Drorbn - User contributions [en]

09-240/Classnotes for Thursday December 3

2009-12-05T04:07:02Z

Kjgry: Ignore

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_December_3rd.pdf|A set of the December 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto. Please note that this set is missing a diagram that could not be converted to PDF form.
Image:Lecture Notes for Dec 3 Pg 1.JPG
Image:Lecture Notes for Dec 3 Pg 2.JPG
</gallery>

~In the above gallery, there is a copy of notes for the lecture given on December 3rd by Professor Natan (in PDF format). Please note that there is an important diagram missing from this file. This is because the diagram was too complex to coherently convert to PDF. Other notes must be referenced for this diagram (otherwise, the PDF is complete)

09-240/Classnotes for Tuesday December 1

2009-12-05T04:06:22Z

Kjgry:

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_December_1st.pdf|A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.
Image:Dec 1 lecture notes Pg 1.JPG|
Image:Dec 1 lecture notes Pg 2.JPG|
Image:Dec 1 lecture notes Pg 3.JPG|
Image:Dec 1 lecture notes Pg 4.JPG|
Image:Dec 1 lecture notes Pg 5.JPG|
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Image:dec1-4.jpg|
Image:dec1-5.jpg|
Image:dec1-6.jpg|
Image:dec1-7.jpg|
Image:dec1-8.jpg|
</gallery>

~In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

--- Wiki Format ---

MAT240 – December 1st

Basic Properties of det: Mnxn→F: 0 det(I) = 1

1. <math>det(E'_{i,j\,\!}A) = -det(A) ; |E'_{i,j\,\!}|= -1. [Note: det(EA) = |E||A|]</math>

* Also, note that exchanging two rows flips the sign.

2. <math>det(E^2_{i,c\,\!}A) = det(A) ; |E^2_{i,j,c\,\!}| = 1</math>

* These are "enough"!

3. <math>det((E_{i,j,c\,\!}A) = det(A) ; |E^3_{i,j,c\,\!}| = 1</math>

* Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If <math> det' : M_{nxn\,\!}</math>→F satisfies properties 0-3 above, then <math>det' = det</math>

<math>det(A) = det'(A)</math>

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function <math>det</math> exists first.

09-240/Classnotes for Tuesday December 1

2009-12-05T04:05:38Z

Kjgry: Uploaded PDF for Dec 1st and created a gallery.

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_December_1st.pdf|A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.
Image:Dec 1 lecture notes Pg 1.JPG|
Image:Dec 1 lecture notes Pg 2.JPG|
Image:Dec 1 lecture notes Pg 3.JPG|
Image:Dec 1 lecture notes Pg 4.JPG|
Image:Dec 1 lecture notes Pg 5.JPG|
Image:dec1-1.jpg|
Image:dec1-2.jpg|
Image:dec1-3.jpg|
Image:dec1-4.jpg|
Image:dec1-5.jpg|
Image:dec1-6.jpg|
Image:dec1-7.jpg|
Image:dec1-8.jpg|
</gallery>

In the above gallery, there is a complete copy of notes for the lecture given on December 1st by Professor Natan (in PDF format).

--- Wiki Format ---

MAT240 – December 1st

Basic Properties of det: Mnxn→F: 0 det(I) = 1

1. <math>det(E'_{i,j\,\!}A) = -det(A) ; |E'_{i,j\,\!}|= -1. [Note: det(EA) = |E||A|]</math>

* Also, note that exchanging two rows flips the sign.

2. <math>det(E^2_{i,c\,\!}A) = det(A) ; |E^2_{i,j,c\,\!}| = 1</math>

* These are "enough"!

3. <math>det((E_{i,j,c\,\!}A) = det(A) ; |E^3_{i,j,c\,\!}| = 1</math>

* Adding a multiple of one row to another does not change the determinant.

The determinant of any matrix can be calculated using the properties above.

Theorem:

If <math> det' : M_{nxn\,\!}</math>→F satisfies properties 0-3 above, then <math>det' = det</math>

<math>det(A) = det'(A)</math>

Philosophical remark: Why not begin our inquiry with the properties above?

We must find an implied need for their use; thus, we must know whether a function <math>det</math> exists first.

File:ALA240-2009 - December 1st.pdf

2009-12-05T04:03:51Z

Kjgry: A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the December 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday December 3

2009-12-05T04:02:50Z

Kjgry: Uploaded PDF for Dec 3rd, created gallery and added a caution.

