Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) is in R if and only if p - q = s - r. Which one of the following is true about R? A. Both reflexive and symmetric B. Reflexive but not symmetric C. Not reflexive but symmetric D. Neither reflexive nor symmetric

GATE 2015 · Discrete Mathematics · Relations · medium

Answer: R is not reflexive but is symmetric. Answer: C. Not reflexive but symmetric.

1. Test reflexivity: p - q = q - p iff 2(p - q) = 0 iff p = q. For p != q (e.g., p=2, q=1): 2 - 1 = 1 but q - p = 1 - 2 = -1, and 1 != -1. So ((2,1),(2,1)) NOT in R. R is NOT reflexive.
2. Test symmetry: Assume p - q = s - r. We need to verify r - s = q - p for the reverse pair. From p - q = s - r, multiply both sides by -1: q - p = r - s. So r - s = q - p, which is exactly the condition for ((r,s),(p,q)) in R. R IS symmetric.