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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 qm=pn}. 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

Correct answer: S is an equivalence relation but R is not an equivalence relation

Solution

S: m/n ~ p/q iff qm=pn means m/n=p/q, i.e. ordinary equality of rationals -> reflexive, symmetric, transitive, so S is an equivalence relation. R: x=w*y for rational w fails symmetry: (0,5) is in R (0=0*5) but (5,0) is not (5=w*0 impossible), so R is not an equivalence relation. Correct: S is an equivalence relation but R is not.

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