Question

Aratati ca legea "*" este asociativa. Legea este: x*y=xy + 3x + 3y + 6.

Answer 1

Pentru a fi asociativa, trebuie sa demonstram ca (a*b)*c = a*(b*c)

(a*b)*c=(ab+3a+3b+6)*c=c(ab+3a+3b+6)+3(ab+3a+3b+6)+3c+6=

=abc+3ac+3bc+6c +3ab+9a+9b+3c+36=abc+3ac+3bc+3ab+9a+9b+9c+24

a*(b*c)=a*(bc+3b+3c+6)=a(bc+3b+3c+6) +3a+3(bc+3b+3c+6)+6=

=abc+3ab+3ac+6a+3a+3bc+9b+9c+24=abc+3ac+3bc+3ab+9a+9b+9c+24

Asadar, q.e.d.

Legea "*" este asociativa.