Drorbn - User contributions [en]

15-344/Classnotes for Tuesday October 20

2015-10-23T02:25:07Z

Zetalda: Created page with "{{15-344/Navigation}} {{15-344:Dror/Students Divider}} == Scanned Lecture Notes for October 20 == <gallery> Image:15-344-Oct20-1.jpg|Page 1 Image:15-344-Oct20-2.jpg|Page 2 ..."

{{15-344/Navigation}}

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

== Scanned Lecture Notes for October 20 ==

<gallery>
Image:15-344-Oct20-1.jpg|Page 1
Image:15-344-Oct20-2.jpg|Page 2

</gallery>

15-344/Classnotes for Thursday October 15

2015-10-23T02:24:20Z

Zetalda: Created page with "{{15-344/Navigation}} {{15-344:Dror/Students Divider}} == Scanned Lecture Notes for October 15 == <gallery> Image:15-344-Oct15-1.jpg|Page 1 Image:15-344-Oct15-2.jpg|Page 2 ..."

{{15-344/Navigation}}

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

== Scanned Lecture Notes for October 15 ==

<gallery>
Image:15-344-Oct15-1.jpg|Page 1
Image:15-344-Oct15-2.jpg|Page 2

</gallery>

File:15-344-Oct20-2.jpg

2015-10-23T02:22:58Z

Zetalda:

File:15-344-Oct20-1.jpg

2015-10-23T02:22:25Z

Zetalda:

File:15-344-Oct15-2.jpg

2015-10-23T02:21:49Z

Zetalda:

File:15-344-Oct15-1.jpg

2015-10-23T02:20:32Z

Zetalda:

15-344/Navigation

2015-10-23T02:19:18Z

Zetalda:

<noinclude>Back to [[15-344]]. </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/Classnotes for Thursday October 15|Thursday]], [[15-344/Homework Assignment 4|HW4]]
|- align=left
|align=center|6
|Oct 19
|[[15-344/Classnotes for Tuesday October 20|Tuesday]], {{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]] [[15-344/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:15-344-Splash.png|310px]]
|}

15-344/Classnotes for Tuesday October 13

2015-10-18T23:14:10Z

Zetalda: Created page with "{{15-344/Navigation}} {{15-344:Dror/Students Divider}} == Scanned Lecture Notes for October 13 == <gallery> Image:15-344-Oct13-1.jpg|Page 1 Image:15-344-Oct13-2.jpg|Page 2 ..."

{{15-344/Navigation}}

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

== Scanned Lecture Notes for October 13 ==

<gallery>
Image:15-344-Oct13-1.jpg|Page 1
Image:15-344-Oct13-2.jpg|Page 2
Image:15-344-Oct13-3.jpg|Page 3

</gallery>

15-344/Navigation

2015-10-18T23:12:55Z

Zetalda:

<noinclude>Back to [[15-344]]. </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
|
|- align=left
|align=center|7
|Oct 26
|[[15-344/Homework Assignment 5|HW5]], [[15-344/Term Test|Term Test]] at classtime 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]] [[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-Oct13-3.jpg

2015-10-18T22:22:59Z

Zetalda:

File:15-344-Oct13-2.jpg

2015-10-18T22:22:11Z

Zetalda:

File:15-344-Oct13-1.jpg

2015-10-18T22:19:39Z

Zetalda:

15-344/Class Photo

2015-10-18T19:38:36Z

Zetalda:

Our class on October 1, 2015:

[[Image:15-344-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{15-344/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}


{{Photo Entry|last=Lin|first=Tong|userid=TongLin|email=tong.lin@ mail. utoronto. ca|location= on the very left of the second row from the front, in a gray and black sweater|comments=}}

{{Photo Entry|last=Rimmer|first=Jack|userid=Jack.rimmer|email=jack.rimmer@ mail. utoronto. ca|location= 2nd row from the front, one person from the left, grey t-shirt and glasses. Looks a bit sleepy.|comments=}}

