a mathematical definition for terms such as maximal and minimal, it does not give the reader an intuitive understanding of these terms, nor do the examples on pages 13 or 14 really help the reader at all. What would be better would be several examples involving numbers and properties, ie maximum of the property multiplication over some set of integers. I only understood the terms for the property division after a lengthy explination, and I am still uncertain as to whether I understand them for other properties. Examples would help me, I do believe. -zozer From yuhsin@MIT.EDU Fri Sep 30 09:25:08 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 j8UDP7w0007759 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:25:07 -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 j8UDP61O002941 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:25:06 -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 j8UDOxfw004661 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:24:59 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-1.mit.edu (8.12.4) id j8UDOx0x022283; Fri, 30 Sep 2005 09:24:59 -0400 Received: from NONE-FOUR-NINETY-SIX.MIT.EDU (NONE-FOUR-NINETY-SIX.MIT.EDU [18.54.6.241]) (User authenticated as yuhsin@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 09:24:59 -0400 Message-ID: <20050930092459.1zplgv7pb94w0o4k@webmail.mit.edu> Date: Fri, 30 Sep 2005 09:24:59 -0400 From: Yuhsin Chen To: 6042-probs@theory.lcs.MIT.EDU Subject: questions MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.541 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 426 Content-Length: 992 X-Keywords: NonJunk About Partial Orders: On page 7, the formal definition of a partial order is given as "A binary relation, R, on a set, A, is a partial order iff there is a g such that R agrees with *squiggly*g for every pair of distinct elements". Since last lecture, I've seen examples of partial orders, and if given a relation I can probably identify it as a partial order or not. However, I'm not convinced I understand how to define what a partial order is if someone asked me. (this is more or less an "afterthought" question) About Isomorphism: Most surprising sentence--"having an efficient isomorphism finding/testing procedure would, for example, make it easy to search for a particular molecule in a database given the molecular bonds". I know this is going to be explained further today, but the implications of isomorphism are very surprising, and I'd like to know more about fields that are deeply related. (The fields could serve as examples to strengthen my concept of isomorphism) Joyce From arup@MIT.EDU Fri Sep 30 09:25:23 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 j8UDPNw0007778 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:23 -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 j8UDPM1O003197 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:22 -0400 (EDT) Received: from aruplaptop.mit.edu (BURTON-TWO-TWENTY-SEVEN.MIT.EDU [18.247.5.227]) (authenticated bits=0) (User authenticated as arup@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDP5s2004704 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:25:06 -0400 (EDT) Message-Id: <6.2.3.4.2.20050930012730.04696c10@po14.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 30 Sep 2005 09:25:03 -0400 To: 6042-probs@theory.lcs.mit.edu From: Arup Sarma Subject: [Jelani] Week 4 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: 427 Content-Length: 315 X-Keywords: NonJunk Section 3.1, page 7: I had trouble understanding partial orders because I wasn't sure what "g(a) is the set of properties that a has" meant, since I confused it with f(a) like in equivalence functions. In general, the notation in this reading confused me for some reason, but I'm getting used to it. |Arup| From veracarr@MIT.EDU Fri Sep 30 09:37:17 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 j8UDbGw0009857 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:17 -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 j8UDbF6K013278 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:15 -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 j8UDbF2G015607 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:15 -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 j8UDbB0v028298 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 09:37:12 -0400 (EDT) Message-ID: <433D5BE5.7010309@mit.edu> Date: Fri, 30 Sep 2005 09:38:13 -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: [Hanson] Binary Relations Reading 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: 428 Content-Length: 288 X-Keywords: NonJunk The section that I found most difficult to understand was the one on Partial Orders (pg 7). I had difficulty recognizing them in the online tutor. Also, when they are defining weak and strict partial orders for both they say iff it is reflexive. For strict did they mean irreflexive? From mwangi@MIT.EDU Fri Sep 30 09:42:07 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 j8UDg7w0010132 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:42:07 -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 j8UDg51O018735 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:42:05 -0400 (EDT) Received: from IBM-1E80C73FC15.mit.edu (NEW-THREE-FOURTEEN.MIT.EDU [18.241.6.59]) (authenticated bits=0) (User authenticated as mwangi@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UDfwtY010481 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 09:41:59 -0400 (EDT) Message-Id: <6.2.1.2.2.20050930092937.04434400@po12.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Fri, 30 Sep 2005 09:41:55 -0400 To: 6042-probs@theory.csail.mit.edu From: Timothy Mwangi Subject: [Sayan] Week 4 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: 429 Content-Length: 281 X-Keywords: NonJunk Passage: 1.2 Images and Inverse Images Page: 2 I found this passage most difficult because the definition of the image of C under R and the definition of the inverse image of C under R was unclear. An example would definitely have helped. Sincerely, Timothy M. Mwangi From kromer@MIT.EDU Fri Sep 30 09:47:36 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 j8UDlaw0010805 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:36 -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 j8UDlZ1O024661 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:35 -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 j8UDlX6L012790 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 09:47:33 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8UDlXdh014719; Fri, 30 Sep 2005 09:47:33 -0400 Received: from BURTON-THREE-TEN.MIT.EDU (BURTON-THREE-TEN.MIT.EDU [18.247.6.55]) (User authenticated as kromer@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 09:47:33 -0400 Message-ID: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> Date: Fri, 30 Sep 2005 09:47:33 -0400 From: Katherine A Romer To: 6042-probs@theory.lcs.MIT.EDU Subject: [David] 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: 0.468 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 430 Content-Length: 345 X-Keywords: NonJunk p. 9: "Theorem: A binary relation is a partial order iff it is transitive and antisymmetric.... To prove the theorem in the right to left direction, assume R is transitive and antisymmetric. Define g(a) ::=R{a} union {a}" I was confused by the notation R{a} union {a}: does this mean: "the set containing a and all the elements that map to a"? From hkhall@MIT.EDU Fri Sep 30 10:19:06 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 j8UEJ6w0019655 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:19:06 -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 j8UEJ41O028324; Fri, 30 Sep 2005 10:19:04 -0400 (EDT) Received: from [18.96.6.143] (SIMMONS-THREE-NINETY-EIGHT.MIT.EDU [18.96.6.143]) (authenticated bits=0) (User authenticated as hkhall@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEIvmo025815 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT); Fri, 30 Sep 2005 10:18:58 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <884bfc7b6180f5205264705c4878ea84@mit.edu> Content-Transfer-Encoding: 7bit From: Harrison King Hall Subject: Reading Comments on LN4 [David]- Harrison Hall Date: Fri, 30 Sep 2005 10:20:57 -0400 To: 6042-probs@theory.lcs.mit.edu X-Mailer: Apple Mail (2.623) X-Spam-Score: -1.036 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 431 Content-Length: 552 X-Keywords: NonJunk While I understand the axioms that define a partial order(e.g. transitivity and antisymmetry) I could not explain to you in a way coherent way what one is. While it is a type of regression I don't own the material until I can explain it to someone else or even myself. Right now I cant do that with partial orders. As such I am not really able to understand all of the proofs in the lecture notes that use partial orders or their properties to prove some property. Some additional explanation of a partial order would be awesome. -Harrison From nedzel@MIT.EDU Fri Sep 30 10:26:53 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEQrw0020339 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:53 