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Question

Let A=Q×Q, where Q is the set of all rational numbers, and is a binary operation on A defined by (a,b)(c,d)=(ac,b+ad) for (a,b),(c,d)A.

Then find (i) The identity element of in A.

(ii) Invertible elements of A, and hence write the inverse of elements (5,3) and (12,4).

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Solution

(i) Let (x,y) be the identity element in A.

(a,b)(x,y)=(a,b)=(x,y)(a,b)

(ax, b+ay)=(a, b)

ax=a, b+ay=b

x=1, y=0

(a, 0) A (set of rational numbers)

(1, 0) is an identity element.

(ii) Here, (1, 0) is the identity element

(a, b) (c, d)=(1, 0)

(ac, b+ad)=(1, 0)

ac=1, b+ad=0

c=1a, d=ba

A1=(1a,ba)

Also, inverse of (5,3)=(15,35) and inverse of (12,4)=(2,8)


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