Question

Let A = N × N and '' be a binary operation on A defined by (a, b) (c, d) = (ad + bc, bd). Show that (A, *) has no identity element.

Answer 1

We have (a, b) * (c, d) = (ad + bc, bd) for (a, b) (c, d) ∈ A

Let (p, q) be the identity element of (A, *)

∴ For (a, b) ∈ A, we have

(a, b) (p, q) = (a, b) = (p, q) (a, b)

⇒ (aq + bp, bq) = (a, b) = (pb + qa, qb)

⇒ aq + bp = α and bq = b

Solving, we get p = 0, q = 1

Since 0 N, (0, 1) A

∴ (A, *) has no identity element. Remark . Since identity element does not exist in the above example, the concept of inverse of an element is not defined in the set A.