Check the commutativity and associativity of each of the following binary operations: (i) ‘*’ on Z defined by a * b = a + b + a b for all a, b ∈ Z (ii) ‘*’ on N defined by a * b = 2ab for all a, b ∈ N

(i)  to check: commutativity of *

   

Now we have to prove associativity of *

   

Thus, * is associative on Z.

(ii) to check : commutativity of *

   

Thus, * is commutative on N

to check: associativity of *

Let a, b, c ∈ N

Then, a * (b * c) = a * (2^bc)

=$2^{a\ast 2^{}}$

2a∗2bc

(a * b) * c = (2^ab) * c

=$2^{ab\ast 2^{}}$

2ab∗2c

=> a * (b * c) ≠ (a * b) * c

Thus, * is not associative on N