Question

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.

Answer 1

$$\begin{gathered} \text{Let}a,b,c\in Z \\ a*\left(b*c\right)=a*\left(bc+1\right) \\ =a\left(bc+1\right)+1 \\ =abc+a+1 \\ \left(a*b\right)*c=\left(ab+1\right)*c \\ =\left(ab+1\right)c+1 \\ =abc+c+1 \\ \text{Thus,}a*\left(b*c\right)\ne \left(a*b\right)*c \end{gathered}$$

Thus, * is not associative on Z.