Determine which of the following binary operations are associative and which are commutative:
*on Q defined by for all a,b Q.
Given that * is a binary operation on N defined by for all a,b∈N.
We know that commutative property is p*q = q*p, where * is a binary operation.
Let’s check the commutativity of given binary operation:
⇒
⇒
⇒ b*a = a*b
∴ The commutative property holds for given binary operation ‘*’ on ‘N’.
We know that associative property is (p*q)*r = p*(q*r)
Let’s check the associativity of given binary operation:
⇒
⇒
⇒ ...... (1)
⇒
⇒
⇒ ...... (2)
From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘N’.