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEQq1O006800 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:52 -0400 (EDT) Received: from ThinkPadT43 (SIMMONS-SEVENTY-SEVEN.MIT.EDU [18.96.5.77]) (authenticated bits=0) (User authenticated as nedzel@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEQos4029247 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:26:50 -0400 (EDT) Message-Id: <200509301426.j8UEQos4029247@outgoing.mit.edu> From: "David A. Nedzel" To: <6042-probs@theory.csail.mit.edu> Subject: [Sayan] Week 4 Comments Date: Fri, 30 Sep 2005 10:26:40 -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: 2.224 X-Spam-Level: ** (2.224) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 432 Content-Length: 107 X-Keywords: NonJunk In section 4.5, I am a little confused by Dilworth's Lemma and its implications to subsequences. - David From mike_a@MIT.EDU Fri Sep 30 10:27:57 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UERvw0020429 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:57 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UERt1O007777 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:56 -0400 (EDT) Received: from [18.238.6.106] (EASTCAMPUS-EIGHT-SEVENTY-THREE.MIT.EDU [18.238.6.106]) (authenticated bits=0) (User authenticated as mike_a@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UERpiE029672 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:27:51 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v734) To: 6042-probs@theory.lcs.mit.edu Message-Id: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> Content-Type: multipart/mixed; boundary=Apple-Mail-1--540584741 From: mike anderson Subject: [David] weekly comment Date: Fri, 30 Sep 2005 10:27:50 -0400 X-Mailer: Apple Mail (2.734) X-Spam-Score: -2.844 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 433 Content-Length: 161029 X-Keywords: NonJunk --Apple-Mail-1--540584741 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Especially the "there is a g such that R agrees w/ ...". What does agrees w/ mean in this context? -mike anderson --Apple-Mail-1--540584741-- From moscicki@MIT.EDU Fri Sep 30 10:28:12 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 j8UESCw0020444 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:12 -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 j8UESB1O007909 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:11 -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 j8UES9dt029783 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 10:28:09 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-5.mit.edu (8.12.4) id j8UES9ZS020773; Fri, 30 Sep 2005 10:28:09 -0400 Received: from NEW-ONE-SIXTY-EIGHT.MIT.EDU (NEW-ONE-SIXTY-EIGHT.MIT.EDU [18.241.5.168]) (User authenticated as moscicki@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 10:28:09 -0400 Message-ID: <20050930102809.v8g0gnfit0cgko8k@webmail.mit.edu> Date: Fri, 30 Sep 2005 10:28:09 -0400 From: Angelique E Moscicki To: 6042-probs@theory.lcs.MIT.EDU Subject: week4comment MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="=_1yozm3wntfnk" Content-Transfer-Encoding: 7bit User-Agent: Internet Messaging Program (IMP) H3 (4.0.3) X-Spam-Score: 0.064 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 434 Content-Length: 4121 X-Keywords: NonJunk This message is in MIME format. --=_1yozm3wntfnk Content-Type: text/plain; charset=ISO-8859-1 Content-Disposition: inline Content-Transfer-Encoding: 7bit attached. --=_1yozm3wntfnk Content-Type: text/plain; charset=ISO-8859-1; name="week4comment.txt" Content-Disposition: attachment; filename="week4comment.txt" Content-Transfer-Encoding: 7bit Get ready for a big substantial comment... did you write me that letter knowing that the next week would be so much more challenging? Relations and Functions: Its a concept we've all been dealing with for many years, but for some reason the idea is hard for me to catch abstractly enough. The notation is confusing. I understand the doman, codomain, and graph, but somehow the ideas do not have concrete meaning for me. This is why all the rest of the reading is so hard. "So a function is a special case of binary relation" is the most helpful sentence of them all I think. It is also hard for me to grasp that the codomain of the function is not defined by the domain and the mapping; that there can be elements that are not used, so there is a certain arbitrariness about the codomain of a relation. Images and Inverse Images: Completely confusing. The notation especially. If CR is a set made from cRb, where B is in the codomain of the relation between A and B, where did A go and why is it relevant? CR is cRb so C acts as the domain RC is aRc C acts as the codomain If I think of it this way, then the idea of images makes sense. Surjective... Again the definitions: surjective -> every element in codomain used at least once. ah! it makes senes! AR= the set of little bs = B -> surjective because the set of bs = the total domain! Sorry for the flame. Equivalence: Symetric, Reflexive, and Transitive. Equivalence by function: Two equal objects give the same hashCode() Partitions: If a set is partitioned, the set of the partitions contains all the elements of the original set and the partitions do not overlap. This one makes sense more than others. Interesting that being in the same block is equivalent. So, an equivalence class is the block the element belongs to. This was very unclear from the lecture. Now it makes a bit more sense. A bit on notation: Saying that reflexivity is defined for R on A by aRa is extremely confusing and I didn't get it all week until now. I think its because there are too many as. Call a q or p or anything but little a and it makes it more clear. Likewise for symmetry: aRa' is not particularly clear I think. The examples are helpful. Partial Orders: The definition give in the first two paragraphs of page 6 is unclear to me. Perhaps it will become clearer as I read on... It seems to be a total order that is not true in all cases, i.e. it does not specify whether elements are related to themselves. weak-> reflexive strict-> irreflexive all are transitive all are antisymmetric Total Orders: pick any element of the domain and any element of the codomain and either the relation or its converse will be true. Note: My summarization of the key ideas I guess is just to show that I understand it, or maybe how the reader might take it. Digraphs: Networking. Searching algorithms, therefore interesting. Noticed the connection that this really is the definition of the relation, just like the equation of a line is only an expression for the actual thing. Path length is the important thing here, as in class problems. Minimal: No edge points to this thing. Its the very bottom, the least element Maximal: The most edges point to this thing. It is farthest up the tree. Topological Sort: Pick the minimal elements first. Many possible orders depending on the set and relation. Critical Path= longest chain= shortest possible number of steps to complete everything The part with chains and antichains is refreshingly clear, and was clear in lecture as well. Ugraphs: Fairly stright forward. The part about isomorphism was not clear at first, but someone explained it to me, so I do understand it... but not from the reading. --=_1yozm3wntfnk-- From yaser@MIT.EDU Fri Sep 30 10:44:43 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 j8UEigw0025095 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:42 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEif1O026131 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:41 -0400 (EDT) Received: from Charizard (NEW-TWO-EIGHTY-NINE.MIT.EDU [18.241.6.34]) (authenticated bits=0) (User authenticated as ykhan@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEidEa007018 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:44:39 -0400 (EDT) Message-Id: <200509301444.j8UEidEa007018@outgoing.mit.edu> From: "Yaser Khan" To: <6042-probs@theory.lcs.mit.edu> Subject: [David] Week 4 comments Date: Fri, 30 Sep 2005 10:44:36 -0400 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C5AB.F43A5430" X-Mailer: Microsoft Office Outlook, Build 11.0.6353 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 Thread-Index: AcXFzXntEats+8hXS2KwkwzUZmB+HQ== X-Spam-Score: 1.015 X-Spam-Level: * (1.015) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 435 Content-Length: 6220 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C5AB.F43A5430 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Hi David, On the topic of isomorphism, I read that there is no generalizable short-cut method to determine if two graphs are isomorphic, short of exhaustively examining each possible bijection. During lecture, could you please elaborate on whether there are some short cuts that hold in certain cases, or what some of the other properties of isomorphic bjiection are? Also, I still don't understand what the take-away message of the height- age demonstration was. Thanks! _Yaser ------=_NextPart_000_0000_01C5C5AB.F43A5430 Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: quoted-printable

