Question

x*y=1 supra 2 (1+x+y-xy)
Determinati el neutru

Answer 1

Se observa ca legea este comutativa; viom rezolva deci doar x*e, nu si
si e*x
x*y = (1./2)* [-(x-1)(y-1)+2]
 x*e=x
(1./2)* [-(x-1)(e-1)+2]=x
-(x-1)(e-1)+2=2x
 2x-2+(x-1)(e-1)=0
2(x-1) +(x-1)(e-1)=0
 (x-1) (2+e-1)=0
(x-1)(1+e)=0 ∀x  ∈ G, unde G este  multimea pe care e definita legea
⇒1+e=0⇒ e=-1
verificare x*e=(1/2) (1+x -1+x)=(1/2) *2x=x , adevarat




Altfel , fara  sa modificam expresia
x*e=x
1/2* (1+x+e-xe)=x
1+x+e-ex=2x
e-ex=x-1
e(1-x)=x-1
e(1-x) + (1-x)=0
 (e+1) (1-x)=0 ∀x∈G, unde G estye multimea pec carec e definita legea
e+1=0
e=-1