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Let R1 = {(a, b) ∈ N × N : |a - b| ≤ 13} and R2 = {(a, b) ∈ N × N : |a - b| ≠ 13}. Then on N: (1) Both R1 and R2 are equivalence relations (2) Neither R1 nor R2 is an equivalence relation (3) R1 is an equivalence relation but R2 is not (4) R2 is an equivalence relation but R1 is not

1. Both R1 and R2 are equivalence relations
2. Neither R1 nor R2 is an equivalence relation
3. R1 is an equivalence relation but R2 is not
4. R2 is an equivalence relation but R1 is not

Correct answer: Neither R1 nor R2 is an equivalence relation

Solution

An equivalence relation must satisfy reflexivity, symmetry, and transitivity. R1 fails transitivity because if (a, b) and (b, c) are in R1, (a, c) may not satisfy |a - c| ≤ 13. R2 fails reflexivity since (a, a) is not in R2 for any a where |a - a| = 0, thus neither relation qualifies as an equivalence relation.

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