{{Photo Entry|last=Semovski|first=Julian|userid=Semovski|email=julian.semovski@ mail. utoronto. ca|location= 3rd row from the front, very left, blue and black stripe sweater. Struggling with life|comments=}}

{{Photo Entry|last=Yoo|first=Seungmin|userid=minheather|email=seungmin.yoo@ mail. utoronto. ca|location= left side of a third row from the back. A girl with short hair|comments=}}

{{Photo Entry|last=Zhang|first=Amy|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location= second row, first person from the right, girl in a white top.|comments=}}

{{Photo Entry|last=Zibert|first=Vincent|userid=orange|email=vincent.zibert@ mail. utoronto. ca|location= front row in the white shirt|comments=}}

User talk:Zetalda

2015-10-05T14:31:45Z

Zetalda: /* Re */ new section

Hi, there!
Sorry to bother you but I have a favor to ask you if you are available..
My name is Amy and I am taking MAT344 course with you this term.
Just having trouble uploading the scanned lecture notes.. It is my first time ever in my life editing a website like this.
So, if you have time, do you mind teaching me how to upload the notes starting from the very very basic steps?
Appreciate it!

== Re ==

Hi Amy,

The only thing worth noting is the 'upload file' button in the 'toolbox' located on the left hand side of the screen. Once you uploaded your notes, just follow the source code of other pages to edit the desired page.

Cheers

15-344/Classnotes for Tuesday September 29

2015-10-01T02:30:54Z

Zetalda:

{{15-344/Navigation}}

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

== Scanned Lecture Notes for September 29 ==

<gallery>
Image:15-344-Sept29-1.jpg|Page 1
Image:15-344-Sept29-2.jpg|Page 2
Image:15-344-Sept29-3.jpg|Page 3

</gallery>

File:15-344-Sept29-3.jpg

2015-10-01T02:30:30Z

Zetalda:

File:15-344-Sept29-2.jpg

2015-10-01T02:29:55Z

Zetalda:

File:15-344-Sept29-1.jpg

2015-10-01T02:29:26Z

Zetalda: Zetalda uploaded a new version of "File:15-344-Sept29-1.jpg"

File:15-344-Sept29-1.jpg

2015-10-01T02:29:14Z

Zetalda:

15-344/Classnotes for Tuesday September 29

2015-10-01T02:28:45Z

Zetalda: Created page with "{{15-344/Navigation}} {{15-344:Dror/Students Divider}} == Scanned Lecture Notes for September 29 == <gallery> Image:15-344-Sept22-1.jpg|Page 1 Image:15-344-Sept22-2.jpg|Pag..."

{{15-344/Navigation}}

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

== Scanned Lecture Notes for September 29 ==

<gallery>
Image:15-344-Sept22-1.jpg|Page 1
Image:15-344-Sept22-2.jpg|Page 2
Image:15-344-Sept22-3.jpg|Page 3
Image:15-344-Sept22-4.jpg|Page 4

</gallery>

15-344/Classnotes for Thursday September 24

2015-10-01T02:26:59Z

Zetalda: Created page with "{{15-344/Navigation}} {{15-344:Dror/Students Divider}} == Scanned Lecture Notes for September 24 == <gallery> Image:15-344-Sept24-1.jpg|Page 1 Image:15-344-Sept24-2.jpg|Pa..."

{{15-344/Navigation}}

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

== Scanned Lecture Notes for September 24 ==

<gallery>
Image:15-344-Sept24-1.jpg|Page 1
Image:15-344-Sept24-2.jpg|Page 2

</gallery>

15-344/Navigation

2015-10-01T02:25:16Z

Zetalda:

<noinclude>Back to [[15-344]]. </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/Homework Assignment 2|HW2]], [[15-344/Class Photo|Class Photo]]
|- align=left
|align=center|4
|Oct 5
|[[15-344/Homework Assignment 3|HW3]]
|- align=left
|align=center|5
|Oct 12
|[[15-344/Homework Assignment 4|HW4]]
|- align=left
|align=center|6
|Oct 19
|
|- align=left
|align=center|7
|Oct 26
|[[15-344/Homework Assignment 5|HW5]], [[15-344/Term Test|Term Test]] in class 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]] [[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-Sept24-2.jpg

