State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation * on a set N, a*a = a ∀ a ∈N.
(ii) If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a.

Consider a binary operation * on N defined as a * b = a³ + b³ . Choose the correct answer.

(A)    Is * both associative and commutative ?
(B)    Is * commutative but not associative?
(C)    Is * associative but not commutative?
(D)    Is * neither commutative nor associative?

Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).

Then

(i) Find the identify element of (A, *)
(ii) Find the invertible elements of (A, *)

Define binary operation ‘*’ on Q as follows : a * b = a + b – ab, a, b∈ Q
(i) Find the identity element of (Q, *).
(ii) Which elements in (Q. *) are invertible?