Hi = David,

 

On the topic of isomorphism, I = read that there is no generalizable short-cut method = to determine if two graphs are isomorphic, short of exhaustively examining = each possible bijection. During lecture, could = you please elaborate on whether there are some short cuts that hold in certain = cases, or what some of the other properties of isomorphic bjiection are?

 

Also, I still don’t = understand what the take-away message of the height- age demonstration = was…

 

Thanks!

 

_Yaser

------=_NextPart_000_0000_01C5C5AB.F43A5430-- From harelw@gmail.com Fri Sep 30 10:49:27 2005 Return-Path: Received: from xproxy.gmail.com (xproxy.gmail.com [66.249.82.207]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UEnQw0027010 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:49:26 -0400 Received: by xproxy.gmail.com with SMTP id h27so104199wxd for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 07:49:21 -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=blJvvddhnLhjfbOI7NWmMlg078svkv603AQcz5QIEFL8eFWsV3ihrv4TVoXbG6HfduyFZKvKNxrBRDIefbXILL6KmpV5+E5Z7u3GDGDv8DWTIGMMc8v7wn3qmfhfOxH90fUPhBzfr0E7MM6wTnKOIkYGns/vHkeS+ulpZX6l8oI= Received: by 10.70.8.2 with SMTP id 2mr914840wxh; Fri, 30 Sep 2005 07:49:20 -0700 (PDT) Received: by 10.70.40.15 with HTTP; Fri, 30 Sep 2005 07:49:20 -0700 (PDT) Message-ID: <8c5248a80509300749s120f6badv679ea2efdfe35ada@mail.gmail.com> Date: Fri, 30 Sep 2005 10:49:20 -0400 From: "Harel M. Williams" Reply-To: "Harel M. Williams" Sender: harelw@gmail.com To: 6042-probs@theory.lcs.mit.edu Subject: TA: Jelani - Comments for Reading MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_Part_2382_27898129.1128091760761" Status: RO X-UID: 436 Content-Length: 726 X-Keywords: NonJunk ------=_Part_2382_27898129.1128091760761 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline As reflected on the tutorial multiple choice problems, I don't understand the notation for surjective and total relation, altough I understand the definition of the terms. ~H ------=_Part_2382_27898129.1128091760761 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Content-Disposition: inline As reflected on the tutorial multiple choice problems, I don't understand the notation for surjective and total relation, altough I understand the definition of the terms.

