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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 on R but T is not (4) T is an equivalence relation on R but S is not

1. (1) neither S nor T is an equivalence relation on R
2. (2) both S and T are equivalence relations on R
3. (3) S is an equivalence relation on R but T is not
4. (4) T is an equivalence relation on R but S is not

Correct answer: (1) neither S nor T is an equivalence relation on R

Solution

Neither S nor T satisfies the properties required for an equivalence relation: reflexivity, symmetry, and transitivity. S is limited to a specific segment of the line and does not relate all elements of R, while T, which relates points based on integer differences, fails to be symmetric and transitive for all real numbers.

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