Let \(R\) be a relation on the set of ordered pairs of positive integers such…
2015
Let \(R\) be a relation on the set of ordered pairs of positive integers such that \(((p,q),(r,s)) ϵ R\) if and only if \(p− s = q− r\). Which one of the following is true about \(R\)?
Answer: C. Not reflexive but symmetric — Short answer: R is symmetric but not reflexive. Reflexive check: For (p,q) to relate to itself we need p−q = q−p, which gives 2(p−q) = 0 and hence p = q.…
- A.
Both reflexive and symmetric
- B.
Reflexive but not symmetric
- C.
Not reflexive but symmetric
- D.
Neither reflexive nor symmetric
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Correct answer: C
Short answer: R is symmetric but not reflexive.
Reflexive check: For (p,q) to relate to itself we need p−q = q−p, which gives 2(p−q) = 0 and hence p = q. Since not every ordered pair of positive integers has equal coordinates, R is not reflexive.
Symmetry check: If p−s = q−r then multiplying both sides by −1 gives r−q = s−p, so ((r,s),(p,q)) satisfies the same condition. Therefore R is symmetric.
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