2015-10-01T02:23:20Z

Zetalda:

File:15-344-Sept24-1.jpg

2015-10-01T02:22:30Z

Zetalda:

15-344/Classnotes for Tuesday September 22

2015-09-23T03:58:16Z

Zetalda: /* Scanned Lecture Note for September 22 */

{{15-344/Navigation}}

The Mathematica notebook we viewed is {{Pensieve link|Classes/15-344-Combinatorics/nb/MountainClimbing.pdf|MountainClimbing.pdf}}.

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

== Scanned Lecture Notes for September 22 ==

<gallery>
Image:15-344-Sept22-1.jpg|Page 1
Image:15-344-Sept22-2.jpg|Page 2
Image:15-344-Sept22-3.jpg|Page 3
Image:15-344-Sept22-4.jpg|Page 4

</gallery>

15-344/Classnotes for Tuesday September 15

2015-09-23T03:57:54Z

Zetalda:

{{15-344/Navigation}}

We mostly went over {{Pensieve link|Classes/15-344-Combinatorics/DayOne.html|Day One Handout}} today.

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

== Lecture Note for September 15 ==

'''DEFINITION 1''' '''Graph''' A graph <math>G = (V,E)</math> is a set <math>V = \{a,b,...\}</math> (usually finite, "vertices")
along with a set <math>E = \{(ab),(bc),(bd),...\}</math> ("edges") of unordered pairs of distinct elements of
<math>V</math>.

'''DEFINITION 2''' '''Incident''' If an edge <math>e = (ab)\in{E}</math>, we say that <math>e</math> is incident to <math>a</math>
''and'' <math>b</math>.

'''DEFINITION 3''' '''N-valent''' In a graph <math>G = (V,E)</math>, a vertex <math>u \in{V}</math> is called
bivalent if it is incident to precisely two edges and n-valent if incident to precisely n edges, where <math>n = 0,1,2,.. </math>.

'''DEFINITION 4''' '''Edge Cover''' An edge cover for graph <math>G = (V,E)</math> is a subset <math>C\subset{V}</math> such that every edge of <math>G</math> incident to at least one vertex in <math>C</math>.

'''DEFINITION 5''' '''Independent''' Let <math>G = (V,E)</math> be a graph. A subset <math>I\subset{V}</math> is called independent if whenever <math>a,b\in{I}</math>, then <math>(ab)\notin{E}</math>.

'''THEOREM 1''' Edge covers are complementary to independent sets. In other words, <math>C\subset{V}</math> is an edge cover if and only if
the complementary subset <math>V-C</math> is an independent set.

'''''Proof'''''
<math>(\rightarrow)</math> Assume <math>C</math> is an edge cover. I assert that <math>I = V-C</math> is independent.
Indeed, if <math>e=(ab)\in{E}</math>, then since <math>C</math> is an edge cover, either <math>a\in{C} \implies a\notin{I}</math> or
<math>b\in{C} \implies b\notin{I}</math>, which implies <math>e</math> does not connect any two elements of <math>I</math>.

<math>(\leftarrow)</math> Assume <math>I=V-C</math> is independent. Pick any edge <math>e = (ab)\in{E}</math>. As <math>I</math> is independent,
<math>(ab)</math> does not connect any two members of <math>I</math>. Hence, either <math>a\notin{I} \implies a\in{C}</math> or <math>b\notin{I} \implies b\in{C}</math>, which implies <math>e</math> is incident to an element of <math>C</math>. QED

'''DEFINITION 6''' '''Matching''' is a subset of edges of a graph where no two edges share a common vertex. A matching is said to be perfect if
all edges in a graph are matched.

