Question

Pe R se considera legea de compozitie x*y = xy +x +y , ∀x , y ∈ R

Aratati ca legea * este asociativa pe R

Answer 1

[tex]x*y = xy +x +y,\quad \forall x , y \in \mathbb{R}.\\ \\ \begin{array}{rcl}(x*y)*z &=& (x*y)\cdot z +(x*y)+z \\ &=& (xy+x+y)\cdot z + (xy+x+y)+z \\ &=& xyz+xz+yz+xy+x+y+z\quad (1) \end{array} \\ \\ \\ \begin{array}{rcl} x*(y*z) &=& x\cdot (y*z) + x+ (y*z) \\&=& x\cdot (yz+y+z) +x+(yz+y+z) \\ &=& xyz+xy+xz+yz+x+y+z \\ &=& xyz+xz+yz+xy+x+y+z \quad (2)\end{array}\\ \\ \\ \text{Din (1) si (2) } \Rightarrow \text{Legea } "*" \text{ este asociativa pe } \mathbb{R}. [/tex]