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

From Drorbn
Jump to navigationJump to search
(Fix some typesetting.)
(fix typesetting, test latex)
Line 1: Line 1:
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 * b^{-1}</math>.
Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.


Theorem:
Theorem:


8. For every <math>a</math> belongs to F , <math>a * 0 = 0</math>.
8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>;
proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>;
By F5 , <math>a * (0 + 0) = a * 0 + a * 0</math>;
By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
By F3 , <math>a * 0 = 0 + a * 0</math>;
By F3 , <math>a \times 0 = 0 + a \times 0</math>;
By Thm P1 ,<math>0 = a * 0</math>.
By Thm P1,<math>0 = a \times 0</math>.
9. There not exists <math>b</math> belongs to F s.t. <math>0 * b = 1</math>;
9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
For every <math>b</math> belongs to F s.t. <math>0 * b </math>is not equal to <math>1</math>.
<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>.
proof of 9: By F3 , <math>\times b = 0 </math>is not equal to <math>1</math>.
10. <math>(-a) * b = a * (-b) = -(a * b)</math>.
10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.
11. <math>(-a) * (-b) = a * b</math>.
11. <math>(-a) \times (-b) = a \times b</math>.
12. <math>a * b = 0 iff a = 0 or b = 0</math>.
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 * b = 0 * b = 0</math>;
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 * b = a * 0 = 0</math>.
By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
=> : Assume <math>a * b = 0 </math> , if a = 0 we have done;
=> : 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 * b = 0 = a * 0</math>;
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) * (a - b) = a^2 - b^2</math>.
<math>(a + b) \times (a - b) = a^2 - b^2</math>.
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a \times a + a \times (-b) + b \times a + (-b) \times b
= a^2 - b^2</math>
= a^2 - b^2</math>
Theorem :
Theorem :
There exists !(unique) iota <math>\iota : \Z \rightarrow F</math> s.t.
<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</math> belong to <math>Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>;
2. For every <math>m ,n \in Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>;
3. For every <math>m ,n</math> belong to <math>Z</math> , <math>\iota(m*n) = \iota(m) * \iota(n)</math>.
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;
Line 43: Line 43:
= iota(26) + iota(1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(26) + 1
= iota(13 * 2) + 1
= iota(13 \times 2) + 1
= iota(2) * iota(13) + 1
= iota(2) \times iota(13) + 1
= (1 + 1) * iota(13) + 1
= (1 + 1) \times iota(13) + 1
= 0 * iota(13) + 1
= 0 \times iota(13) + 1
= 1</math>
= 1</math>

Revision as of 12:31, 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 \times b^{-1} }[/math].

Theorem: 

        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  }[/math]is not equal to [math]\displaystyle{ 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 or 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  }[/math] is not equal to [math]\displaystyle{ 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))
                                                = a \times a + a \times (-b) + b \times a + (-b) \times b
                                                = 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. For every [math]\displaystyle{ m ,n \in Z }[/math] , [math]\displaystyle{ \iota(m+n) = \iota(m) + \iota(n) }[/math];
              3. For every [math]\displaystyle{ m ,n \in  }[/math] , [math]\displaystyle{ \iota(m\times n) = \iota(m) \times \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 \times 2) + 1
                                         = iota(2) \times iota(13) + 1
                                         = (1 + 1) \times iota(13) + 1
                                         = 0 \times iota(13) + 1
                                         = 1 }[/math]
Retrieved from "https://drorbn.net/index.php?title=14-240/Classnotes_for_Monday_September_15&oldid=13312"