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_December_3rd.pdf|A set of the December 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto. Please note that this set is missing a diagram that could not be converted to PDF form.
Image:Lecture Notes for Dec 3 Pg 1.JPG
Image:Lecture Notes for Dec 3 Pg 2.JPG
</gallery>

In the above gallery, there is a copy of notes for the lecture given on December 3rd by Professor Natan (in PDF format). Please note that there is an important diagram missing from this file. This is because the diagram was too complex to coherently convert to PDF. Other notes must be referenced for this diagram (otherwise, the PDF is complete)

File:ALA240-2009 - December 3rd.pdf

2009-12-05T03:54:04Z

Kjgry: A complete set of the December 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the December 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday November 3

2009-12-05T03:53:23Z

Kjgry: Uploaded PDF for Nov 3rd, created gallery

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_3rd.pdf|A complete set of the November 26th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto
Image:09-240-3nov-1.jpg|
Image:09-240-3nov-2.jpg|
Image:09-240-3nov-3.jpg|
Image:09-240-3nov-4.jpg|
Image:09-240-3nov-5.jpg|
</gallery>

In the above gallery, there is a complete copy of notes for the lecture given on November 3rd by Professor Natan (in PDF format).

09-240/Classnotes for Tuesday November 17

2009-12-05T03:52:12Z

Kjgry:

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_17th.pdf|A complete set of the November 17th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.
Image:Lecture Notes for Nov 17 Page 1.JPG|
Image:Lecture Notes for Nov 17 Page 4.JPG|
Image:Lecture Notes for Nov 17 Page 2.JPG|
Image:Lecture Notes for Nov 17 Page 3.JPG|
Image:Lecture Notes for Nov 17 Page 5.JPG|
</gallery>

In the above gallery, there is a complete copy of notes for the lecture given on November 17th by Professor Natan (in PDF format).

File:ALA240-2009 - November 3rd.pdf

2009-12-05T03:50:25Z

Kjgry: A complete set of the November 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 3rd lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday November 17

2009-12-05T03:49:33Z

Kjgry: Uploaded notes for Nov 17th and created a gallery.

09-240/Classnotes for Thursday November 19

2009-12-05T03:48:47Z

Kjgry:

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_19th.pdf
Image:Notes for Nov 19 Page 1b.JPG|
Image:Notes for Nov 19 Page 2b.JPG|
Image:Notes for Nov 19 Page 3.JPG|
</gallery>
In the above gallery, there is a complete copy of notes for the lecture given on November 19th by Professor Natan (in PDF format).
 
 

'''Tutorial Notes: Alan's group'''
 
 
<gallery>
Image:Image002.jpg
Image:Image003.jpg
</gallery>

09-240/Classnotes for Thursday November 19

2009-12-05T03:48:31Z

Kjgry:

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_19th.pdf
Image:Notes for Nov 19 Page 1b.JPG|600px
Image:Notes for Nov 19 Page 2b.JPG|600px
Image:Notes for Nov 19 Page 3.JPG|600px
</gallery>
In the above gallery, there is a complete copy of notes for the lecture given on November 19th by Professor Natan (in PDF format).
 
 

'''Tutorial Notes: Alan's group'''
 
 
<gallery>
Image:Image002.jpg
Image:Image003.jpg
</gallery>

File:ALA240-2009 - November 17th.pdf

2009-12-05T03:45:50Z

Kjgry: A complete set of the November 17th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 17th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday November 5

2009-11-27T21:27:12Z

Kjgry: Uploaded PDF, added captions

{{09-240/Navigation}}
{{09-240/Class Notes Warning}}

<gallery>
Image:ALA240_-_November_5th_(PDF).PDF|A complete copy of notes for the lecture given on November 5th by Professor Natan (in PDF format).
Image:ALA240-2009_-_November_5th(Tutorial).pdf|A complete copy of notes for the tutorial given on November 5th by Nevena Francetic (in PDF format).
Image:Notes for Nov 5 Page 1.JPG
Image:Notes for Nov 5 Page 2.JPG
</gallery>

(In the above gallery, there is a complete copy of notes for the lecture given on November 5th by Professor Natan, as well as a one of the tutorials.)

File:ALA240-2009 - November 5th(Tutorial).pdf

2009-11-27T21:24:07Z

Kjgry: A complete set of the November 5th tutorial notes given by Nevena Francetic for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 5th tutorial notes given by Nevena Francetic for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday September 15

2009-11-27T02:44:33Z

Kjgry: Update file caption

{{09-240/Navigation}}
{{09-240/Class Notes Warning}}

<gallery>
Image:09-240 Classnotes for Tuesday September 15 2009 page 1.jpg|[[User:Yangjiay|Yangjiay]] - Page 1
Image:09-240 Classnotes for Tuesday September 15 2009 page 2.jpg|Page 2
Image:09-240 Classnotes for Tuesday September 15 2009 page 3.jpg|Page 3
Image:09-240 Classnotes for Tuesday September 15 2009 page 4.jpg|Page 4
Image:09-240 Classnotes for Tuesday September 15 2009 page 5.jpg|Page 5
Image:ALA240-2009_-_September_15th.pdf|A complete copy of notes for the lecture given on September 15th by Professor Natan (in PDF format)
</gallery>
(In the above gallery, there is a complete set of notes for the lecture given by Professor Natan on September 15th in PDF form.)