== Scanned Lecture Notes for September 15 ==

<gallery>
Image:15-344-Sept15-1.jpg|Page 1
Image:15-344-Sept15-2.jpg|Page 2
Image:15-344_Note_1.jpg|Summary
</gallery>

15-344/Classnotes for Tuesday September 22

2015-09-23T03:57:17Z

Zetalda:

{{15-344/Navigation}}

The Mathematica notebook we viewed is {{Pensieve link|Classes/15-344-Combinatorics/nb/MountainClimbing.pdf|MountainClimbing.pdf}}.

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

== Scanned Lecture Note for September 22 ==

<gallery>
Image:15-344-Sept22-1.jpg|Page 1
Image:15-344-Sept22-2.jpg|Page 2
Image:15-344-Sept22-3.jpg|Page 3
Image:15-344-Sept22-4.jpg|Page 4

</gallery>

File:15-344-Sept22-4.jpg

2015-09-23T03:54:33Z

Zetalda:

File:15-344-Sept22-3.jpg

2015-09-23T03:53:13Z

Zetalda:

File:15-344-Sept22-2.jpg

2015-09-23T03:52:20Z

Zetalda:

File:15-344-Sept22-1.jpg

2015-09-23T03:51:49Z

Zetalda:

15-344/Classnotes for Tuesday September 15

2015-09-16T13:07:32Z

Zetalda: /* Lecture Note for September 15 */

{{15-344/Navigation}}

We mostly went over {{Pensieve link|Classes/15-344-Combinatorics/DayOne.html|Day One Handout}} today.

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

== Lecture Note for September 15 ==

'''DEFINITION 1''' '''Graph''' A graph <math>G = (V,E)</math> is a set <math>V = \{a,b,...\}</math> (usually finite, "vertices")
along with a set <math>E = \{(ab),(bc),(bd),...\}</math> ("edges") of unordered pairs of distinct elements of
<math>V</math>.

'''DEFINITION 2''' '''Incident''' If an edge <math>e = (ab)\in{E}</math>, we say that <math>e</math> is incident to <math>a</math>
''and'' <math>b</math>.

'''DEFINITION 3''' '''N-valent''' In a graph <math>G = (V,E)</math>, a vertex <math>u \in{V}</math> is called
bivalent if it is incident to precisely two edges and n-valent if incident to precisely n edges, where <math>n = 0,1,2,.. </math>.

'''DEFINITION 4''' '''Edge Cover''' An edge cover for graph <math>G = (V,E)</math> is a subset <math>C\subset{V}</math> such that every edge of <math>G</math> incident to at least one vertex in <math>C</math>.

'''DEFINITION 5''' '''Independent''' Let <math>G = (V,E)</math> be a graph. A subset <math>I\subset{V}</math> is called independent if whenever <math>a,b\in{I}</math>, then <math>(ab)\notin{E}</math>.

'''THEOREM 1''' Edge covers are complementary to independent sets. In other words, <math>C\subset{V}</math> is an edge cover if and only if
the complementary subset <math>V-C</math> is an independent set.

'''''Proof'''''
<math>(\rightarrow)</math> Assume <math>C</math> is an edge cover. I assert that <math>I = V-C</math> is independent.
Indeed, if <math>e=(ab)\in{E}</math>, then since <math>C</math> is an edge cover, either <math>a\in{C} \implies a\notin{I}</math> or
<math>b\in{C} \implies b\notin{I}</math>, which implies <math>e</math> does not connect any two elements of <math>I</math>.

