Question 51

The relation $R$ on positive integer pairs is defined by:

1. Test Reflexivity:
   $R$ is reflexive if $((p, q), (p, q)) \in R$ for every pair of positive integers $(p, q)$.
   Here, $((p, q), (p, q)) \in R \iff p + p = q + q \iff 2p = 2q \iff p = q$.
   This holds only if $p = q$. For any pair where $p \ne q$ (e.g., $(1, 2)$), $1 + 1 \ne 2 + 2$, so $((1, 2), (1, 2)) \notin R$.
   Thus, $R$ is not reflexive.

2. Test Symmetry:
   $R$ is symmetric if $((p, q), (r, s)) \in R \implies ((r, s), (p, q)) \in R$.
   Suppose $((p, q), (r, s)) \in R$. Then $p + r = q + s$.
   For $((r, s), (p, q)) \in R$, the condition is $r + p = s + q$.
   Since integer addition is commutative ($p + r = r + p$ and $q + s = s + q$), $r + p = s + q$ holds whenever $p + r = q + s$.
   Thus, $R$ is symmetric.

Therefore, the relation $R$ is not reflexive but symmetric (Option C).