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>
# <math>\,\!F_7 = \{ 0, 1,2,3,4,5,6 \}</math>
# <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field because not every element has a multiplicative inverse.
#: Let <math>a = 2.</math>
#: Then <math>a \cdot 0 = 0, a \cdot 1 = 2, a \cdot 3 = 0, a \cdot 4 = 2, a \cdot 5 = 4</math>
#: Therefore F4 fails; there is '''no''' number ''b'' in ''F''6 s.t. ''a · b'' = 1

{|
|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! + !! 0 !! 1
|-
! 0
| 0 || 1
|-
! 1
| 1 || 0
|-
|}

|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! × !! 0 !! 1
|-
! 0
| 0 || 0
|-
! 1
| 0 || 1
|}

|-
|
{| border="1" cellspacing="0"
|+ Ex. 5
|-
! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 6 || 0
|-
! 2
| 2 || 3 || 4 || 5 || 6 || 0 || 1
|-
! 3
| 3 || 4 || 5 || 6 || 0 || 1 || 2
|-
! 4
| 4 || 5 || 6 || 0 || 1 || 2 || 3
|-
! 5
| 5 || 6 || 0 || 1 || 2 || 3 || 4
|-
! 6
| 6 || 0 || 1 || 2 || 3 || 4 || 5
|}

|
{|border="1" cellspacing="0"
|+ Ex. 5
|-
! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
! 1
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 2
| 0 || 2 || 4 || 6 || 1 || 3 || 5
|-
! 3
| 0 || 3 || 6 || 2 || 5 || 1 || 4
|-
! 4
| 0 || 4 || 1 || 5 || 2 || 6 || 3
|-
! 5
| 0 || 5 || 3 || 1 || 6 || 4 || 2
|-
! 6
| 0 || 6 || 5 || 4 || 3 || 2 || 1
|}
|}

'''Theorem''': ''F''2 is a field.

In order to prove that the associative property holds, make a table (similar to a [http://en.wikipedia.org/wiki/Truth_table truth table]) for ''a'', ''b'' and ''c''.

{| border="1" cellspacing="0" style="text-align: center;"
! a !! b !! c !!  
|-
| 0 || 0 || 0 ||  
|-
| 0 || 0 || 1 ||  
|-
| 0 || 1 || 0 ||  
|-
| 0 || 1 || 1 || (0 + 1) + 1 =? 0 + (1 + 1) 1 + 1 =? 0 + 0 0 = 0
|-
| 1 || 0 || 0 ||  
|-
| 1 || 0 || 1 ||  
|-
| 1 || 1 || 0 ||  
|-
| 1 || 1 || 1 ||  
|}

'''Theorem''': <math>\,\! F_p </math> for <math>p > 1</math> is a field ''iff'' ([http://en.wikipedia.org/wiki/If_and_only_if if and only if]) <math>p</math> is a prime number

Proof:

<math>a \in \mathbb Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.

This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:

: <math>\exists x, y \mbox{ s.t. } ax + my = 1</math>
: <math>\left(ax + my\right) \pmod{m} = 1\pmod{m}</math>
: <math>ax = 1</math>
: <math>x = a^{-1}</math>

We have shown that <math>a</math> has a multiplicative inverse modulo m if <math>a</math> and <math>m</math> are relatively prime. It is therefore a natural conclusion that if <math>m</math> is a prime number all elements in the set will be relatively prime to m.

----

Multiplication is repeated addition.

<math>23 \times 27 = \begin{matrix} 27 \\ \overbrace{23 + 23 + 23 + \cdots + 23} \end{matrix} = 621</math>

<math>27 \times 23 = \begin{matrix} 23 \\ \overbrace{27 + 27 + 27 + \cdots + 27} \end{matrix} = 621</math>

One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.

Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 23 = 8, but 32 = 9.

