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

Given that * is a binary operation on Z defined by a*b = ab + 1 for all a,b∈Z. 

We know that associative property is (p*q)*r = p*(q*r), where x is a binary operation. 

Let’s check the associativity of given binary operation: 

⇒ (a*b)*c = (ab + 1)*c 

⇒ (a*b)*c = ((ab + 1)×c) + 1 

⇒ (a*b)*c = 1 + c + abc ...... (1) 

⇒ a*(b*c) = a*(bc + 1) 

⇒ a*(b*c) = (a×(bc + 1)) + 1 

⇒ a*(b*c) = abc + a + 1 ...... (2) 

From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘Z’