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Question

Consider the following relations:
R = {(x, y) | x, y are real numbers and x = wy for some rational number w};
S = {(mn, pq)| m, n, p and q are integers such that n, q 0 and qm = pn}.
Then

A
Neither R nor S is an equivalence relation
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B
S is an equivalence relation but R is not an equivalence relation.
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C
R and S both are equivalence relations
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D
R is an equivalence relation but S is not an equivalence relation
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Solution

The correct option is B S is an equivalence relation but R is not an equivalence relation.

X Ry need not implies yRx
R is not symmetric and hence not an equivalence relation.
S:mnspq
Given qm=pnpq=mnmnsmn (reflexive)mnspqpqsmn (symmetric)mnspq,pqsrs qm=pm, ps=rqpq=mn=rsms=rn (transitive)
S is an equivalence relation.


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