Let A = Z × Z and * be a binary operation on A defined by (a, b)*(c, d) = (ad + bc, bd).

Find the identity element for * in the set A.

Let, an element (e, f)Z × Z be the identity element, if

(a, b) * (e, f) = (a, b) = (e, f) * (a, b) ∀ (a, b) Z × Z

⇒ (af + be, bf) = (a, b) = (eb + fa, fb)

⇒ bf = b = fb … (1)

⇒ f = 1 … (2)

Also, af + be = a = eb + fa

As, f = 1

⇒ a + be = a

⇒ be = 0

⇒ e = 0

So, e = 0, f = 1

Hence, (0, 1) is the identity element.