14-240/Classnotes for Monday September 15

Definition:

           Subtraction: 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, b \in F, a - b = a + (-b)}
.
           Division: 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, b \in F, a / b = a * b^{-1}}
.

Theorem:

        8. For every 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}
 belongs to F , 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 = 0}
.
                   proof of 8: 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 a * 0 = a * (0 + 0)}
;
                               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 * (0 + 0) = a * 0 + a * 0}
;
                               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 a * 0 = 0 + a * 0}
;
                               By Thm P1 ,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 = a * 0}
.
       
        9. There not exists $b$ belongs to F 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 * b = 1}
;
           For every 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}
 belongs to F 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 * b }
is not equal to $1$.
                   proof of 9: By F3 , $0*b=0$is not equal to $1$.
       
       10. $(-a)*b=a*(-b)=-(a*b)$.
     
       11. $(-a)*(-b)=a*b$.
      
       12. $a*b=0iffa=0orb=0$.
                   proof of 12: <= : By P8 , if $a=0$ , then $a*b=0*b=0$;
                                     By P8 , if $b=0$ , then $a*b=a*0=0$.
                                => : Assume $a*b=0$ , if a = 0 we have done;
                                     Otherwise , by P8 , $a$ is not equal to $0$and we have $a*b=0=a*0$;  
                                                 by cancellation (P2) , $b=0$.
       

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) * (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) * (a - b) = a * (a + (-b)) + b * (a + (-b))                                                 = a * a + a * (-b) + b * a + (-b) * b                                                 = a^2 - b^2}

Theorem :

        There exists !(unique) iota 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 : \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. For every 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 m ,n}
 belong to 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 Z}
 , 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(m+n) = \iota(m) + \iota(n)}
;
              3. For every 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 m ,n}
 belong to 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 Z}
 , $\iota (m*n)=\iota (m)*\iota (n)$.

        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 , 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 27 ----> 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}