Question

Check the commutativity and associativity of each of the following binary operations: ‘*’ on Z defined by a*b = a – b for all a,b∈Z

Answer 1

Given that * is a binary operation on Z defined by a*b = a – b 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 – b 

⇒ b*a = b – a 

⇒ b*a≠a*b 

∴ Commutative property doesn’t holds 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 – b)*c 

⇒ (a*b)*c = (a – b) – c 

⇒ (a*b)*c = a – b – c ...... (1) 

⇒ a*(b*c) = a*(b – c) 

⇒ a*(b*c) = a – (b – c) 

⇒ a*(b*c) = a – b + c ...... (2) 

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