== Tedious Theorem ==
# <math>a + b = c + b \Rightarrow a = c </math> "cancellation property"
#: Proof:
#: By F4, <math>\exists d \mbox{ s.t. } b + d = 0</math>
#: <math>\,\! (a + b) + d = (c + b) + d</math>
#: <math>\Rightarrow a + (b + d) = c + (b + d)</math> by F2
#: <math>\Rightarrow a + 0 = c + 0</math> by choice of ''d''
#: <math>\Rightarrow a = c</math> by F3
# <math> a \cdot b = c \cdot b , (b \ne 0) \Rightarrow a = c </math>
# <math>a + O' = a \Rightarrow O' = 0</math>
#: Proof:
#: <math>\,\! a + O' = a</math>
#: <math>\Rightarrow a + O' = a + 0</math> by F3
#: <math>\Rightarrow O' = 0</math> by adding the additive inverse of ''a'' to both sides
# <math>a \cdot l' = a, a \ne 0 \Rightarrow l' = 1</math>
# <math>a + b = 0 = a + b' \Rightarrow b = b'</math>
# <math>a \cdot b = 1 = a \cdot b' \Rightarrow b = b' = a^{-1}</math>
#: <math>\,\! \mbox{Aside: } a - b = a + (-b)</math>
#: <math>\frac ab = a \cdot b^{-1}</math>
# <math>\,\! -(-a) = a = (a^{-1})^{-1}</math>
# <math>a \cdot 0 = 0</math>
#: Proof:
#: <math>a \cdot 0 = a(0 + 0)</math> by F3
#: <math>= a \cdot 0 + a \cdot 0</math> by F5
#: <math>= 0 = a \cdot 0</math>
# <math>\forall b, 0 \cdot b \ne 1</math>
#: So there is no 0−1
# <math>(-a) \cdot b = a \cdot (-b) = -(a \cdot b)</math>
# <math>(-a) \cdot (-b) = a \cdot b</math>
# (Bonus) <math>\,\! (a + b)(a - b) = a^2 - b^2</math>

== Quotation of the Day ==
......

09-240/Classnotes for Tuesday November 24

2009-11-27T02:36:33Z

Kjgry: Reorder

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_24th.pdf|A complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format)
Image:09240_Nov24_Notes_Page_1.png|600px
Image:09240_Nov24_Notes_Page_2.png|600px
Image:09240_Nov24_Notes_Page_3.png|600px
Image:09240_Nov24_Notes_Page_4.png|600px
Image:09240_Nov24_Notes_Page_5.png|600px
</gallery>

In the above gallery, there is a complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format).

09-240/Classnotes for Thursday October 29

2009-11-27T02:36:06Z

Kjgry: Updated file, ordered gallery.

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_October_29th.pdf|A complete copy of notes for the lecture given on October 29th by Professor Natan (in PDF format)
Image:Image-Oct. 29th classnotes pg0.jpg
Image:Oct. 29th classnotes pg1.jpg
</gallery>

A complete set of notes for the lecture given by Professor Natan on October 29th is included in the above gallery (in PDF form).

09-240/Classnotes for Thursday November 5

2009-11-27T02:33:39Z

Kjgry: Updated file caption, ordered gallery

{{09-240/Navigation}}
{{09-240/Class Notes Warning}}

<gallery>
Image:ALA240_-_November_5th_(PDF).PDF|A complete copy of notes for the lecture given on November 5th by Professor Natan (in PDF format)
Image:Notes for Nov 5 Page 1.JPG
Image:Notes for Nov 5 Page 2.JPG
</gallery>

(In the above gallery, there is a complete copy of notes for the lecture given on November 5th by Professor Natan.)

09-240/Classnotes for Thursday October 1

2009-11-27T02:32:24Z

Kjgry: Updated file information.

{{09-240/Navigation}}

==A Message from Accessibility Services==
:"Accessibility Services requires dependable volunteer note-takers in this course to assist students with disabilities. Those who are interested in assisting with this essential service will gain valuable volunteer experience and a certificate of recognition. If you are interested in becoming a volunteer note-taker, please take an information form and register online, or visit the Accessibility Services office at 215 Huron Street."

You can also sign up online, at http://www.accessibility.utoronto.ca/newreturn/note_taking_accommodation.htm.

{{09-240/Class Notes Warning}}
==Class notes for today==
<gallery>
Image:09-240_OCT01_NOTES.JPG
Image:ALA240-2009 - October 1st.pdf|A complete copy of notes for the lecture given on October 1st by Professor Natan (in PDF format)
</gallery>
(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan (in PDF format).

09-240/Classnotes for Tuesday November 24

2009-11-27T02:31:25Z

Kjgry: Minor change

{{09-240/Navigation}}
<gallery>
Image:09240_Nov24_Notes_Page_1.png|600px
Image:09240_Nov24_Notes_Page_2.png|600px
Image:09240_Nov24_Notes_Page_3.png|600px
Image:09240_Nov24_Notes_Page_4.png|600px
Image:09240_Nov24_Notes_Page_5.png|600px
Image:ALA240-2009_-_November_24th.pdf|A complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format)
</gallery>

In the above gallery, there is a complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format).

09-240/Classnotes for Thursday November 26

2009-11-27T02:30:57Z

Kjgry: Uploaded PDF, created an ordered gallery.

{{09-240/Navigation}}
<gallery>
Image:ALA240-2009_-_November_26th.pdf|A complete copy of notes for the lecture given on November 26th by Professor Natan (in PDF format)
Image:09-240-26.10-1.jpg
Image:09-240-26.10-2.jpg
Image:09-240-26.10-3.jpg
</gallery>
In the above gallery, there is a complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format).

