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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: Neither S nor T is an equivalence relation on R

Solution

An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. The set S does not satisfy reflexivity since not every element in R is related to itself, and T fails to meet the transitivity requirement, as the integer difference condition does not guarantee that if x-y and y-z are integers, then x-z is also an integer.

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