14-240/Classnotes for Monday September 15: Difference between revisions
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Definition: |
Definition: |
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Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>. |
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Division: if <math>a, b \in F, a / b = a * b^{-1}</math>. |
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Theorem: |
Theorem: |
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by cancellation (P2) , <math>b = 0</math>. |
by cancellation (P2) , <math>b = 0</math>. |
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<math>(a + b) * (a - b) = a |
<math>(a + b) * (a - b) = a^2 - b^2</math>. |
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proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b)) |
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b)) |
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= a * a + a * (-b) + b * a + (-b) * b |
= a * a + a * (-b) + b * a + (-b) * b |
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= a |
= a^2 - b^2</math> |
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Theorem : |
Theorem : |
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There exists !(unique) <math>iota : Z |
There exists !(unique) iota <math>\iota : \Z \rightarrow F</math> s.t. |
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1. <math>iota(0) = 0 , iota(1) = 1</math>; |
1. <math>\iota(0) = 0 , \iota(1) = 1</math>; |
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2. For every <math>m ,n</math> belong to <math>Z</math> , <math>iota(m+n) = iota(m) + iota(n)</math>; |
2. For every <math>m ,n</math> belong to <math>Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>; |
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3. |
3. For every <math>m ,n</math> belong to <math>Z</math> , <math>\iota(m*n) = \iota(m) * \iota(n)</math>. |
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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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Revision as of 08:42, 16 September 2014
Definition:
Subtraction: if [math]\displaystyle{ a, b \in F, a - b = a + (-b) }[/math]. Division: if [math]\displaystyle{ a, b \in F, a / b = a * b^{-1} }[/math].
Theorem:
8. For every [math]\displaystyle{ a }[/math] belongs to F , [math]\displaystyle{ a * 0 = 0 }[/math]. proof of 8: By F3 , [math]\displaystyle{ a * 0 = a * (0 + 0) }[/math]; By F5 , [math]\displaystyle{ a * (0 + 0) = a * 0 + a * 0 }[/math]; By F3 , [math]\displaystyle{ a * 0 = 0 + a * 0 }[/math]; By Thm P1 ,[math]\displaystyle{ 0 = a * 0 }[/math]. 9. There not exists [math]\displaystyle{ b }[/math] belongs to F s.t. [math]\displaystyle{ 0 * b = 1 }[/math]; For every [math]\displaystyle{ b }[/math] belongs to F s.t. [math]\displaystyle{ 0 * b }[/math]is not equal to [math]\displaystyle{ 1 }[/math]. proof of 9: By F3 , [math]\displaystyle{ 0 * b = 0 }[/math]is not equal to [math]\displaystyle{ 1 }[/math]. 10. [math]\displaystyle{ (-a) * b = a * (-b) = -(a * b) }[/math]. 11. [math]\displaystyle{ (-a) * (-b) = a * b }[/math]. 12. [math]\displaystyle{ a * b = 0 iff a = 0 or b = 0 }[/math]. proof of 12: <= : By P8 , if [math]\displaystyle{ a = 0 }[/math] , then [math]\displaystyle{ a * b = 0 * b = 0 }[/math]; By P8 , if [math]\displaystyle{ b = 0 }[/math] , then [math]\displaystyle{ a * b = a * 0 = 0 }[/math]. => : Assume [math]\displaystyle{ a * b = 0 }[/math] , if a = 0 we have done; Otherwise , by P8 , [math]\displaystyle{ a }[/math] is not equal to [math]\displaystyle{ 0 }[/math]and we have [math]\displaystyle{ a * b = 0 = a * 0 }[/math]; by cancellation (P2) , [math]\displaystyle{ b = 0 }[/math].
[math]\displaystyle{ (a + b) * (a - b) = a^2 - b^2 }[/math].
proof: By F5 , [math]\displaystyle{ (a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a^2 - b^2 }[/math]
Theorem :
There exists !(unique) iota [math]\displaystyle{ \iota : \Z \rightarrow F }[/math] s.t. 1. [math]\displaystyle{ \iota(0) = 0 , \iota(1) = 1 }[/math]; 2. For every [math]\displaystyle{ m ,n }[/math] belong to [math]\displaystyle{ Z }[/math] , [math]\displaystyle{ \iota(m+n) = \iota(m) + \iota(n) }[/math]; 3. For every [math]\displaystyle{ m ,n }[/math] belong to [math]\displaystyle{ Z }[/math] , [math]\displaystyle{ \iota(m*n) = \iota(m) * \iota(n) }[/math].
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 , [math]\displaystyle{ 27 ----\gt iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 * 2) + 1
= iota(2) * iota(13) + 1
= (1 + 1) * iota(13) + 1
= 0 * iota(13) + 1
= 1 }[/math]
