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Consider the following relations: R = {(...

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 , p and q are integers such that n ,q"!="0 and q m = p n"}`. then,

A

neither R nor S is an equivalence relation

B

S is an equivalence relation but R is not an equivalence relation

C

R and S both are equivalence relations

D

R is an equivalence relation but S is not an equivalence relation

Text Solution

Verified by Experts

The correct Answer is:
B
xRy need not implies yRx.
`S:(m)/(n)S(P)/(q)iffqm="pm"implies(m)/(s)S(m)/(n)` is reflexive. `(m)/(s)S(p)/(q)implies(p)/(q)S(m)/(n)` is symmetric.
and `(m)/(n)S(p)/(q),(p)/(q)S(r)/(t)impliesqm=pn,pt=qr`
`implies mt=nrimplies(m)/(n)S(r)/(t)` is transitive.
`therefore` S is an equivalence relation.
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