~H
------=_Part_2382_27898129.1128091760761-- From mrivas03@MIT.EDU Fri Sep 30 10:52:20 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 j8UEqKw0027293 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:20 -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 j8UEqJ1O004157 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:19 -0400 (EDT) Received: from el-ternero.mit.edu (TDCIP95.MIT.EDU [18.237.0.95]) (authenticated bits=0) (User authenticated as mrivas03@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UEqEh2010132 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:52:15 -0400 (EDT) Message-Id: <6.2.3.4.2.20050930104220.02eaecb0@po10.mit.edu> X-Mailer: QUALCOMM Windows Eudora Version 6.2.3.4 Date: Fri, 30 Sep 2005 10:46:45 -0400 To: 6042-probs@theory.lcs.mit.edu From: Manuel Rivas Subject: [Sayan] Reading Course Notes Week 4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Spam-Score: 0.978 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 437 Content-Length: 401 X-Keywords: NonJunk The concept of Bijection in Isomorphic and non-isomorphic graphs was not too clear from the course notes. Such as finding the number of bijections between two isomorphic graphs. It would be particulary helpful if that is gone over in class. I found topological sorting of the section notes interesting, especially the claim that all partial orders have topological sorts. Thanks, Manuel Rivas From rshearer@MIT.EDU Fri Sep 30 10:53:31 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UErVw0027408 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:31 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UErT1O005376 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:29 -0400 (EDT) Received: from [18.240.6.247] (MCCORMICK-FIVE-O-TWO.MIT.EDU [18.240.6.247]) (authenticated bits=0) (User authenticated as rshearer@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UErM0p010641 (version=TLSv1/SSLv3 cipher=RC4-SHA bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 10:53:22 -0400 (EDT) Mime-Version: 1.0 (Apple Message framework v623) Content-Transfer-Encoding: 7bit Message-Id: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=US-ASCII; format=flowed To: 6042-probs@theory.lcs.mit.edu From: Rachel Shearer Subject: [Jelani] Week 4 Comment Date: Fri, 30 Sep 2005 10:53:09 -0400 X-Mailer: Apple Mail (2.623) X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 438 Content-Length: 317 X-Keywords: NonJunk I had a lot of trouble with the tutor problem relating to Isomorphism, especially the one about how to prove that two graphs are not isomorphic. I feel like it's actually not very hard but I am just assuming it's more complicated than it actually is. Hopefully class today will make things more clear. Rachel From nancyk@MIT.EDU Fri Sep 30 10:54:59 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 j8UEsxw0027474 for <6042-probs@theory.csail.MIT.EDU>; Fri, 30 Sep 2005 10:54:59 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UEsw1O007088; Fri, 30 Sep 2005 10:54:58 -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 j8UEspj3011286; Fri, 30 Sep 2005 10:54:52 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-2.mit.edu (8.12.4) id j8UEsprV001156; Fri, 30 Sep 2005 10:54:51 -0400 Received: from WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU (WAREHOUSE-FOUR-NINETY-TWO.MIT.EDU [18.139.6.237]) (User authenticated as nancyk@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 10:54:51 -0400 Message-ID: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> Date: Fri, 30 Sep 2005 10:54:51 -0400 From: Nancy L Keuss To: 6042-probs@theory.csail.MIT.EDU Subject: [Hanson] Week 4 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: 439 Content-Length: 386 X-Keywords: NonJunk Hi, On page 7 of the Week 4 notes, directly under equation (7) is some discussion about the identity function. I do not understand why it follows that the partial order binary relation of I_A is the same relation as the proper subset relation. I wished there was more explanation between those two statements, as I could not make the logical jump from one to the other. Thanks, Nancy From letrec@MIT.EDU Fri Sep 30 10:56:29 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 j8UEuTw0027574 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:29 -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 j8UEuS1O008795 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:28 -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 j8UEuKr9011979 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 10:56:21 -0400 (EDT) Message-ID: <433D520F.7010207@mit.edu> Date: Fri, 30 Sep 2005 10:56:15 -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.csail.mit.edu Subject: 6042-student: Hanson Week 4 comments X-Enigmail-Version: 0.90.0.0 X-Enigmail-Supports: pgp-inline, pgp-mime Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -0.924 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 440 Content-Length: 238 X-Keywords: NonJunk Could I have the section on bijective mappings, and their counting made more explicit. Also, in relation to the tutor problems, could the meaning of: RA=B AR=B, etc...be explained more fully, as the syntax appeared rather foreign. ~ From dshin@MIT.EDU Fri Sep 30 11:02:16 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 j8UF2Gw0027980 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:02:16 -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 j8UF2EAW000848; Fri, 30 Sep 2005 11:02:14 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF2B0v003288; Fri, 30 Sep 2005 11:02:12 -0400 (EDT) Message-ID: <433D536A.1070806@mit.edu> Date: Fri, 30 Sep 2005 11:02:02 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: mike anderson CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] weekly comment References: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> In-Reply-To: <5B7D6F99-9F9E-4306-B7CA-326B336C68BD@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.625 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 441 Content-Length: 343 X-Keywords: Try the very next sentence in the notes. g agrees with R iff for all distinct a,b, aRb iff g(a) is a proper subset of g(b) DS mike anderson wrote: > From Page 7 (section 3.1) > > I don't really understand this. Especially the "there is a g such > that R agrees w/ ...". What does agrees w/ mean in this context? > > > -mike anderson From dshin@MIT.EDU Fri Sep 30 11:03:15 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 j8UF3Fw0028260 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:03:15 -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 j8UF3E6K023996 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:03:14 -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 j8UF3ErH019531; Fri, 30 Sep 2005 11:03:14 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF360v003342; Fri, 30 Sep 2005 11:03:06 -0400 (EDT) Message-ID: <433D53A1.2080004@mit.edu> Date: Fri, 30 Sep 2005 11:02:57 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Harrison King Hall CC: 6042-probs@theory.lcs.mit.edu Subject: Re: Reading Comments on LN4 [David]- Harrison Hall References: <884bfc7b6180f5205264705c4878ea84@mit.edu> In-Reply-To: <884bfc7b6180f5205264705c4878ea84@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 442 Content-Length: 668 X-Keywords: Hopefully, you will get the practice you need in the next pset. DS Harrison King Hall wrote: > While I understand the axioms that define a partial order(e.g. > transitivity and antisymmetry) I could not explain to you in a way > coherent way what one is. While it is a type of regression I don't > own the material until I can explain it to someone else or even > myself. Right now I cant do that with partial orders. As such I am > not really able to understand all of the proofs in the lecture notes > that use partial orders or their properties to prove some property. > Some additional explanation of a partial order would be awesome. > -Harrison > From dshin@MIT.EDU Fri Sep 30 11:06:27 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 j8UF6Rw0029551 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:06:27 -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 j8UF6P6K027841 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:06:25 -0400 (EDT) Received: from outgoing-legacy.mit.edu (OUTGOING-LEGACY.MIT.EDU [18.7.22.104]) by central-city-carrier-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UF6PrH019686; Fri, 30 Sep 2005 11:06:25 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF6M3a008178; Fri, 30 Sep 2005 11:06:23 -0400 (EDT) Message-ID: <433D5465.5050706@mit.edu> Date: Fri, 30 Sep 2005 11:06:13 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Katherine A Romer CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] Email comments for reading References: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> In-Reply-To: <20050930094733.2gko3s8aj808kw8s@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -1.782 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 443 Content-Length: 512 X-Keywords: For any set set S, R(S) is the set {R(s) : s in S}. So, R{a} (I guess R({a}) would be better notation...) is just {R(a)}. DS Katherine A Romer wrote: >p. 9: >"Theorem: A binary relation is a partial order iff it is transitive and >antisymmetric.... To prove the theorem in the right to left direction, assume R >is transitive and antisymmetric. Define g(a) ::=R{a} union {a}" > >I was confused by the notation R{a} union {a}: does this mean: "the set >containing a and all the elements that map to a"? > > From dshin@MIT.EDU Fri Sep 30 11:08:52 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 j8UF8qw0029709 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:08:52 -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 j8UF8nAW001170; Fri, 30 Sep 2005 11:08:49 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UF8f3a008311; Fri, 30 Sep 2005 11:08:42 -0400 (EDT) Message-ID: <433D54F0.2010809@mit.edu> Date: Fri, 30 Sep 2005 11:08:32 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: jacques CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 comments References: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> In-Reply-To: <20050930040736.fk9hd1m30p8oowwc@webmail.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 444 Content-Length: 375 X-Status: A X-Keywords: asymmetric simply means "not symmetric". If you know what "symmetric" means, you understand asymmetry as well. DS jacques wrote: >In the lecture notes, page 8, there is a very good definition of the property of >antisymmetry. However, asymmetry is mentioned in the lecture slides but never >really defined in the lecture notes. That's something I'd like to see. > > From dshin@MIT.EDU Fri Sep 30 11:14:30 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 j8UFETw0030019 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:14:29 -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 j8UFESwb004236 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:14:28 -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 j8UFESrH020111; Fri, 30 Sep 2005 11:14:28 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UFEK0v004143; Fri, 30 Sep 2005 11:14:20 -0400 (EDT) Message-ID: <433D5643.8060607@mit.edu> Date: Fri, 30 Sep 2005 11:14:11 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yaser Khan CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] Week 4 comments References: <200509301444.j8UEidEa007018@outgoing.mit.edu> In-Reply-To: <200509301444.j8UEidEa007018@outgoing.mit.edu> Content-Type: multipart/alternative; boundary="------------020809080009010409040107" X-Spam-Score: -2.098 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 445 Content-Length: 8614 X-Keywords: This is a multi-part message in MIME format. --------------020809080009010409040107 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Great question! There CERTAINLY are short cuts that hold in certain cases. For example, one short cut is to just count the number of vertices and edges in each graph. If the numbers don't match up, I've come up with a short cut to show they are not isomorphic. Or, let's say I'm trying to show they are isomorphic, and each graph has exactly one node of degree 10. Well, if there is a bijection, those two nodes have to map to each other, so that narrows down my search for bijections. In fact, it's interesting to note that we've never been able to come up with a "difficult class" of graph-pairs, for which we're not able to solve graph-isomorphism by short-cuts! As for the height-age demonstration, imagine that all the people in the class are standing in order according to age. Then, given that fixed list of people, we define the following partial order: a Hi David, > > > > On the topic of isomorphism, I read that there is no generalizable > short-cut method to determine if two graphs are isomorphic, short of > exhaustively examining each possible bijection. During lecture, could > you please elaborate on whether there are some short cuts that hold in > certain cases, or what some of the other properties of isomorphic > bjiection are? > > > > Also, I still don't understand what the take-away message of the > height- age demonstration was... > > > > Thanks! > > > > _Yaser > --------------020809080009010409040107 Content-Type: text/html; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Great question!

