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?

Let the relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined by:

1. Reflexivity (Proposition P): For any $(a, b) \in \mathbb{N} \times \mathbb{N}$, we test if $(a, b)R(a, b)$. Since $a \le a$ (and $b \le b$) is always true for any natural number, $(a, b)R(a, b)$ holds for all $(a, b)$. Therefore, $R$ is reflexive, so proposition P is true.

2. Transitivity (Proposition Q): For $R$ to be transitive, if $(a, b)R(c, d)$ and $(c, d)R(e, f)$, then it must follow that $(a, b)R(e, f)$. Let us construct a counterexample:

   • Let $(a, b) = (5, 2)$

   • Let $(c, d) = (2, 3)$

   • Let $(e, f) = (1, 1)$

   Check the pairs:

   • $(a, b)R(c, d) \implies (5, 2)R(2, 3)$ is true because $2 \le 3$ ($b \le d$).

   • $(c, d)R(e, f) \implies (2, 3)R(1, 1)$ is false for this specific pair, so let's choose another:

   • Let $(a, b) = (4, 1)$, $(c, d) = (2, 3)$, and $(e, f) = (1, 2)$:

     • $(a, b)R(c, d) \implies (4, 1)R(2, 3)$ is true since $1 \le 3$.

     • $(c, d)R(e, f) \implies (2, 3)R(1, 2)$ is false since neither $2 \le 1$ nor $3 \le 2$.

   • Better counterexample:

     • Let $(a, b) = (5, 2)$

     • Let $(c, d) = (6, 1)$

     • Let $(e, f) = (2, 0)$ (or in $\mathbb{N} = \{1, 2, \dots\}$, $(e, f) = (2, 1)$):

     • $(5, 2)R(6, 1)$ is true because $5 \le 6$.

     • $(6, 1)R(2, 2)$ is true because $1 \le 2$.

     • However, $(5, 2)R(2, 2)$ requires $5 \le 2$ or $2 \le 2$, which holds.

   • Consider $(a, b) = (10, 2)$, $(c, d) = (12, 1)$, $(e, f) = (4, 1)$:

     • $(10, 2)R(12, 1)$ holds because $10 \le 12$.

     • $(12, 1)R(4, 1)$ holds because $1 \le 1$.

     • For $(10, 2)R(4, 1)$ to hold, we need $10 \le 4$ or $2 \le 1$, both of which are false!

   • Thus, $(10, 2)R(4, 1)$ does not hold.

   Therefore, $R$ is not transitive, so proposition Q is false.

Thus, P is true and Q is false, which matches option (B).