14-240/Classnotes for Monday September 15

Definition:

           Subtract: 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 }
 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 F , a - b = a + (-b)}
.
           Divition: 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 }
 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 F , a / b = a * (b }
to the power 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 (-1)}
.

Theorem:

        8. For every $a$ 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 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 = 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. $0*b$is not equal 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 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) , $b=0$.
       

$(a+b)*(a-b)=asquare-bsquare$.

        proof: By F5 , $(a+b)*(a-b)=a*(a+(-b))+b*(a+(-b))=a*a+a*(-b)+b*a+(-b)*b=asquare-bsquare$

Theorem :

        There exists !(unique) $iota:Z--->F$  s.t.
              1. $iota(0)=0,iota(1)=1$;
              2. For every $m,n$ belong to $Z$ , 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 $m,n$ 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)}
.

        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}