File:ALA240-2009 - November 26th.pdf

2009-11-27T02:27:49Z

Kjgry: A complete set of the November 26th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 26th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday November 24

2009-11-26T04:07:41Z

Kjgry:

{{09-240/Navigation}}
<gallery>
Image:09240_Nov24_Notes_Page_1.png|600px
Image:09240_Nov24_Notes_Page_2.png|600px
Image:09240_Nov24_Notes_Page_3.png|600px
Image:09240_Nov24_Notes_Page_4.png|600px
Image:09240_Nov24_Notes_Page_5.png|600px
Image:ALA240-2009_-_November_24th.pdf|A complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format)
</gallery>

(In the above gallery, there is a complete copy of notes for the lecture given on November 24th by Professor Natan (in PDF format).

09-240/Classnotes for Tuesday November 24

2009-11-26T04:07:09Z

Kjgry:

File:ALA240-2009 - November 24th.pdf

2009-11-26T04:03:33Z

Kjgry: A complete set of the November 24th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 24th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday November 19

2009-11-26T02:18:21Z

Kjgry:

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(In the above gallery, there is a complete copy of notes for the lecture given on November 19th by Professor Natan (in PDF format).
 
 

'''Tutorial Notes: Alan's group'''
 
 
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09-240/Classnotes for Thursday November 19

2009-11-26T02:17:39Z

Kjgry:

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(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan (in PDF format).
 
 

'''Tutorial Notes: Alan's group'''
 
 
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09-240/Classnotes for Thursday October 1

2009-11-26T02:13:55Z

Kjgry:

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==Class notes for today==
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(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan (in PDF format).

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2009-11-26T02:11:58Z

Kjgry: A complete set of the November 19th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 19th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday October 1

2009-11-16T20:53:00Z

Kjgry:

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{{09-240/Class Notes Warning}}
==Class notes for today==
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(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan (in PDF format).)

09-240/Classnotes for Tuesday November 10

2009-11-16T20:51:01Z

Kjgry:

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(In the above gallery, there is a complete copy of notes for the lecture given on November 10th by Professor Natan.)

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2009-11-16T20:49:34Z

Kjgry: A complete set of the November 10th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the November 10th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday November 10

2009-11-14T00:57:06Z

Kjgry:

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(In the above gallery, there is a complete copy of notes for the lecture given on November 5th by Professor Natan.)

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2009-11-14T00:54:35Z

Kjgry: A complete set of the October 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the October 1st lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday October 8

2009-11-08T19:18:44Z

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A complete set of notes for the lecture given by Professor Natan on October 8th is included in the above gallery (in PDF form).

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2009-11-08T19:14:19Z

Kjgry: A complete set of the October 8th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the October 8th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday October 29

2009-11-08T03:54:11Z

Kjgry:

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A complete set of notes for the lecture given by Professor Natan on October 29th is included in the above gallery (in PDF form).

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2009-11-08T03:48:50Z

Kjgry: A complete set of the October 30th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

A complete set of the October 30th lecture notes given by Professor Dror Bar-Natan for the Fall Session of MAT240 at the University of Toronto.

09-240/Classnotes for Thursday November 5

2009-11-06T01:16:31Z

Kjgry:

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09-240/Classnotes for Tuesday September 15

2009-11-06T01:15:38Z

Kjgry:

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(In the above gallery, there is a complete set of notes for the lecture given by Professor Natan on September 15th in PDF form.)

The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>
# <math>\,\!F_7 = \{ 0, 1,2,3,4,5,6 \}</math>
# <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field because not every element has a multiplicative inverse.
#: Let <math>a = 2.</math>
#: Then <math>a \cdot 0 = 0, a \cdot 1 = 2, a \cdot 3 = 0, a \cdot 4 = 2, a \cdot 5 = 4</math>
#: Therefore F4 fails; there is '''no''' number ''b'' in ''F''6 s.t. ''a · b'' = 1

{|
|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! + !! 0 !! 1
|-
! 0
| 0 || 1
|-
! 1
| 1 || 0
|-
|}

|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! × !! 0 !! 1
|-
! 0
| 0 || 0
|-
! 1
| 0 || 1
|}

|-
|
{| border="1" cellspacing="0"
|+ Ex. 5
|-
! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 6 || 0
|-
! 2
| 2 || 3 || 4 || 5 || 6 || 0 || 1
|-
! 3
| 3 || 4 || 5 || 6 || 0 || 1 || 2
|-
! 4
| 4 || 5 || 6 || 0 || 1 || 2 || 3
|-
! 5
| 5 || 6 || 0 || 1 || 2 || 3 || 4
|-
! 6
| 6 || 0 || 1 || 2 || 3 || 4 || 5
|}

|
{|border="1" cellspacing="0"
|+ Ex. 5
|-
! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
! 1
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 2
| 0 || 2 || 4 || 6 || 1 || 3 || 5
|-
! 3
| 0 || 3 || 6 || 2 || 5 || 1 || 4
|-
! 4
| 0 || 4 || 1 || 5 || 2 || 6 || 3
|-
! 5
| 0 || 5 || 3 || 1 || 6 || 4 || 2
|-
! 6
| 0 || 6 || 5 || 4 || 3 || 2 || 1
|}
|}

'''Theorem''': ''F''2 is a field.

