Let R be a symmetric and transitive relation on a set A. Then A. R is reflexive and hence an equivalence relation B. R is reflexive and hence a partial order C. R is reflexive and hence not an equivalence relation D. None of the above

GATE 1995 · Discrete Mathematics · Relations · medium

Answer: D. None of the above — a symmetric and transitive relation need not be reflexive.

1. Identify the faulty argument: The 'proof' that symmetric + transitive implies reflexive goes: if (a,b) in R then (b,a) in R (symmetry), so (a,a) in R (transitivity). But this requires an element b with (a,b) in R — an element with NO related pairs is not covered.
2. Construct a counterexample: Symmetric: (1,1) has its reverse (1,1) in R. Transitive: (1,1)+(1,1) gives (1,1) in R. But (2,2) is not in R, so R is not reflexive. Hence symmetric + transitive does not imply reflexive.