Consider the following relations R= {(x, y): x, y∈ R} and x = ω y, for some rational no. ω S = {(mn,pq) : m, n, pand q are integers such that n, q ≠ 0 and qm = pn} Then
A
R is equivalence but S is not an equivalence
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B
Neither R nor S is an equivalence relation
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C
S is equivalence but R is not
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D
R & S both are equivalence
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Solution
The correct option is B S is equivalence but R is not For R xRy ⇒x=ωy, for reflexive xRx⇒x=ωy which is true when ω = 1 For symmetric consider x=0, y≠ 0, xRy ⇒ 0Ry ⇒ 0= ω y, which is true when ω =0. Now yRx ⇒yR0⇒y=ω.0. There is no rational value of ω for which y=ω×0⇒R is not symmetric and hence not equivalence. For S mnSmn⇒mn=nm⇒S is reflexive For symmetric let mnSpq⇒qm=np pqSmn⇒pn = mq, which is true. ⇒ S is symmetric For transitive, letmnSpq⇒ qm = pn .......(i) pqSrs⇒ps=rq ......(ii) From (i) and (ii), we conclude that ms = nr ⇒mnSrs ⇒ S is transitive Hence, S is an equivalence relation.