In order to prove that the associative property holds, make a table (similar to a [http://en.wikipedia.org/wiki/Truth_table truth table]) for ''a'', ''b'' and ''c''.

{| border="1" cellspacing="0" style="text-align: center;"
! a !! b !! c !!  
|-
| 0 || 0 || 0 ||  
|-
| 0 || 0 || 1 ||  
|-
| 0 || 1 || 0 ||  
|-
| 0 || 1 || 1 || (0 + 1) + 1 =? 0 + (1 + 1) 1 + 1 =? 0 + 0 0 = 0
|-
| 1 || 0 || 0 ||  
|-
| 1 || 0 || 1 ||  
|-
| 1 || 1 || 0 ||  
|-
| 1 || 1 || 1 ||  
|}

'''Theorem''': <math>\,\! F_p </math> for <math>p > 1</math> is a field ''iff'' ([http://en.wikipedia.org/wiki/If_and_only_if if and only if]) <math>p</math> is a prime number

Proof:

<math>a \in \mathbb Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.

This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:

: <math>\exists x, y \mbox{ s.t. } ax + my = 1</math>
: <math>\left(ax + my\right) \pmod{m} = 1\pmod{m}</math>
: <math>ax = 1</math>
: <math>x = a^{-1}</math>

We have shown that <math>a</math> has a multiplicative inverse modulo m if <math>a</math> and <math>m</math> are relatively prime. It is therefore a natural conclusion that if <math>m</math> is a prime number all elements in the set will be relatively prime to m.

----

Multiplication is repeated addition.

<math>23 \times 27 = \begin{matrix} 27 \\ \overbrace{23 + 23 + 23 + \cdots + 23} \end{matrix} = 621</math>

<math>27 \times 23 = \begin{matrix} 23 \\ \overbrace{27 + 27 + 27 + \cdots + 27} \end{matrix} = 621</math>

One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.

Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 23 = 8, but 32 = 9.

== Tedious Theorem ==
# <math>a + b = c + b \Rightarrow a = c </math> "cancellation property"
#: Proof:
#: By F4, <math>\exists d \mbox{ s.t. } b + d = 0</math>
#: <math>\,\! (a + b) + d = (c + b) + d</math>
#: <math>\Rightarrow a + (b + d) = c + (b + d)</math> by F2
#: <math>\Rightarrow a + 0 = c + 0</math> by choice of ''d''
#: <math>\Rightarrow a = c</math> by F3
# <math> a \cdot b = c \cdot b , (b \ne 0) \Rightarrow a = c </math>
# <math>a + O' = a \Rightarrow O' = 0</math>
#: Proof:
#: <math>\,\! a + O' = a</math>
#: <math>\Rightarrow a + O' = a + 0</math> by F3
#: <math>\Rightarrow O' = 0</math> by adding the additive inverse of ''a'' to both sides
# <math>a \cdot l' = a, a \ne 0 \Rightarrow l' = 1</math>
# <math>a + b = 0 = a + b' \Rightarrow b = b'</math>
# <math>a \cdot b = 1 = a \cdot b' \Rightarrow b = b' = a^{-1}</math>
#: <math>\,\! \mbox{Aside: } a - b = a + (-b)</math>
#: <math>\frac ab = a \cdot b^{-1}</math>
# <math>\,\! -(-a) = a = (a^{-1})^{-1}</math>
# <math>a \cdot 0 = 0</math>
#: Proof:
#: <math>a \cdot 0 = a(0 + 0)</math> by F3
#: <math>= a \cdot 0 + a \cdot 0</math> by F5
#: <math>= 0 = a \cdot 0</math>
# <math>\forall b, 0 \cdot b \ne 1</math>
#: So there is no 0−1
# <math>(-a) \cdot b = a \cdot (-b) = -(a \cdot b)</math>
# <math>(-a) \cdot (-b) = a \cdot b</math>
# (Bonus) <math>\,\! (a + b)(a - b) = a^2 - b^2</math>

== Quotation of the Day ==
......

09-240/Classnotes for Thursday November 5

2009-11-06T01:14:11Z

Kjgry:

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In the above gallery, there is a complete copy of the notes given on November 5th by Professor Natan in PDF form.

