Let A be any set containing more than one element. Let ‘*’ be a binary operation on A defined by a*b = b for all a,b∈A. Is ‘*’ commutative or associative on A?
Given that * is a binary operation on set A defined by a*b = b for all a,b∈A.
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 = b
⇒ b*a = a
⇒ b*a≠a*b
∴ The commutative property does not hold for given binary operation ‘*’ on ‘A’.
We know that associative property is (p*q)*r = p*(q*r)
Let’s check the associativity of given binary operation:
⇒ (a*b)*c = (b)*c
⇒ (a*b)*c = b*c
⇒ (a*b)*c = c ...... (1)
⇒ a*(b*c) = a*(c)
⇒ a*(b*c) = a*c
⇒ a*(b*c) = c ...... (2)
From (1) and (2) we can clearly say that associativity holds for the binary operation ‘*’ on ‘A’.