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Let R be the real line. Consider the following subsets of the plane R×R: S={(x,y): y=x+1 and 0<x<2} T={(x,y): x−y is an integer}. Which one of the following is true?

1. Neither S nor T is an equivalence relation on R
2. Both S and T are equivalence relations on R
3. S is an equivalence relation but T is not
4. T is an equivalence relation but S is not

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

Solution

S = {(x,y): y=x+1} is not reflexive (no x satisfies x=x+1), so it is not an equivalence relation. T = {(x,y): x-y is an integer} is reflexive, symmetric and transitive, so T is an equivalence relation but S is not.

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