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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. 

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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. 

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