<math>(\leftarrow)</math> Assume <math>I=V-C</math> is independent. Pick any edge <math>e = (ab)\in{E}</math>. As <math>I</math> is independent,
<math>(ab)</math> does not connect any two members of <math>I</math>. Hence, either <math>a\notin{I} \implies a\in{C}</math> or <math>b\notin{I} \implies b\in{C}</math>, which implies <math>e</math> is incident to an element of <math>C</math>. QED

== Scanned Lecture Note for September 15 ==

<gallery>
Image:15-344-Sept15-1.jpg|Page 1
Image:15-344-Sept15-2.jpg|Page 2
</gallery>

File:15-344-Sept15-2.jpg

2015-09-16T13:05:38Z

Zetalda:

File:15-344-Sept15-1.jpg

2015-09-16T13:04:50Z

Zetalda:

12-240/Classnotes for Tuesday November 20

2012-11-21T03:26:06Z

Zetalda:

12-240/Navigation

2012-11-21T03:25:22Z

Zetalda:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, [[12-240/Homework Assignment 7|HW7]], [[12-240/Classnotes for Tuesday November 15|Thursday]]
|- align=left
|align=center|11
|Nov 19
|[[12-240/Homework Assignment 8|HW8]], [[12-240/Classnotes for Tuesday November 20|Tuesday]]
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

File:12-240-Nov20-4.jpeg

2012-11-21T03:23:59Z

Zetalda:

File:12-240-Nov20-3.jpeg

2012-11-21T03:23:50Z

Zetalda:

File:12-240-Nov20-2.jpeg

2012-11-21T03:23:39Z

Zetalda:

File:12-240-Nov20-1.jpeg

2012-11-21T03:23:30Z

Zetalda:

12-240/Navigation

2012-11-21T02:56:19Z

Zetalda:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, [[12-240/Homework Assignment 7|HW7]], [[12-240/Classnotes for Tuesday November 15|Thursday]]
|- align=left
|align=center|11
|Nov 19
|[[12-240/Homework Assignment 8|HW8]], [[12-240/Classnotes for Tuesday November 6|Tuesday]]
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/Classnotes for Tuesday November 15

2012-11-16T01:42:32Z

Zetalda:

{{12-240/Navigation}}
'''Problem.''' Find the rank the matrix
<center><math>A=\begin{pmatrix}0&2&4&2&2\\4&4&4&8&0\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>.</center>

'''Solution.''' Using (invertible!) row/column operations we aim to bring <math>A</math> to look as close as possible to an identity matrix:

{| border="1px" cellspadding="5" cellspacing=0 style="font-size:90%;"
|+
|align=center|'''Do'''
|align=center|'''Get'''
|- valign=top
|1. Bring a <math>1</math> to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by <math>1/4</math>.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\8&2&0&10&2\\6&3&2&9&1\end{pmatrix}</math>
|- valign=top
|2. Add <math>(-8)</math> times the first row to the third row, in order to cancel the <math>8</math> in position 3-1.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&-6&-8&-6&2\\6&3&2&9&1\end{pmatrix}</math>
|- valign=top
|3. Likewise add <math>(-6)</math> times the first row to the fourth row, in order to cancel the <math>6</math> in position 4-1.
|align=center|<math>\begin{pmatrix}1&1&1&2&0\\0&2&4&2&2\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|- valign=top
|4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling).
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&2&4&2&2\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|- valign=top
|5. Turn the 2-2 entry to a <math>1</math> by multiplying the second row by <math>1/2</math>.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&-6&-8&-6&2\\0&-3&-4&-3&1\end{pmatrix}</math>
|- valign=top
|6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" <math>1</math> at position 2-2.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&2&1&1\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}</math>
|- valign=top
|7. Using three column operations clean the second row except the pivot.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&4&0&8\\0&0&2&0&4\end{pmatrix}</math>
|- valign=top
|8. Clean up the row and the column of the <math>4</math> in position 3-3 by first multiplying the third row by <math>1/4</math> and then performing the appropriate row and column transformations. Notice that by pure luck, the <math>4</math> at position 4-5 of the matrix gets killed in action.
|align=center|<math>\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&0&0\end{pmatrix}</math>
|}
Thus the rank of our matrix is 3.

== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==
<gallery>
Image:12-240-Nov15-1.jpeg|Page 1
Image:12-240-Nov15-2.jpeg|Page 2
</gallery>

File:12-240-Nov15-2.jpeg

2012-11-16T01:41:38Z

Zetalda:

File:12-240-Nov15-1.jpeg

2012-11-16T01:41:28Z

Zetalda:

12-240/Navigation

2012-11-09T14:10:38Z

Zetalda:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]], [[12-240/Classnotes for Tuesday November 8|Thursday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, HW7 on web Wednesday
|- align=left
|align=center|11
|Nov 19
|HW8
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}

12-240/Classnotes for Tuesday November 6

2012-11-07T02:55:40Z

Zetalda:

== Lecture notes scanned by [[User:Zetalda|Zetalda]] ==
<gallery>
Image:12-240-Nov6-1.jpeg|Page 1
Image:12-240-Nov6-2.jpeg|Page 2
Image:12-240-Nov6-3.jpeg|Page 3
</gallery>

File:12-240-Nov6-3.jpeg

2012-11-07T02:53:53Z

Zetalda:

File:12-240-Nov6-2.jpeg

2012-11-07T02:53:41Z

Zetalda:

File:12-240-Nov6-1.jpeg

2012-11-07T02:53:32Z

Zetalda:

12-240/Navigation

2012-11-07T02:53:05Z

Zetalda:

<noinclude>Back to [[12-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 10
|[[12-240/About This Class|About This Class]], [[12-240/Classnotes for Tuesday September 11|Tuesday]], [[12-240/Classnotes for Thursday September 13|Thursday]]
|- align=left
|align=center|2
|Sep 17
|[[12-240/Homework Assignment 1|HW1]], [[12-240/Classnotes for Tuesday September 18|Tuesday]], [[12-240/Classnotes for Thursday September 20|Thursday]]
|- align=left
|align=center|3
|Sep 24
|[[12-240/Homework Assignment 2|HW2]], [[12-240/Classnotes for Tuesday September 25|Tuesday]], [[12-240/Class Photo|Class Photo]], [[12-240/Classnotes for Thursday September 27|Thursday]]
|- align=left
|align=center|4
|Oct 1
|[[12-240/Homework Assignment 3|HW3]], [[12-240/Classnotes for Tuesday October 2|Tuesday]], [[12-240/Classnotes for Thursday October 4|Thursday]]
|- align=left
|align=center|5
|Oct 8
|[[12-240/Homework Assignment 4|HW4]], [[12-240/Classnotes for Tuesday October 09|Tuesday]], [[12-240/Classnotes for Thursday October 11|Thursday]]
|- align=left
|align=center|6
|Oct 15
|[[12-240/Classnotes for Tuesday October 16|Tuesday]], [[12-240/Classnotes for Thursday October 18|Thursday]]
|- align=left
|align=center|7
|Oct 22
|[[12-240/Homework Assignment 5|HW5]], [[12-240/Classnotes for Tuesday October 23|Tuesday]], [[12-240/Term Test|Term Test]] was on Thursday.
|- align=left
|align=center|8
|Oct 29
|[[12-240/Linear Algebra - Why We Care|Why LinAlg?]], [[12-240/Homework Assignment 6|HW6]], [[12-240/Classnotes for Tuesday October 30|Tuesday]], [[12-240/Classnotes for Thursday November 1|Thursday]], Nov 4 is the last day to drop this class
|- align=left
|align=center|9
|Nov 5
|[[12-240/Classnotes for Tuesday November 6|Tuesday]]
|- align=left
|align=center|10
|Nov 12
|Monday-Tuesday is UofT November break, HW7 on web Wednesday
|- align=left
|align=center|11
|Nov 19
|HW8
|- align=left
|align=center|12
|Nov 26
|HW9
|- align=left
|align=center|13
|Dec 3
|UofT Fall Semester ends Wednesday
|- align=left
|align=center|F1
|Dec 10
|Finals
|- align=left
|align=center|F2
|Dec 17
|Finals
|- align=left
|colspan=3 align=center|[[12-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:12-240-ClassPhoto.jpg|310px]] [[12-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:12-240-Splash.png|310px]]
|}