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’.