Given that * is a binary operation on Q defined by a*b = ab + 1 for all a,b∈Q.

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 = ab + 1

⇒ b*a = ba + 1 = ab + 1

⇒ b*a = a*b

∴ Commutative property holds for given binary operation ‘*’ on ‘Q’.

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

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (ab + 1)*c

⇒ (a*b)*c = ((ab + 1)×c) + 1

⇒ (a*b)*c = abc + c + 1 ...... (1)

⇒ a*(b*c) = a*(bc + 1)

⇒ a*(b*c) = (a×(bc + 1)) + 1

⇒ a*(b*c) = abc + a + 1 ...... (2)

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