Question

On the set Z of integers a binary operation * is defined by a 8b = ab + 1 for all a, b ∈ Z. Prove that * is not associative on Z.

Answer 1

Let a, b, c ∈ Z

a * (b * c) = a * (bc + 1)

= a(bc + 1) + 1

= abc + a + 1

(a * b) * c = (ab + 1) * c

= (ab + 1)c + 1

= abc + c + 1

So, a * (b * c) ≠ (a * b) * c

Hence, * is not associative on Z.