There CERTAINLY are short cuts that hold in certain cases.  For example, one short cut is to just count the number of vertices and edges in each graph.  If the numbers don't match up, I've come up with a short cut to show they are not isomorphic. 

Or, let's say I'm trying to show they are isomorphic, and each graph has exactly one node of degree 10.  Well, if there is a bijection, those two nodes have to map to each other, so that narrows down my search for bijections.

In fact, it's interesting to note that we've never been able to come up with a "difficult class" of graph-pairs, for which we're not able to solve graph-isomorphism by short-cuts! 

As for the height-age demonstration, imagine that all the people in the class are standing in order according to age.  Then, given that fixed list of people, we define the following partial order:  a<b if a is shorter than b.  Dilworth's Lemma claims that we should be able to find either a long increasing subsequence or a long decreasing subsequence.  Those were the two sets of people you saw during the demonstration.

DS

Yaser Khan wrote:

Hi David,

 

On the topic of isomorphism, I read that there is no generalizable short-cut method to determine if two graphs are isomorphic, short of exhaustively examining each possible bijection. During lecture, could you please elaborate on whether there are some short cuts that hold in certain cases, or what some of the other properties of isomorphic bjiection are?

 

Also, I still don’t understand what the take-away message of the height- age demonstration was…

 

Thanks!

 

