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

1.	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» An element (e, f) ∈ Z × Z bethe identity element, if (a, b) * (e, f) = (a, b) = (e, f) * (a, b) ∀ (a, b) ∈ 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.

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»

An element (e, f) ∈ Z × Z bethe identity element, if

(a, b) * (e, f) = (a, b) = (e, f) * (a, b) ∀ (a, b) ∈ 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.

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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.

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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.