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2024-03-28T16:58:55Z
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http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_November_26
15-344/Classnotes for Thursday November 26
2015-11-27T16:24:02Z
<p>Roroyap: </p>
<hr />
<div>{{15-344/Navigation}}<br />
<br />
{{15-344:Dror/Students Divider}}<br />
<br />
== Scanned Lecture Notes for November 26 == <br />
<gallery><br />
15-344_Nov_26_Lecture_(1).jpg | Page 1<br />
15-344_Nov_26_Lecture_(2).jpg | Page 2<br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_Nov_26_Lecture_(2).jpg
File:15-344 Nov 26 Lecture (2).jpg
2015-11-27T16:23:26Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_Nov_26_Lecture_(1).jpg
File:15-344 Nov 26 Lecture (1).jpg
2015-11-27T16:22:24Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_November_19
15-344/Classnotes for Thursday November 19
2015-11-23T17:09:51Z
<p>Roroyap: /* Handwritten Lecture Notes in PDF */</p>
<hr />
<div>{{15-344/Navigation}}<br />
<br />
{{15-344:Dror/Students Divider}}<br />
<br />
== Handwritten Lecture Notes in PDF ==<br />
[[Media:15-344(lecture18).PDF|MAT344 - Lecture18 (Nov 19)]]<br />
<br />
== Scanned Tutorial Notes for September 17 == <br />
<gallery><br />
15-344 Tutorial Week 10 (1).jpg|Page 1<br />
15-344 Tutorial Week 10 (2).jpg|Page 2<br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_Tutorial_Week_10_(2).jpg
File:15-344 Tutorial Week 10 (2).jpg
2015-11-23T17:09:38Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_Tutorial_Week_10_(1).jpg
File:15-344 Tutorial Week 10 (1).jpg
2015-11-23T17:09:12Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_November_19
15-344/Classnotes for Thursday November 19
2015-11-20T15:28:53Z
<p>Roroyap: /* Handwritten Lecture Notes in PDF */</p>
<hr />
<div>{{15-344/Navigation}}<br />
<br />
{{15-344:Dror/Students Divider}}<br />
<br />
== Handwritten Lecture Notes in PDF ==<br />
[[Media:15-344(lecture18).PDF|MAT344 - Lecture18 (Nov 19)]]<br />
<br />
== Scanned Tutorial Notes for November 19 ==<br />
<gallery><br />
November 19 Tutorial 1.jpg|Page 1<br />
November 19 Tutorial 2.jpeg|Page 2<br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Navigation
15-344/Navigation
2015-10-22T23:22:01Z
<p>Roroyap: </p>
<hr />
<div><noinclude>Back to [[15-344]].<br/></noinclude><br />
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"<br />
|- align=center style="color: red;"<br />
|colspan=3|'''Welcome to Math 344!'''<br />
|- align=left<br />
!#<br />
!Week of...<br />
!Notes and Links<br />
|- align=left<br />
|align=center|1<br />
|Sep 14<br />
|[[15-344/About This Class|About This Class]], {{Pensieve link|Classes/15-344-Combinatorics/DayOne.html|Day One Handout}}, [[15-344/Classnotes for Tuesday September 15|Tuesday]], {{Pensieve link|Classes/15-344-Combinatorics/Hour3.html|Hour 3 Handout}}, [[15-344/Classnotes for Thursday September 17|Thursday]], {{Pensieve link|Classes/15-344-Combinatorics/Tutorial1.jpg|Tutorial 1 Handout}}<br />
|- align=left<br />
|align=center|2<br />
|Sep 21<br />
|[[15-344/Classnotes for Tuesday September 22|Tuesday]], {{Pensieve link|Classes/15-344-Combinatorics/Tutorial2-1.jpg|Tutorial 2 Page 1}}, {{Pensieve link|Classes/15-344-Combinatorics/Tutorial2-2.jpg|Tutorial 2 Page 2}}, [[15-344/Classnotes for Thursday September 24|Thursday]], [[15-344/Homework Assignment 1|HW1]]<br />
|- align=left<br />
|align=center|3<br />
|Sep 28<br />
