14-240/Classnotes for Monday September 15: Difference between revisions

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Definition:
==Definition of Subtraction and Division==
* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.
* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.


==Basic Properties of a Field (cont'd)==
⚫
Theorem:


* 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
'''8.''' <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
;Proof of 8
⚫
proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>;
By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
:By F3 , <math>a \times 0 = a \times (0 + 0)</math>
By F3 , <math>a \times 0 = 0 + a \times 0</math>;
:By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
By Thm P1,<math>0 = a \times 0</math>.
:By F3 , <math>a \times 0 = 0 + a \times 0</math>;
:By Thm P1, <math>0 = a \times 0</math>.
9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
'''9.''' <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
:<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
;Proof of 9
proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.
⚫
:By F3 , <math>\times b = 0 \neq 1</math>.

10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.
'''10.''' <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.
11. <math>(-a) \times (-b) = a \times b</math>.
'''11.''' <math>(-a) \times (-b) = a \times b</math>.
12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
'''12.''' <math>a \times b = 0 \iff a = 0</math> or <math>b = 0</math>.
;Proof of 12
⚫
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
:'''<= :'''
By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
=> : Assume <math>a \times b = 0 </math> , if a = 0 we are done;
:By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
:By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
:'''=> :''' Assume <math>a \times b = 0 </math> , if a = 0 we are done;
⚫
by cancellation (P2) , <math>b = 0</math>.
⚫
:Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
:by cancellation (P2) , <math>b = 0</math>.
<math>(a + b) \times (a - b) = a^2 - b^2</math>.
<math>(a + b) \times (a - b) = a^2 - b^2</math>.
;Proof
proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
<math>= a \times a + a \times (-b) + b \times a + (-b) \times b</math>
:By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
<math>= a^2 - b^2</math>
:<math>= a^2 - b^2</math>
Theorem :
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
⚫
1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
⚫
2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
⚫
3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.


⚫
==Theorem==
⚫
<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
⚫
:1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
......
⚫
:2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
⚫
:3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.

;Examples
⚫
<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>

⚫
......
In F2:
⚫
In F2 , <math>27 ----> \iota(27) = \iota(26 + 1)</math>
<math>
<math>= \iota(26) + \iota(1)</math>
\begin{align}
<math>= \iota(26) + 1</math>
⚫
27 ----> \iota(27) &= \iota(26 + 1)\\
<math>= \iota(13 \times 2) + 1</math>
<math>= \iota(2) \times \iota(13) + 1</math>
&= \iota(26) + \iota(1)\\
&= \iota(26) + 1\\
<math>= (1 + 1) \times \iota(13) + 1</math>
&= \iota(13 \times 2) + 1\\
<math>= 0 \times \iota(13) + 1</math>
&= \iota(2) \times \iota(13) + 1\\
<math>= 1</math>
&= (1 + 1) \times \iota(13) + 1\\
&= 0 \times \iota(13) + 1\\
&= 1
\end{align}
</math>

==Scanned Lecture Notes by [[User:AM|AM]]==

<gallery>
File:MAT 240 lecture 3 (1 of 2).pdf|page 1
File:MAT240 lectuire 3 (2 of 2).pdf|page 2
</gallery>
==Scanned Lecture Notes by [[User Boyang.wu|Boyang.wu]]==

[[File:W21.pdf]]

Latest revision as of 00:57, 8 December 2014

DBN: Classes: 2014-15: 14-240 / Navigation
Welcome to Math 240!
(additions to this web site no longer count towards good deed points)
# Week of... Notes and Links
1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday
2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf
3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf
4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf
5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf
6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial
7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday
8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial
9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial
10 Nov 10 HW8, Monday, Tutorial
11 Nov 17 Monday-Tuesday is UofT November break
12 Nov 24 HW9
13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
F Dec 8 The Final Exam
Register of Good Deeds
Class Photo
Add your name / see who's in!
14-240-Splash.png

Definition of Subtraction and Division

  • Subtraction: if .
  • Division: if .

Basic Properties of a Field (cont'd)

8. , .

Proof of 8
By F3 ,
By F5 , ;
By F3 , ;
By Thm P1, .

9. s.t. ;

s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \times b \neq 1} .
Proof of 9
By F3 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times b = 0 \neq 1} .

10. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-a) \times b = a \times (-b) = -(a \times b)} .

11. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-a) \times (-b) = a \times b} .

12. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \times b = 0 \iff a = 0} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} .

Proof of 12
<= :
By P8 , if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 0} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \times b = 0 \times b = 0} ;
By P8 , if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} , then .
=> : Assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \times b = 0 } , if a = 0 we are done;
Otherwise , by P8 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq 0 } and we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \times b = 0 = a \times 0} ;
by cancellation (P2) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 0} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + b) \times (a - b) = a^2 - b^2} .

Proof
By F5 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = a^2 - b^2}

Theorem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists! \iota : \Z \rightarrow F} s.t.

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota(0) = 0 , \iota(1) = 1} ;
2. ;
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)} .
Examples

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;}

......

In F2:

Scanned Lecture Notes by AM

  • page 1

    page 1

  • page 2

    page 2

Scanned Lecture Notes by Boyang.wu

File:W21.pdf

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