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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: R and S both are equivalence relations

Solution

Both R and S satisfy the properties of reflexivity, symmetry, and transitivity, which are essential for equivalence relations. For R, any real number can be expressed as a multiple of itself, ensuring reflexivity, while the relationship defined by a rational multiplier ensures symmetry and transitivity. Similarly, S, defined through integer ratios, also maintains these properties, confirming that both relations are indeed equivalence relations.

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