Difference between revisions of "14-240/Classnotes for Monday September 15"
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Definition: | Definition: | ||
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 | + | Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>. |
Theorem: | Theorem: | ||
− | 8. | + | 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>. |
− | proof of 8: By F3 , <math>a | + | proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>; |
− | By F5 , <math>a | + | By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>; |
− | By F3 , <math>a | + | By F3 , <math>a \times 0 = 0 + a \times 0</math>; |
− | By Thm P1 ,<math>0 = a | + | By Thm P1,<math>0 = a \times 0</math>. |
− | 9. | + | 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: By F3 , <math> | + | proof of 9: By F3 , <math>\times b = 0 </math>is not equal to <math>1</math>. |
− | 10. <math>(-a) | + | 10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>. |
− | 11. <math>(-a) | + | 11. <math>(-a) \times (-b) = a \times b</math>. |
− | 12. <math>a | + | 12. <math>a \times b = 0 \iff a = 0 or b = 0</math>. |
− | proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a | + | 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 | + | By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>. |
− | => : Assume <math>a | + | => : Assume <math>a \times b = 0 </math> , if a = 0 we are done; |
− | Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a | + | Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a \times b = 0 = a \times 0</math>; |
by cancellation (P2) , <math>b = 0</math>. | by cancellation (P2) , <math>b = 0</math>. | ||
− | <math>(a + b) | + | <math>(a + b) \times (a - b) = a^2 - b^2</math>. |
− | proof: By F5 , <math>(a + b) | + | proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b)) |
− | = a | + | = a \times a + a \times (-b) + b \times a + (-b) \times b |
= a^2 - b^2</math> | = a^2 - b^2</math> | ||
Theorem : | Theorem : | ||
− | + | <math>\exists! \iota : \Z \rightarrow F</math> s.t. | |
1. <math>\iota(0) = 0 , \iota(1) = 1</math>; | 1. <math>\iota(0) = 0 , \iota(1) = 1</math>; | ||
− | 2. For every <math>m ,n | + | 2. For every <math>m ,n \in Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>; |
− | 3. For every <math>m ,n | + | 3. For every <math>m ,n \in </math> , <math>\iota(m\times n) = \iota(m) \times \iota(n)</math>. |
iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1; | iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1; | ||
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= iota(26) + iota(1) | = iota(26) + iota(1) | ||
= iota(26) + 1 | = iota(26) + 1 | ||
− | = iota(13 | + | = iota(13 \times 2) + 1 |
− | = iota(2) | + | = iota(2) \times iota(13) + 1 |
− | = (1 + 1) | + | = (1 + 1) \times iota(13) + 1 |
− | = 0 | + | = 0 \times iota(13) + 1 |
= 1</math> | = 1</math> |
Revision as of 13:31, 16 September 2014
Definition:
Subtraction: if . Division: if .
Theorem:
8. , . proof of 8: By F3 , ; By F5 , ; By F3 , ; By Thm P1,. 9. s.t. ; s.t. . proof of 9: By F3 , is not equal to . 10. . 11. . 12. . proof of 12: <= : By P8 , if , then ; By P8 , if , then . => : Assume , if a = 0 we are done; Otherwise , by P8 , is not equal to and we have ; by cancellation (P2) , .
.
proof: By F5 ,
Theorem :
s.t. 1. ; 2. For every , ; 3. For every , .
iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1; iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1; ...... In F2 ,