|[[15-344/Classnotes for Tuesday September 29|Tuesday]], [[15-344/Class Photo|Class Photo]], {{Pensieve link|Classes/15-344-Combinatorics/TutorialOct1-1.jpg|Tutorial 3 Page 1}}, {{Pensieve link|Classes/15-344-Combinatorics/TutorialOct1-2.jpg|Tutorial 3 Page 2}}, [[15-344/Classnotes for Thursday October 1|Thursday]], [[15-344/Homework Assignment 2|HW2]]<br />
|- align=left<br />
|align=center|4<br />
|Oct 5<br />
|[[15-344/Classnotes for Tuesday October 6|Tuesday]], {{Pensieve link|Classes/15-344-Combinatorics/nb/Cubes.pdf|Drawing <math>n</math>-cubes}}, [[15-344/Classnotes for Thursday October 8|Thursday]], [[15-344/Homework Assignment 3|HW3]]<br />
|- align=left<br />
|align=center|5<br />
|Oct 12<br />
|[[15-344/Classnotes for Tuesday October 13|Tuesday]], {{Pensieve link|Classes/15-344-Combinatorics/TutorialOct15.jpg|Tutorial Handout}}, [[15-344/Homework Assignment 4|HW4]]<br />
|- align=left<br />
|align=center|6<br />
|Oct 19<br />
|{{Pensieve link|Classes/15-344-Combinatorics/TutorialOct22.jpg|Tutorial Handout}},[[15-344/Classnotes for Thursday October 22|Thursday]]<br />
|- align=left<br />
|align=center|7<br />
|Oct 26<br />
|[[15-344/Homework Assignment 5|HW5]], [[15-344/Term Test|Term Test]] on Tuesday<br />
|- align=left<br />
|align=center|8<br />
|Nov 2<br />
|Sunday November 8 is the last day to drop this class, [[15-344/Homework Assignment 6|HW6]]<br />
|- align=left<br />
|align=center|9<br />
|Nov 9<br />
|Monday-Tuesday is UofT Fall Break, [[15-344/Homework Assignment 7|HW7]]<br />
|- align=left<br />
|align=center|10<br />
|Nov 16<br />
|[[15-344/Homework Assignment 8|HW8]]<br />
|- align=left<br />
|align=center|11<br />
|Nov 23<br />
|[[15-344/Homework Assignment 9|HW9]]<br />
|- align=left<br />
|align=center|12<br />
|Nov 30<br />
|[[15-344/Homework Assignment 10|HW10]]<br />
|- align=left<br />
|align=center|13<br />
|Dec 7<br />
|Semester ends on Wednesday - no class Thursday<br />
|- align=left<br />
|align=center|F<br />
|Dec 11-22<br />
|[[15-344/The Final Exam|The Final Exam]]<br />
|- align=left<br />
|colspan=3 align=center|[[15-344/Register of Good Deeds|Register of Good Deeds]]<br />
|- align=left<br />
|colspan=3 align=center|[[Image:15-344-ClassPhoto.jpg|310px|Class Photo]]<br/>[[15-344/Class Photo|Add your name / see who's in!]]<br />
|- align=left<br />
|colspan=3 align=center|[[Image:15-344-Splash.png|310px]]<br />
|}</div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_October_22.jpg
File:15-344 October 22.jpg
2015-10-22T23:19:25Z
<p>Roroyap: Roroyap uploaded a new version of &quot;File:15-344 October 22.jpg&quot;</p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_October_22
15-344/Classnotes for Thursday October 22
2015-10-22T23:17:53Z
<p>Roroyap: Created page with "== Scanned Lecture Notes for October 22 == <gallery> 15-344_October_22.jpg|Lecture </gallery>"</p>
<hr />
<div>== Scanned Lecture Notes for October 22 == <br />
<gallery><br />
15-344_October_22.jpg|Lecture <br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_October_22.jpg
File:15-344 October 22.jpg
2015-10-22T23:17:05Z
<p>Roroyap: </p>
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<div></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_September_17
15-344/Classnotes for Thursday September 17
2015-10-09T03:18:47Z
<p>Roroyap: /* Scanned Lecture Note for September 17 */</p>
<hr />
<div>{{15-344/Navigation}}<br />
== Lecture Notes for September 17 ==<br />
'''DEFINITION 7 Isomorphism''' A graph <math>G_1 = (V_1, E_1)</math> is called ''isomorphic'' to a graph <math>G_2 = (V_2, E_2)</math> whenever<br />