_Yaser

--------------020809080009010409040107-- From antonk@MIT.EDU Fri Sep 30 11:33:25 2005 Return-Path: Received: from biscayne-one-station.mit.edu (BISCAYNE-ONE-STATION.MIT.EDU [18.7.7.80]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UFXPw0031511 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:25 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UFXNrh017209 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:24 -0400 (EDT) Received: from quick-w20-copy2.mit.edu (QUICK-W20-COPY2.MIT.EDU [18.187.0.202]) (authenticated bits=56) (User authenticated as antonk@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UFXJA4027896 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 11:33:20 -0400 (EDT) Received: (from antonk@localhost) by quick-w20-copy2.mit.edu (8.12.9) id j8UFXIZ0013576; Fri, 30 Sep 2005 11:33:18 -0400 (EDT) Subject: [David] 3 From: Anton Katz To: 6042-probs@theory.lcs.MIT.EDU Content-Type: text/plain Content-Transfer-Encoding: 7bit Date: Fri, 30 Sep 2005 11:33:16 -0400 Message-Id: <1128094396.13555.3.camel@quick-w20-copy2.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: 446 Content-Length: 338 X-Keywords: NonJunk Hi, chapter 5.2 discusses isomorphism. I understand the first example with the square graph, but I want to hear more about the star graph. The same problem was on the tutorial questions and I had to try a lot of numbers before getting the answer. Thank you and sorry for being late, i simply forgot about the email, hope its ok. Anton From dshin@MIT.EDU Fri Sep 30 11:47:03 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 j8UFl3w0001948 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 11:47:03 -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 j8UFl1AW002960; Fri, 30 Sep 2005 11:47:01 -0400 (EDT) Received: from [192.168.1.6] (c-65-96-190-39.hsd1.ma.comcast.net [65.96.190.39]) ) by outgoing-legacy.mit.edu (8.12.4/8.12.4) with ESMTP id j8UFkn0v006168; Fri, 30 Sep 2005 11:46:50 -0400 (EDT) Message-ID: <433D5DE0.1020305@mit.edu> Date: Fri, 30 Sep 2005 11:46:40 -0400 From: David Shin User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Anton Katz CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [David] 3 References: <1128094396.13555.3.camel@quick-w20-copy2.mit.edu> In-Reply-To: <1128094396.13555.3.camel@quick-w20-copy2.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.099 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 447 Content-Length: 500 X-Keywords: I can show you in more detail if you come by office hours today at 1 (24-319), or if you come to class a little early. DS Anton Katz wrote: >Hi, > >chapter 5.2 discusses isomorphism. >I understand the first example with the square graph, but I want to hear >more about the star graph. >The same problem was on the tutorial questions and I had to try a lot of >numbers before getting the answer. > >Thank you and sorry for being late, i simply forgot about the email, >hope its ok. > >Anton > > From alisonc@MIT.EDU Fri Sep 30 11:59:23 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 j8UFxNw0005749 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:23 -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 j8UFxLrh016780 for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:22 -0400 (EDT) Received: from [18.238.1.185] (ALISONC.MIT.EDU [18.238.1.185]) (authenticated bits=0) (User authenticated as alisonc@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UFxGC5009082 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT) for <6042-probs@theory.csail.mit.edu>; Fri, 30 Sep 2005 11:59:16 -0400 (EDT) Message-ID: <433D60E2.70403@mit.edu> Date: Fri, 30 Sep 2005 11:59:30 -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: [Hanson] Lecture notes 4 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: 448 Content-Length: 214 X-Keywords: NonJunk I found the parallel task scheduling and topological sorting most interesting. It seems to have really fascinating direct applications to computer science, parallel processing and so forth, which is really neat. From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:07:37 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UG7bw0007066; Fri, 30 Sep 2005 12:07:37 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UG7anX014059; Fri, 30 Sep 2005 12:07:37 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UG7a1m014056; Fri, 30 Sep 2005 12:07:36 -0400 Date: Fri, 30 Sep 2005 12:07:36 -0400 (EDT) From: Hanson Zhou To: "Xiaoran (Sharon) Zhang" cc: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [Hanson] Week 4 comments In-Reply-To: <20050930003434.peberlpwka0osskk@webmail.mit.edu> Message-ID: References: <20050930003434.peberlpwka0osskk@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 449 Content-Length: 867 X-Keywords: Go to office hours to get this cleared up. One helpful thing might be to review the definition of a bijection, and to reexamine how it applies in the case of graph isomorphism. -Hanson On Fri, 30 Sep 2005, Xiaoran (Sharon) Zhang wrote: > I was confused about isomorphism for undirected graphs. Page 17, section 5. I am > still unsure which characteristics to evaluate when determining whether two > graphs are isomorphic or not. The correspondence between the vertex of the > graphs are not clear to me. > > Sharon > > ************************************************** > 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 hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:08:56 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UG8uw0007161; Fri, 30 Sep 2005 12:08:56 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UG8uU6014070; Fri, 30 Sep 2005 12:08:56 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UG8u55014067; Fri, 30 Sep 2005 12:08:56 -0400 Date: Fri, 30 Sep 2005 12:08:56 -0400 (EDT) From: Hanson Zhou To: Lohith Kini cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Week 4 Reading Comments In-Reply-To: <433CC791.7090107@mit.edu> Message-ID: References: <433CC791.7090107@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 450 Content-Length: 360 X-Keywords: This is might be a bit tricky...go to office hours if you're still confused. -Hanson On Fri, 30 Sep 2005, Lohith Kini wrote: > Hi, > > I was totally confused about parallel task scheduling and theorem 4.11. > I didn't understand the corollaries either (because of this). I would > like it if you could go over that a little bit more. > > Thanks, > Lohith > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:12:24 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGCOw0007426; Fri, 30 Sep 2005 12:12:24 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGCOI6014093; Fri, 30 Sep 2005 12:12:24 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGCOH3014090; Fri, 30 Sep 2005 12:12:24 -0400 Date: Fri, 30 Sep 2005 12:12:24 -0400 (EDT) From: Hanson Zhou To: Elizabeth Reid cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Reading Comments In-Reply-To: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> Message-ID: References: <348E403C-2681-4A5D-BE9B-0F623B985801@MIT.EDU> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 