A binary relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) is defined as…
2016
A binary relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) is defined as follows: \((a, b) R(c, d)\) if \(a \leq c\) or \(b \leq d\). Consider the following propositions:
P: \(R\) is reflexive
Q: \(R\) is transitive
Which one of the following statements is TRUE?
- A.
Both P and Q are true.
- B.
P is true and Q is false.
- C.
P is false and Q is true.
- D.
Both P and Q are false.
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Correct answer: B
Answer: P is true and Q is false.
Reflexive: For any (a,b) in N×N we have a ≤ a (and b ≤ b). Therefore (a,b)R(a,b) holds for every pair, so the relation is reflexive.
Not transitive: Provide a counterexample. Take (a,b) = (2,3), (c,d) = (3,1), (e,f) = (1,2).
(2,3)R(3,1) holds because 2 ≤ 3.
(3,1)R(1,2) holds because 1 ≤ 2.
But (2,3)R(1,2) is false since 2 ≤ 1 and 3 ≤ 2 are both false. Hence transitivity fails.
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