File:ALA240 - November 5th (PDF).PDF

2009-11-06T01:08:11Z

Kjgry: Full notes for the November 5th lecture given by Professor Dror Bar-Natan for the Fall 2009 course MAT240 at the University of Toronto.

Full notes for the November 5th lecture given by Professor Dror Bar-Natan for the Fall 2009 course MAT240 at the University of Toronto.

09-240/Classnotes for Tuesday September 15

2009-11-05T23:35:41Z

Kjgry:

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The real numbers A set <math>\mathbb R</math> with two binary operators and two special elements <math>0, 1 \in \mathbb R</math> s.t.

: <math>F1.\quad \forall a, b \in \mathbb R, a + b = b + a \mbox{ and } a \cdot b = b \cdot a</math>
: <math>F2.\quad \forall a, b, c, (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>\mbox{(So for any real numbers } a_1, a_2, ..., a_n, \mbox{ one can sum them in any order and achieve the same result.}</math>
: <math>F3.\quad \forall a, a + 0 = a \mbox{ and } a \cdot 1 = a</math>
: <math>F4.\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>\mbox{So } a + (-a) = 0 \mbox{ and } a \cdot a^{-1} = 1</math>
: <math>\mbox{(So } (a + b) \cdot (a - b) = a^2 - b^2)</math>
: <math>\forall a, \exists x, x \cdot x = a \mbox{ or } a + x \cdot x = 0</math>
: Note: '''or''' means '''inclusive or''' in math.
: <math>F5.\quad (a + b) \cdot c = a \cdot c + b \cdot c</math>

Definition: A '''field''' is a set ''F'' with two binary operators <math>\,\!+</math>: ''F''×''F'' → ''F'', <math>\times\,\!</math>: ''F''×''F'' → ''F'' and two elements <math>0, 1 \in \mathbb R</math> s.t.
: <math>F1\quad \mbox{Commutativity } a + b = b + a \mbox{ and } a \cdot b = b \cdot a \forall a, b \in F</math>
: <math>F2\quad \mbox{Associativity } (a + b) + c = a + (b + c) \mbox{ and } (a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
: <math>F3\quad a + 0 = a, a \cdot 1 = a</math>
: <math>F4\quad \forall a, \exists b, a + b = 0 \mbox{ and } \forall a \ne 0, \exists b, a \cdot b = 1</math>
: <math>F5\quad \mbox{Distributivity } (a + b) \cdot c = a \cdot c + b \cdot c</math>

== Examples ==

# <math>F = \mathbb R</math>
# <math>F = \mathbb Q</math>
# <math>\mathbb C = \{ a + bi : a, b \in \mathbb R \}</math>
#: <math>i = \sqrt{-1}</math>
#: <math>\,\!(a + bi) + (c + di) = (a + c) + (b + d)i</math>
#: <math>\,\!0 = 0 + 0i, 1 = 1 + 0i</math>
# <math>\,\!F_2 = \{ 0, 1 \}</math>
# <math>\,\!F_7 = \{ 0, 1,2,3,4,5,6 \}</math>
# <math>\,\!F_6 = \{ 0, 1,2,3,4,5 \}</math> is not a field because not every element has a multiplicative inverse.
#: Let <math>a = 2.</math>
#: Then <math>a \cdot 0 = 0, a \cdot 1 = 2, a \cdot 3 = 0, a \cdot 4 = 2, a \cdot 5 = 4</math>
#: Therefore F4 fails; there is '''no''' number ''b'' in ''F''6 s.t. ''a · b'' = 1

{|
|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! + !! 0 !! 1
|-
! 0
| 0 || 1
|-
! 1
| 1 || 0
|-
|}

|
{| border="1" cellspacing="0"
|+ Ex. 4
|-
! × !! 0 !! 1
|-
! 0
| 0 || 0
|-
! 1
| 0 || 1
|}

|-
|
{| border="1" cellspacing="0"
|+ Ex. 5
|-
! + !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 6 || 0
|-
! 2
| 2 || 3 || 4 || 5 || 6 || 0 || 1
|-
! 3
| 3 || 4 || 5 || 6 || 0 || 1 || 2
|-
! 4
| 4 || 5 || 6 || 0 || 1 || 2 || 3
|-
! 5
| 5 || 6 || 0 || 1 || 2 || 3 || 4
|-
! 6
| 6 || 0 || 1 || 2 || 3 || 4 || 5
|}

|
{|border="1" cellspacing="0"
|+ Ex. 5
|-
! × !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
! 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
! 1
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
! 2
| 0 || 2 || 4 || 6 || 1 || 3 || 5
|-
! 3
| 0 || 3 || 6 || 2 || 5 || 1 || 4
|-
! 4
| 0 || 4 || 1 || 5 || 2 || 6 || 3
|-
! 5
| 0 || 5 || 3 || 1 || 6 || 4 || 2
|-
! 6
| 0 || 6 || 5 || 4 || 3 || 2 || 1
|}
|}

'''Theorem''': ''F''2 is a field.

