Question

On Z, the set of all integers, a binary operation * is defined by a*b = a + 3b – 4. Prove that * is neither commutative nor associative on Z.

Answer 1

Given that * is a binary operation on Z defined by a*b = a + 3b – 4 for all a,b∈Z.

We know that commutative property is p*q = q*p, where * is a binary operation.

Let’s check the commutativity of given binary operation: 

⇒ a*b = a + 3b – 4 

⇒ b*a = b + 3a – 4 

⇒ b*a ≠ a*b 

∴ The commutative property doesn’t hold for given binary operation ‘*’ on ‘Z’. 

We know that associative property is (p*q)*r = p*(q*r) 

Let’s check the associativity of given binary operation: 

⇒ (a*b)*c = (a + 3b – 4)*c 

⇒ (a*b)*c = a + 3b – 4 + 3c – 4 

⇒ (a*b)*c = a + 3b + 3c – 8 ...... (1) 

⇒ a*(b*c) = a*(b + 3c – 4) 

⇒ a*(b*c) = a + (3×(b + 3c – 4)) – 4 

⇒ a*(b*c) = a + 3b + 9c – 12 – 4 

⇒ a*(b*c) = a + 3b + 9c – 16 ...... (2) 

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