451 Content-Length: 288 X-Keywords: Why would you want to maximize it? -Hanson On Fri, 30 Sep 2005, Elizabeth Reid wrote: > If they could talk a bit about parallel task scheduling in lecture > that would be nice. I was wondering if there was any mathematical way > to maximize it, or if it's more of a guess and check? > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:15:10 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGF9w0007984; Fri, 30 Sep 2005 12:15:09 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGF9GZ014110; Fri, 30 Sep 2005 12:15:09 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGF9kT014107; Fri, 30 Sep 2005 12:15:09 -0400 Date: Fri, 30 Sep 2005 12:15:09 -0400 (EDT) From: Hanson Zhou To: Alton Torregano cc: 6042-probs@theory.csail.mit.edu Subject: Re: 6042-student: Hanson Week 4 comments In-Reply-To: <433D520F.7010207@mit.edu> Message-ID: References: <433D520F.7010207@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 452 Content-Length: 399 X-Keywords: Read section 1.2 to understand that notation. Go to office hours if you are still confused. -Hanson On Fri, 30 Sep 2005, Alton Torregano wrote: > Could I have the section on bijective mappings, and their counting made > more explicit. Also, in relation to the tutor problems, could the > meaning of: > > RA=B > AR=B, etc...be explained more fully, as the syntax appeared rather foreign. > > ~ > From pjs@MIT.EDU Fri Sep 30 12:18:30 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 j8UGIUw0008880 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:30 -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 j8UGITrh006051 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:29 -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 j8UGILp0016426 for <6042-probs@theory.lcs.MIT.EDU>; Fri, 30 Sep 2005 12:18:22 -0400 (EDT) Received: (from nobody@localhost) by w92-130-webmail-4.mit.edu (8.12.4) id j8UGILZj032653; Fri, 30 Sep 2005 12:18:21 -0400 Received: from STRATTON-FOUR-TWENTY-EIGHT.MIT.EDU (STRATTON-FOUR-TWENTY-EIGHT.MIT.EDU [18.187.6.173]) (User authenticated as pjs@ATHENA.MIT.EDU) by webmail.mit.edu (Horde MIME library) with HTTP for ; Fri, 30 Sep 2005 12:18:21 -0400 Message-ID: <20050930121821.ldch0z2q034s4goo@webmail.mit.edu> Date: Fri, 30 Sep 2005 12:18:21 -0400 From: Paul J Steiner To: 6042-probs@theory.lcs.MIT.EDU Subject: email comments [sayan] 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: 453 Content-Length: 233 X-Keywords: NonJunk I found the notion of topological sorting on page 13 the most surprising in the notes. It allows for the surprisingly elegant notions of minimum/maximum on a graph, such as 1/0 for divisibility on the positive integers. PJ Steiner From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:22:50 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGMow0010575; Fri, 30 Sep 2005 12:22:50 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGMo3v014152; Fri, 30 Sep 2005 12:22:50 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGMoZp014149; Fri, 30 Sep 2005 12:22:50 -0400 Date: Fri, 30 Sep 2005 12:22:50 -0400 (EDT) From: Hanson Zhou To: Nancy L Keuss cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Week 4 comments In-Reply-To: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> Message-ID: References: <20050930105451.9n56e01s07c4ckc4@webmail.mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 454 Content-Length: 553 X-Keywords: Go to office hours to get this clarified. It is essentially the definition of partial order. -Hanson On Fri, 30 Sep 2005, Nancy L Keuss wrote: > Hi, > > On page 7 of the Week 4 notes, directly under equation (7) is some discussion > about the identity function. I do not understand why it follows that the > partial order binary relation of I_A is the same relation as the proper subset > relation. I wished there was more explanation between those two statements, as > I could not make the logical jump from one to the other. > > Thanks, > Nancy > From hmzhou@blackbird.csail.mit.edu Fri Sep 30 12:24:33 2005 Return-Path: Received: from blackbird.csail.mit.edu (blackbird.csail.mit.edu [128.30.51.72]) by theory.csail.mit.edu (8.12.10/8.12.1) with ESMTP id j8UGOXw0011446; Fri, 30 Sep 2005 12:24:33 -0400 Received: from blackbird.csail.mit.edu (localhost.localdomain [127.0.0.1]) by blackbird.csail.mit.edu (8.12.11/8.12.11) with ESMTP id j8UGOXjM014165; Fri, 30 Sep 2005 12:24:33 -0400 Received: from localhost (hmzhou@localhost) by blackbird.csail.mit.edu (8.12.11/8.12.11/Submit) with ESMTP id j8UGOXc2014162; Fri, 30 Sep 2005 12:24:33 -0400 Date: Fri, 30 Sep 2005 12:24:33 -0400 (EDT) From: Hanson Zhou To: Vera Carr cc: 6042-probs@theory.lcs.mit.edu Subject: Re: [Hanson] Binary Relations Reading In-Reply-To: <433D5BE5.7010309@mit.edu> Message-ID: References: <433D5BE5.7010309@mit.edu> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-UID: 455 Content-Length: 421 X-Keywords: Yes strict is irreflexive...I believe that's what it also says on page 7. -Hanson On Fri, 30 Sep 2005, Vera Carr wrote: > The section that I found most difficult to understand was the one on > Partial Orders (pg 7). I had difficulty recognizing them in the online > tutor. > > Also, when they are defining weak and strict partial orders for both > they say iff it is reflexive. For strict did they mean irreflexive? > From scholtz@MIT.EDU Fri Sep 30 13:35:48 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 j8UHZlw0026948 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:47 -0400 Received: from outgoing.mit.edu (OUTGOING-AUTH.MIT.EDU [18.7.22.103]) by biscayne-one-station.mit.edu (8.12.4/8.9.2) with ESMTP id j8UHZkrh022648 for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:46 -0400 (EDT) Received: from smed (MACGREGOR-ONE-O-FIVE.MIT.EDU [18.239.5.105]) (authenticated bits=0) (User authenticated as scholtz@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j8UHZdVu012450 (version=TLSv1/SSLv3 cipher=RC4-MD5 bits=128 verify=NOT) for <6042-probs@theory.lcs.mit.edu>; Fri, 30 Sep 2005 13:35:39 -0400 (EDT) Message-Id: <200509301735.j8UHZdVu012450@outgoing.mit.edu> Reply-To: From: "Eddie Scholtz" To: <6042-probs@theory.lcs.mit.edu> Subject: [Hanson] Week 4 Comments Date: Fri, 30 Sep 2005 13:35:38 -0400 Organization: MIT MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0000_01C5C5C3.D85FAA90" X-Mailer: Microsoft Office Outlook, Build 11.0.5510 Thread-Index: AcXF5V7dinHKt8iBQBGAULZqwMcDow== X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-Spam-Score: 2.371 X-Spam-Level: ** (2.371) X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 456 Content-Length: 1908 X-Keywords: NonJunk This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C5C5C3.D85FAA90 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit I found section 3.4 on page 9 to be most difficult. Its not very easy to understand by just reading it, but I think doing more practice problems will help me understand it a lot easier. ------=_NextPart_000_0000_01C5C5C3.D85FAA90 Content-Type: text/html; charset="us-ascii" Content-Transfer-Encoding: quoted-printable

