14-240/Classnotes for Monday September 15: Difference between revisions
From Drorbn
Jump to navigationJump to search
No edit summary |
|||
| (20 intermediate revisions by 7 users not shown) | |||
| Line 1: | Line 1: | ||
{{14-240/Navigation}} |
|||
Definition: |
|||
==Definition of Subtraction and Division== |
|||
| ⚫ | |||
* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>. |
|||
| ⚫ | |||
==Basic Properties of a Field (cont'd)== |
|||
| ⚫ | |||
'''8.''' <math>\forall a \in F</math>, <math>a \times 0 = 0</math>. |
|||
;Proof of 8 |
|||
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>; |
|||
: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 = 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>; |
|||
:<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>. |
|||
;Proof of 9 |
|||
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>. |
|||
:By F3 , <math>\times b = 0 \neq 1</math>. |
|||
'''10.''' <math>(-a) \times b = 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</math> or <math>b = 0</math>. |
|||
;Proof of 12 |
|||
proof of 12: <= : By P8 , <math>if a = 0 , then a * b = 0 * b = 0</math>; |
|||
:'''<= :''' |
|||
| ⚫ | |||
: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; |
|||
| ⚫ | |||
:Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>; |
|||
| ⚫ | |||
<math>(a + b) |
<math>(a + b) \times (a - b) = a^2 - b^2</math>. |
||
;Proof |
|||
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b)) |
|||
:By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math> |
|||
:<math>= a^2 - b^2</math> |
|||
= a square - b square</math> |
|||
Theorem : |
|||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
...... |
|||
| ⚫ | |||
| ⚫ | |||
;Examples |
|||
| ⚫ | |||
| ⚫ | |||
| ⚫ | |||
In F2: |
|||
| ⚫ | |||
<math> |
|||
= iota(26) + iota(1) |
|||
\begin{align} |
|||
= iota(26) + 1 |
|||
| ⚫ | |||
= iota(13 * 2) + 1 |
|||
&= \iota(26) + \iota(1)\\ |
|||
&= \iota(26) + 1\\ |
|||
= (1 + 1) * iota(13) + 1 |
|||
&= \iota(13 \times 2) + 1\\ |
|||
= 0 * iota(13) + 1 |
|||
&= \iota(2) \times \iota(13) + 1\\ |
|||
= 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
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Definition of Subtraction and Division
- Subtraction: if [math]\displaystyle{ a, b \in F, a - b = a + (-b) }[/math].
- Division: if [math]\displaystyle{ a, b \in F, a / b = a \times b^{-1} }[/math].
Basic Properties of a Field (cont'd)
8. [math]\displaystyle{ \forall a \in F }[/math], [math]\displaystyle{ a \times 0 = 0 }[/math].
- Proof of 8
- By F3 , [math]\displaystyle{ a \times 0 = a \times (0 + 0) }[/math]
- By F5 , [math]\displaystyle{ a \times (0 + 0) = a \times 0 + a \times 0 }[/math];
- By F3 , [math]\displaystyle{ a \times 0 = 0 + a \times 0 }[/math];
- By Thm P1, [math]\displaystyle{ 0 = a \times 0 }[/math].
9. [math]\displaystyle{ \nexists b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b = 1 }[/math];
- [math]\displaystyle{ \forall b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b \neq 1 }[/math].
- Proof of 9
- By F3 , [math]\displaystyle{ \times b = 0 \neq 1 }[/math].
10. [math]\displaystyle{ (-a) \times b = a \times (-b) = -(a \times b) }[/math].
11. [math]\displaystyle{ (-a) \times (-b) = a \times b }[/math].
12. [math]\displaystyle{ a \times b = 0 \iff a = 0 }[/math] or [math]\displaystyle{ b = 0 }[/math].
- Proof of 12
- <= :
- By P8 , if [math]\displaystyle{ a = 0 }[/math] , then [math]\displaystyle{ a \times b = 0 \times b = 0 }[/math];
- By P8 , if [math]\displaystyle{ b = 0 }[/math] , then [math]\displaystyle{ a \times b = a \times 0 = 0 }[/math].
- => : Assume [math]\displaystyle{ a \times b = 0 }[/math] , if a = 0 we are done;
- Otherwise , by P8 , [math]\displaystyle{ a \neq 0 }[/math] and we have [math]\displaystyle{ a \times b = 0 = a \times 0 }[/math];
- by cancellation (P2) , [math]\displaystyle{ b = 0 }[/math].
[math]\displaystyle{ (a + b) \times (a - b) = a^2 - b^2 }[/math].
- Proof
- By F5 , [math]\displaystyle{ (a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b)) }[/math]
- [math]\displaystyle{ = a^2 - b^2 }[/math]
Theorem
[math]\displaystyle{ \exists! \iota : \Z \rightarrow F }[/math] s.t.
- 1. [math]\displaystyle{ \iota(0) = 0 , \iota(1) = 1 }[/math];
- 2. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n) }[/math];
- 3. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n) }[/math].
- Examples
[math]\displaystyle{ \iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1; }[/math] [math]\displaystyle{ \iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1; }[/math]
......
In F2: [math]\displaystyle{ \begin{align} 27 ----\gt \iota(27) &= \iota(26 + 1)\\ &= \iota(26) + \iota(1)\\ &= \iota(26) + 1\\ &= \iota(13 \times 2) + 1\\ &= \iota(2) \times \iota(13) + 1\\ &= (1 + 1) \times \iota(13) + 1\\ &= 0 \times \iota(13) + 1\\ &= 1 \end{align} }[/math]
Scanned Lecture Notes by AM
page 1
page 2
