Drorbn - User contributions [en]

15-344/Classnotes for Thursday November 26

2015-11-27T16:24:02Z

Roroyap:

{{15-344/Navigation}}

{{15-344:Dror/Students Divider}}

== Scanned Lecture Notes for November 26 ==
<gallery>
15-344_Nov_26_Lecture_(1).jpg | Page 1
15-344_Nov_26_Lecture_(2).jpg | Page 2
</gallery>

File:15-344 Nov 26 Lecture (2).jpg

2015-11-27T16:23:26Z

Roroyap:

File:15-344 Nov 26 Lecture (1).jpg

2015-11-27T16:22:24Z

Roroyap:

15-344/Classnotes for Thursday November 19

2015-11-23T17:09:51Z

Roroyap: /* Handwritten Lecture Notes in PDF */

{{15-344/Navigation}}

{{15-344:Dror/Students Divider}}

== Handwritten Lecture Notes in PDF ==
[[Media:15-344(lecture18).PDF|MAT344 - Lecture18 (Nov 19)]]

== Scanned Tutorial Notes for September 17 ==
<gallery>
15-344 Tutorial Week 10 (1).jpg|Page 1
15-344 Tutorial Week 10 (2).jpg|Page 2
</gallery>

File:15-344 Tutorial Week 10 (2).jpg

2015-11-23T17:09:38Z

Roroyap:

File:15-344 Tutorial Week 10 (1).jpg

2015-11-23T17:09:12Z

Roroyap:

15-344/Classnotes for Thursday November 19

2015-11-20T15:28:53Z

Roroyap: /* Handwritten Lecture Notes in PDF */

{{15-344/Navigation}}

{{15-344:Dror/Students Divider}}

== Handwritten Lecture Notes in PDF ==
[[Media:15-344(lecture18).PDF|MAT344 - Lecture18 (Nov 19)]]

== Scanned Tutorial Notes for November 19 ==
<gallery>
November 19 Tutorial 1.jpg|Page 1
November 19 Tutorial 2.jpeg|Page 2
</gallery>

15-344/Navigation

2015-10-22T23:22:01Z

Roroyap:

<noinclude>Back to [[15-344]].<br/></noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 344!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 14
|[[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}}
|- align=left
|align=center|2
|Sep 21
|[[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]]
|- align=left
|align=center|3
|Sep 28
|[[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]]
|- align=left
|align=center|4
|Oct 5
|[[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]]
|- align=left
|align=center|5
|Oct 12
|[[15-344/Classnotes for Tuesday October 13|Tuesday]], {{Pensieve link|Classes/15-344-Combinatorics/TutorialOct15.jpg|Tutorial Handout}}, [[15-344/Homework Assignment 4|HW4]]
|- align=left
|align=center|6
|Oct 19
|{{Pensieve link|Classes/15-344-Combinatorics/TutorialOct22.jpg|Tutorial Handout}},[[15-344/Classnotes for Thursday October 22|Thursday]]
|- align=left
|align=center|7
|Oct 26
|[[15-344/Homework Assignment 5|HW5]], [[15-344/Term Test|Term Test]] on Tuesday
|- align=left
|align=center|8
|Nov 2
|Sunday November 8 is the last day to drop this class, [[15-344/Homework Assignment 6|HW6]]
|- align=left
|align=center|9
|Nov 9
|Monday-Tuesday is UofT Fall Break, [[15-344/Homework Assignment 7|HW7]]
|- align=left
|align=center|10
|Nov 16
|[[15-344/Homework Assignment 8|HW8]]
|- align=left
|align=center|11
|Nov 23
|[[15-344/Homework Assignment 9|HW9]]
|- align=left
|align=center|12
|Nov 30
|[[15-344/Homework Assignment 10|HW10]]
|- align=left
|align=center|13
|Dec 7
|Semester ends on Wednesday - no class Thursday
|- align=left
|align=center|F
|Dec 11-22
|[[15-344/The Final Exam|The Final Exam]]
|- align=left
|colspan=3 align=center|[[15-344/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:15-344-ClassPhoto.jpg|310px|Class Photo]]<br/>[[15-344/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:15-344-Splash.png|310px]]
|}

File:15-344 October 22.jpg

2015-10-22T23:19:25Z

Roroyap: Roroyap uploaded a new version of "File:15-344 October 22.jpg"

15-344/Classnotes for Thursday October 22

2015-10-22T23:17:53Z

Roroyap: Created page with "== Scanned Lecture Notes for October 22 == <gallery> 15-344_October_22.jpg|Lecture </gallery>"

== Scanned Lecture Notes for October 22 ==
<gallery>
15-344_October_22.jpg|Lecture
</gallery>

File:15-344 October 22.jpg

2015-10-22T23:17:05Z

Roroyap:

15-344/Classnotes for Thursday September 17

2015-10-09T03:18:47Z

Roroyap: /* Scanned Lecture Note for September 17 */

{{15-344/Navigation}}
== Lecture Notes for September 17 ==
'''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
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
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.

*A ''bijection'' is a one-to-one and on-to function. https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

*Isomorphism does not mean two things are identical but means they are mathematically the same.

The relationship of isomorphisms:

1. '''Reflexive''': <math>G \sim G</math> A graph is isomorphic to itself

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
<math>\phi ^{-1} :V_2 \rightarrow V_1</math>

3. '''Transitive:''' <math>G_1 \sim G_2 , G_2 \sim G_3 \implies G_1 \sim G_3 </math>

CLAIM If two graphs are isomorphic, then they have:

1. same number of vertices

2. same number of edges

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,
then the other graph should have the same

4. same number of subgraphs

5. same number of complements denoted by <math>G^c = (V,E^c)</math>

*Complement means <math>(ab)\in E^c \Longleftrightarrow (ab)\notin E</math>

'''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
<math>E'\subset E</math>.

*Checking if two graphs are isomorphic is a hard problem

== Scanned Lecture Note for September 17 ==

<gallery>
Image:15-344_Note_2.jpg|Answers of sample question
Image:15-344-Sept17-1.jpg|Class notes page 1
Image:15-344-Sept17-2.jpg|Class notes page 2
Image:15-344-Sept17-3.jpg|Class notes page 3
</gallery>

== Scanned Tutorial Notes for September 17 ==

<gallery>
15-344 Tutorial 1.jpg|Page 1
</gallery>

File:15-344 Tutorial 1.jpg

2015-10-09T03:16:25Z

Roroyap:

15-344/Classnotes for Thursday October 1

2015-10-02T06:50:26Z

Roroyap: /* Scanned Lecture Notes for October 1 */

== Scanned Lecture Notes for October 1 ==

<gallery>
15-344 October 1 Page 1.jpg|Page 1
15-344 October 1 Page 2.jpg|Page 2
</gallery>

15-344/Classnotes for Thursday October 1

2015-10-02T06:49:44Z

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"

== Scanned Lecture Notes for October 1 ==

[[File:15-344 October 1 Page 1.jpg]]
[[File:15-344 October 1 Page 2.jpg]]

File:15-344 October 1 Page 2.jpg

2015-10-02T06:49:01Z

Roroyap:

File:15-344 October 1 Page 1.jpg

2015-10-02T06:47:58Z

Roroyap: