Check the commutativity and associativity of each of the following binary operations:

‘*’ on Z defined by a*b = a + b + ab for all a,b∈Z

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

⇒ b*a = b + a + ba

⇒ b*a = a + b + ab

⇒ b*a = a*b

∴ Commutative property 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 + ab)*c

⇒ (a*b)*c = (a + b + ab + c + ((a + b + ab)×c))

⇒ (a*b)*c = a + b + c + ab + ac + ac + abc ...... (1)

⇒ a*(b*c) = a*(b + c + bc)

⇒ a*(b*c) = (a + b + c + bc + (a×(b + c + bc)))

⇒ a*(b*c) = a + b + c + ab + bc + ac + abc ...... (2)

From (1) and (2) we can clearly say that associativity holds for the binary operation ‘*’ on ‘Z’.