there exists a bijection <math>\phi:V_1 \rightarrow V_2</math> such that <math>\forall a,b\in V_1</math> we have <math>(ab)\in E_1</math> if and <br />
only if <math>( \phi (a) \phi (b))\in E_2</math>. <math>G_1\sim G_2</math> means they are isomorphic to each other.<br />
<br />
*A ''bijection'' is a one-to-one and on-to function. https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection<br />
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*Isomorphism does not mean two things are identical but means they are mathematically the same. <br />
<br />
The relationship of isomorphisms:<br />
<br />
1. '''Reflexive''': <math>G \sim G</math> A graph is isomorphic to itself<br />
<br />
2. '''Symmetric:''' <math>G_1 \sim G_2 \implies G_2 \sim G_1 </math> In other words, for every <math>\phi:V_1 \rightarrow V_2</math> we have <br />
<math>\phi ^{-1} :V_2 \rightarrow V_1</math><br />
<br />
3. '''Transitive:''' <math>G_1 \sim G_2 , G_2 \sim G_3 \implies G_1 \sim G_3 </math><br />
<br />
CLAIM If two graphs are isomorphic, then they have:<br />
<br />
1. same number of vertices<br />
<br />
2. same number of edges<br />
<br />
3. vertex degrees (valencies) are the same between the two. For example, if one graph has 3 vertices of degree 2, and 2 vertices of degree 1, <br />
then the other graph should have the same<br />
<br />
4. same number of subgraphs<br />
<br />
5. same number of complements denoted by <math>G^c = (V,E^c)</math> <br />
<br />
*Complement means <math>(ab)\in E^c \Longleftrightarrow (ab)\notin E</math><br />
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'''DEFINITION 8''' '''Subgraph''' A subgraph of a graph <math>G = (V,E)</math> is a graph <math>G' = (V',E')</math> such that <math>V'\subset V</math> and<br />
<math>E'\subset E</math>.<br />
<br />
*Checking if two graphs are isomorphic is a hard problem<br />
<br />
<br />
<br />
== Scanned Lecture Note for September 17 ==<br />
<br />
<gallery><br />
Image:15-344_Note_2.jpg|Answers of sample question<br />
Image:15-344-Sept17-1.jpg|Class notes page 1<br />
Image:15-344-Sept17-2.jpg|Class notes page 2<br />
Image:15-344-Sept17-3.jpg|Class notes page 3<br />
</gallery><br />
<br />
== Scanned Tutorial Notes for September 17 == <br />
<br />
<gallery><br />
15-344 Tutorial 1.jpg|Page 1<br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_Tutorial_1.jpg
File:15-344 Tutorial 1.jpg
2015-10-09T03:16:25Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_October_1
15-344/Classnotes for Thursday October 1
2015-10-02T06:50:26Z
<p>Roroyap: /* Scanned Lecture Notes for October 1 */</p>
<hr />
<div>== Scanned Lecture Notes for October 1 == <br />
<br />
<br />
<gallery><br />
15-344 October 1 Page 1.jpg|Page 1<br />
15-344 October 1 Page 2.jpg|Page 2<br />
</gallery></div>
Roroyap
http://drorbn.net/index.php?title=15-344/Classnotes_for_Thursday_October_1
15-344/Classnotes for Thursday October 1
2015-10-02T06:49:44Z
<p>Roroyap: Created page with "== Scanned Lecture Notes for October 1 == File:15-344 October 1 Page 1.jpg File:15-344 October 1 Page 2.jpg"</p>
<hr />
<div>== Scanned Lecture Notes for October 1 == <br />
<br />
[[File:15-344 October 1 Page 1.jpg]]<br />
[[File:15-344 October 1 Page 2.jpg]]</div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_October_1_Page_2.jpg
File:15-344 October 1 Page 2.jpg
2015-10-02T06:49:01Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap
http://drorbn.net/index.php?title=File:15-344_October_1_Page_1.jpg
File:15-344 October 1 Page 1.jpg
2015-10-02T06:47:58Z
<p>Roroyap: </p>
<hr />
<div></div>
Roroyap