In order to prove that the associative property holds, make a table (similar to a [http://en.wikipedia.org/wiki/Truth_table truth table]) for ''a'', ''b'' and ''c''.

{| border="1" cellspacing="0" style="text-align: center;"
! a !! b !! c !!  
|-
| 0 || 0 || 0 ||  
|-
| 0 || 0 || 1 ||  
|-
| 0 || 1 || 0 ||  
|-
| 0 || 1 || 1 || (0 + 1) + 1 =? 0 + (1 + 1) 1 + 1 =? 0 + 0 0 = 0
|-
| 1 || 0 || 0 ||  
|-
| 1 || 0 || 1 ||  
|-
| 1 || 1 || 0 ||  
|-
| 1 || 1 || 1 ||  
|}

'''Theorem''': <math>\,\! F_p </math> for <math>p > 1</math> is a field ''iff'' ([http://en.wikipedia.org/wiki/If_and_only_if if and only if]) <math>p</math> is a prime number

Proof:

<math>a \in \mathbb Z</math> has a multiplicative inverse modulo <math>m</math> if and only if a and m are relatively prime.

This can be shown using [http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]:

: <math>\exists x, y \mbox{ s.t. } ax + my = 1</math>
: <math>\left(ax + my\right) \pmod{m} = 1\pmod{m}</math>
: <math>ax = 1</math>
: <math>x = a^{-1}</math>

We have shown that <math>a</math> has a multiplicative inverse modulo m if <math>a</math> and <math>m</math> are relatively prime. It is therefore a natural conclusion that if <math>m</math> is a prime number all elements in the set will be relatively prime to m.

----

Multiplication is repeated addition.

<math>23 \times 27 = \begin{matrix} 27 \\ \overbrace{23 + 23 + 23 + \cdots + 23} \end{matrix} = 621</math>

<math>27 \times 23 = \begin{matrix} 23 \\ \overbrace{27 + 27 + 27 + \cdots + 27} \end{matrix} = 621</math>

One may interpret this as counting the units in a 23×27 rectangle; one may choose to count along either 23 rows or 27 columns, but both ways lead to the same answer.

Exponentiation is repeated multiplication, but it does not have the same properties as multiplication; 23 = 8, but 32 = 9.

== Tedious Theorem ==
# <math>a + b = c + b \Rightarrow a = c </math> "cancellation property"
#: Proof:
#: By F4, <math>\exists d \mbox{ s.t. } b + d = 0</math>
#: <math>\,\! (a + b) + d = (c + b) + d</math>
#: <math>\Rightarrow a + (b + d) = c + (b + d)</math> by F2
#: <math>\Rightarrow a + 0 = c + 0</math> by choice of ''d''
#: <math>\Rightarrow a = c</math> by F3
# <math> a \cdot b = c \cdot b , (b \ne 0) \Rightarrow a = c </math>
# <math>a + O' = a \Rightarrow O' = 0</math>
#: Proof:
#: <math>\,\! a + O' = a</math>
#: <math>\Rightarrow a + O' = a + 0</math> by F3
#: <math>\Rightarrow O' = 0</math> by adding the additive inverse of ''a'' to both sides
# <math>a \cdot l' = a, a \ne 0 \Rightarrow l' = 1</math>
# <math>a + b = 0 = a + b' \Rightarrow b = b'</math>
# <math>a \cdot b = 1 = a \cdot b' \Rightarrow b = b' = a^{-1}</math>
#: <math>\,\! \mbox{Aside: } a - b = a + (-b)</math>
#: <math>\frac ab = a \cdot b^{-1}</math>
# <math>\,\! -(-a) = a = (a^{-1})^{-1}</math>
# <math>a \cdot 0 = 0</math>
#: Proof:
#: <math>a \cdot 0 = a(0 + 0)</math> by F3
#: <math>= a \cdot 0 + a \cdot 0</math> by F5
#: <math>= 0 = a \cdot 0</math>
# <math>\forall b, 0 \cdot b \ne 1</math>
#: So there is no 0−1
# <math>(-a) \cdot b = a \cdot (-b) = -(a \cdot b)</math>
# <math>(-a) \cdot (-b) = a \cdot b</math>
# (Bonus) <math>\,\! (a + b)(a - b) = a^2 - b^2</math>

== Quotation of the Day ==
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File:ALA240-2009 - September 15th.pdf

2009-11-05T23:28:19Z

Kjgry: Lecture 1 by Professor Dror Bar-Natan for MAT240 Fall Session at the University of Toronto

Lecture 1 by Professor Dror Bar-Natan for MAT240 Fall Session at the University of Toronto