Let N² = N × N, where N is the set of natural numbers, and let R be the relation on N² with ⟨x, y⟩ R ⟨u, v⟩ ⇔ xv = yu. Which of the following are TRUE? (A) Reflexive (B) Symmetric (C) Transitive (D) Asymmetric
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Correct answer: D. (A), (B) and (C) Only
Explanation
xy = yx gives reflexivity; xv = yu ⇔ uy = vx gives symmetry; xv = yu and uq = vp give xq = yp (cancel the non-zero u, v), so it is transitive. It is an equivalence relation (like equal fractions x/y), not asymmetric.
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