I found section 3.4 on page 9 to be most = difficult. Its not very easy to understand by just reading it, but I think doing = more practice problems will help me understand it a lot = easier.

------=_NextPart_000_0000_01C5C5C3.D85FAA90-- From minilek@theory.csail.mit.edu Sat Oct 1 11:58:26 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 j91FwQw0011245 for <6042-probs@theory.lcs.mit.edu>; Sat, 1 Oct 2005 11:58:26 -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 j91FwNTF023261; Sat, 1 Oct 2005 11:58:23 -0400 (EDT) Received: from [18.250.6.245] (ASHDOWN-FIVE-HUNDRED.MIT.EDU [18.250.6.245]) (authenticated bits=0) (User authenticated as minilek@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j91FwHRX004195 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 1 Oct 2005 11:58:22 -0400 (EDT) Message-ID: <433EB23B.2030705@theory.csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:51 -0400 From: Jelani Nelson User-Agent: Mozilla Thunderbird 1.0.6 (X11/20050912) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Rachel Shearer Subject: Re: [Jelani] Week 4 Comment References: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> In-Reply-To: <5f81ece2a4a43a5294c91cd8f813405c@mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Spam-Score: -2.564 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 457 Content-Length: 746 X-Keywords: NonJunk Is it clearer now after Friday's lecture? One of the strategies, for instance, could be to count vertices of a certain degree (then apply what you proved in problem 1 in the class problems). This can only catch some non-isomorphic graphs and may produce false positives, but it can usually at least cut down on the n! different potential isomorphisms that need eliminating. -Jelani Rachel Shearer wrote: > I had a lot of trouble with the tutor problem relating to Isomorphism, > especially the one about how to prove that two graphs are not > isomorphic. I feel like it's actually not very hard but I am just > assuming it's more complicated than it actually is. Hopefully class > today will make things more clear. > > Rachel > From meyer@csail.mit.edu Sat Oct 1 11:58:40 2005 Return-Path: Received: from smtp105.mail.sc5.yahoo.com (smtp105.mail.sc5.yahoo.com [66.163.169.225]) by theory.csail.mit.edu (8.12.10/8.12.1) with SMTP id j91Fwdw0011253 for <6042-probs@theory.lcs.mit.edu>; Sat, 1 Oct 2005 11:58:39 -0400 Received: (qmail 64281 invoked from network); 1 Oct 2005 15:58:38 -0000 Received: from unknown (HELO ?192.168.1.100?) (armeyer10@71.245.230.197 with plain) by smtp105.mail.sc5.yahoo.com with SMTP; 1 Oct 2005 15:58:38 -0000 Message-ID: <433EB22E.5040000@csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:38 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yasin Ozcan CC: 6042-probs@theory.lcs.mit.edu, Hanson M Zhou Subject: Re: [hanson]reading comments References: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> In-Reply-To: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO X-UID: 458 Content-Length: 781 X-Keywords: The notation and answers to TP4.4 are given explicitly in Week 4 Notes, sections 1.2 & 1.3. Did you see that and not understand, or miss it? I have to agree that the Week 4 Notes were longer than I like (despite substantial cutting from earlier versions :-( ). But just saying "too long" is not the kind of comment we look for: we expect you to identify a specific topic that caught your attention. I'll look forward to a more detailed email comment next week. Regards, A. Yasin Ozcan wrote: > Hi, > > The only comments I have about this week's readings is that it was too > long. > > Also, I am a little bit confused by the notation in the True/False > questions in the Tutor problems, i.e. Question 4.4. Can we go over it > in the lecture > > thanks > > yasin From hmzhou@MIT.EDU Sat Oct 1 13:23:13 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 j91HNDw0021719 for <6042-probs@theory.csail.MIT.EDU>; Sat, 1 Oct 2005 13:23:13 -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 j91HNBTF025653; Sat, 1 Oct 2005 13:23:12 -0400 (EDT) Received: from all-night-tool.mit.edu (ALL-NIGHT-TOOL.MIT.EDU [18.7.16.70]) (authenticated bits=56) (User authenticated as hmzhou@ATHENA.MIT.EDU) by outgoing.mit.edu (8.13.1/8.12.4) with ESMTP id j91HN9Na013821 (version=TLSv1/SSLv3 cipher=DHE-RSA-AES256-SHA bits=256 verify=NOT); Sat, 1 Oct 2005 13:23:10 -0400 (EDT) Received: (from hmzhou@localhost) by all-night-tool.mit.edu (8.12.9) id j91HN9TR008305; Sat, 1 Oct 2005 13:23:09 -0400 (EDT) Date: Sat, 1 Oct 2005 13:23:09 -0400 (EDT) From: Hanson M Zhou To: ereid@MIT.EDU cc: 6042-probs@theory.csail.MIT.EDU Subject: Re: [Hanson] Reading Comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Score: -2.599 X-Spam-Flag: NO X-Scanned-By: MIMEDefang 2.42 Status: RO X-UID: 459 Content-Length: 872 X-Keywords: Depends on number of processors. If you have as many as you want, you are only constrained by the longest chain, and it is also the best you can do. Otherwise, it gets more complicated but I'm sure there are algorithms out there for parallel task scheduling. -Hanson On Fri, 30 Sep 2005, Elizabeth Reid wrote: > Sorry, I meant *minimize* the time. Is there any method that will > always find the shortest time possible, or is it just kind of logic > mixed with guess and check? > > > On Sep 30, 2005, at 12:12 PM, Hanson Zhou wrote: > > > Why would you want to maximize it? > > > > -Hanson > > > > On Fri, 30 Sep 2005, Elizabeth Reid wrote: > > > > > >> If they could talk a bit about parallel task scheduling in lecture > >> that would be nice. I was wondering if there was any mathematical way > >> to maximize it, or if it's more of a guess and check? > >> > > > > From meyer@imap.theory.csail.mit.edu Sat Oct 01 11:49:25 2005 Message-ID: <433EB004.2070606@csail.mit.edu> Date: Sat, 01 Oct 2005 11:49:24 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: Jelani Nelson Subject: Re: Jelani: Reading #4 References: <200509300509.j8U59WYL022496@outgoing.mit.edu> In-Reply-To: <200509300509.j8U59WYL022496@outgoing.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit Status: RO Content-Length: 350 X-UID: 460 X-Keywords: sounds like we should get together and talk about this. Are you free Wednesday after class? (I'd offer to meet sooner, but have religious holidays Mon eve through Wed afternoon). Clinton Blackburn wrote: > I’m just totally confused by this subject. I don’t know… > > --- > > Clinton Blackburn > > DLP – Have you seen it? > From meyer@imap.theory.csail.mit.edu Thu Sep 29 12:02:30 2005 Message-ID: <433C1019.6070903@csail.mit.edu> Date: Thu, 29 Sep 2005 12:02:33 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Jing He CC: 6042-probs@theory.lcs.MIT.EDU Subject: Re: [David] relations and graph theory reading comments References: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> In-Reply-To: <20050929010953.czhxhaunzi80wso4@webmail.mit.edu> Content-Type: text/plain; charset=GB2312 Content-Transfer-Encoding: 7bit Status: RO Content-Length: 693 X-UID: 461 X-Keywords: 4.16 says there is a chain of size t, or an antichain of size n/t, for ANY t. So let t=sqrt(n) , and you get a chain of size sqrt(n) or an antichain of size n/sqrt(n), which also equals sqrt(n). So you 're sure to have a sqrt(n) size chain or antichain, which is what 4.17 says. Does that help? regards, A. Jing He wrote: >The most difficult part of this section for me to understand was what a total >order was, since it wasn't given the formal definitions that the partial order >came with in the reading. I also didn't understand the reasoning for the >"proof" of Corollary 4.17 on page 16. The last section on graphs (and >applications to computing) was the most interesting. > > From meyer@imap.theory.csail.mit.edu Sat Oct 01 22:08:53 2005 BCC: Ronitt Rubinfeld Message-ID: <433F4136.2080206@csail.mit.edu> Date: Sat, 01 Oct 2005 22:08:54 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Clinton Blackburn CC: Jelani Nelson Subject: Re: Jelani: Reading #4 References: <200510011821.j91ILIFq020002@outgoing.mit.edu> In-Reply-To: <200510011821.j91ILIFq020002@outgoing.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 764 X-UID: 462 X-Keywords: OK, see you wed after class regards, A. Clinton Blackburn wrote: > Yes, sir. I am free. > > Thanks. > --- > Clinton Blackburn > DLP - Have you seen it? > http://www.dlp.com > > -----Original Message----- > From: Prof. Albert R. Meyer [mailto:meyer@csail.mit.edu] > Sent: Saturday, October 01, 2005 11:49 AM > To: Clinton Blackburn > Cc: Jelani Nelson > Subject: Re: Jelani: Reading #4 > > sounds like we should get together and talk about this. Are you free > Wednesday after class? (I'd offer to meet sooner, but have religious > holidays Mon eve through Wed afternoon). > > Clinton Blackburn wrote: > > >>I'm just totally confused by this subject. I don't know. >> >>--- >> >>Clinton Blackburn >> >>DLP - Have you seen it? >> From meyer@imap.theory.csail.mit.edu Sat Oct 01 11:58:39 2005 Message-ID: <433EB22E.5040000@csail.mit.edu> Date: Sat, 01 Oct 2005 11:58:38 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Yasin Ozcan CC: 6042-probs@theory.lcs.mit.edu, Hanson M Zhou Subject: Re: [hanson]reading comments References: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> In-Reply-To: <5.2.1.1.2.20050930001519.02477570@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 782 X-UID: 463 X-Keywords: The notation and answers to TP4.4 are given explicitly in Week 4 Notes, sections 1.2 & 1.3. Did you see that and not understand, or miss it? I have to agree that the Week 4 Notes were longer than I like (despite substantial cutting from earlier versions :-( ). But just saying "too long" is not the kind of comment we look for: we expect you to identify a specific topic that caught your attention. I'll look forward to a more detailed email comment next week. Regards, A. Yasin Ozcan wrote: > Hi, > > The only comments I have about this week's readings is that it was too > long. > > Also, I am a little bit confused by the notation in the True/False > questions in the Tutor problems, i.e. Question 4.4. Can we go over it > in the lecture > > thanks > > yasin From meyer@imap.theory.csail.mit.edu Thu Sep 29 18:17:10 2005 Message-ID: <433C67E9.1050704@csail.mit.edu> Date: Thu, 29 Sep 2005 18:17:13 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Chieu Nguyen CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Jelani] Week 4 comments References: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> In-Reply-To: <029201c5c542$a7b0ea70$e905ee12@addressisp.com> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 254 X-UID: 464 X-Keywords: Glad you think so, but we'd like more specific reactions to particular sections of the notes that caught your attention. I'll look forward to fuller comments next week. regards, A. Chieu Nguyen wrote: > (Pages 11-21) Graphs are cool. > > --Chieu From meyer@imap.theory.csail.mit.edu Thu Sep 29 18:14:32 2005 Message-ID: <433C674B.7010806@csail.mit.edu> Date: Thu, 29 Sep 2005 18:14:35 -0400 From: "Prof. Albert R. Meyer" Organization: MIT CSAIL User-Agent: Mozilla Thunderbird 1.0.6 (Windows/20050716) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Praveen Pamidimukkala CC: 6042-probs@theory.lcs.mit.edu Subject: Re: [Sayan] Week 4 Comments References: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> In-Reply-To: <6.2.1.2.2.20050929173734.01f08050@po12.mit.edu> Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: RO Content-Length: 789 X-UID: 465 X-Keywords: If they had the same elements, they'd be the same set, not two sets :-) But I grant you it would have been clearer to say "two distinct sets," and I'm editing the notes to say that right now. Thanks for the comment. Regards, A. Praveen Pamidimukkala wrote: > In Section 3.2, the last 2 sentences read, "So < and < are total > orders on [the set of all Real Numbers]. On the other hand, the subset > relation is generally not total: any two finite sets of the same size > will be incomparable under [the contained-in-or-equal subset > relation]." Don't they mean just the [proper subset relation]? If > the two finite sets contain the exact same elements, then aren't they > comparable under [the contain-in-or-equal subset relation]? > > --Praveen Pamidimukkala > Team 7