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 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 / b = a * (b to the power (-1)}
.

Theorem:

        8. For every $abelongstoF,a*0=0$.
                   proof of 8: By F3 , $a*0=a*(0+0)$;
                               By F5 , $a*(0+0)=a*0+a*0$;
                               By F3 , $a*0=0+a*0$;
                               By Thm P1 ,$0=a*0$.
       
        9. There not exists $bbelongstoFs.t.0*b=1$;
           For every $bbelongstoFs.t.0*b$is not equal to $1$.
                   proof of 9: By F3 , $0*b=0$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}
.
       
       10. $(-a)*b=a*(-b)=-(a*b)$.
     
       11. 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}
.
      
       12. 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 = 0 iff a = 0 or b = 0}
.
                   proof of 12: <= : By P8 , 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 if a = 0 , then a * b = 0 * b = 0}
;
                                     By P8 , 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 if b = 0 , then a * b = a * 0 = 0}
.
                                => : Assume 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 = 0}
 , if a = 0 we have done;
                                     Otherwise , by P8 , 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 }
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 0 }
and we have $a*b=0=a*0$;  
                                                 by cancellation (P2) , 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 = 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 square - b square} .

        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 square - b square}

Theorem :

        There exists !(unique) 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 ---> 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 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 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 , $27---->iota(27)=iota(26+1)=iota(26)+iota(1)=iota(26)+1=iota(13*2)+1=iota(2)*iota(13)+1=(1+1)*iota(13)+1=0*iota(13)+1=1$