(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 * (2bc)
=
(a * b) * c = (2ab) * c
=
=> a * (b * c) ≠ (a * b) * c
Thus, * is not associative on N