Question

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.​

Answer 1

Answer:

An element (e, f) ϵ Z × Z be the identity element, if

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

i.e., if, (af + be, bf) = (a, b) = (eb + fa, fb)

i.e., if, af + be = a = eb + fa and bf = b = fb …(1)

i.e., if, f = 1, e = 0 …(2)

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

Answer 2

Answer:

An element (e, f) ϵ Z × Z be the identity element, if

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

i.e., if, (af + be, bf) = (a, b) = (eb + fa, fb)

i.e., if, af + be = a = eb + fa and bf = b = fb …(1)

i.e., if, f = 1, e = 0 …(2)

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