Question

‘*’ be a binary operation on N × N defined as (a, b) * (c, d) – (ac, bd)

1. Show that * is commutative.

2. Find the identity element of * if any

3. Write an element of N × N which has an inverse.

Answer 1

1. (a, b) * (c, d) = (ac, bd) = (ca, db)

= (c, d) * (a, b)

Hence * is commutative.

2. (a, b) * (e, e) = (a, b)

⇒ (ae, be) = (a, b)

⇒ ae = a ⇒ e = 1

⇒ be = b ⇒ e = 1

Therefore (1,1) is the identity element.

3. Invertible element is (1,1).