Discussion
A binary relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) is defined as follows: \((a,b)R(c,d)\) if \(a \le c\) or \(b \le d\). Consider the following propositions:
P: \(R\) is reflexive
Q: \(R\) is transitive
Which one of the following statements is TRUE?
Answer: B
P is true and Q is false: \(R\) is reflexive but not transitive.
Reflexive: \((a,b)R(a,b)\) holds because \(a \le a\).
Not transitive: take \((1,5)R(2,1)\), true since \(1 \le 2\), and \((2,1)R(0,3)\), true since \(1 \le 3\). But \((1,5)R(0,3)\) needs \(1 \le 0\) or \(5 \le 3\), and both fail. The “or” lets each step use a different coordinate, so the chain does not carry through.
Hence options calling \(R\) transitive, or not reflexive, are wrong.
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