Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; `S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"}` . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Options

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

Answer

D

Explanation

xRy need not imply yRx.
`(m)/(n) S(p)/(q)hArrqm=pn`
`(m)/(n)S(m)/(n)impliesS` is reflexive
`(m)/(n)S(p)/(q)`
`implies (p)/(q) S(m)/(n)implies S` is symmetric
`(m)/(n)S(p)/(q),(p)/(q) S(r)/(s)`
`implies qm=pn,ps=rqimpliesms=rn`
`implies (m)/(n) S(r)/(s)implies R ` is transitive.
Hence, S is an equivalence relation.