Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p - s = q - r$. Which one of the following is true about $R$?

Given the condition for $((p, q), (r, s)) \in R$:

1. Reflexivity: For $R$ to be reflexive, $((p, q), (p, q)) \in R$ must hold for all pairs of positive integers $(p, q)$. This requires:

   $$p + p = q + q \iff 2p = 2q \iff p = q$$

   This is not true for arbitrary positive integers (e.g., for $(1, 2)$, $1 + 1 \neq 2 + 2$). Thus, $R$ is not reflexive.

2. Symmetry: Assume $((p, q), (r, s)) \in R$, which means $p + r = q + s$. For symmetry, $((r, s), (p, q)) \in R$ must hold, which requires:

   $$r + p = s + q$$

   Since addition of integers is commutative, $p + r = q + s$ implies $r + p = s + q$. Thus, $R$ is symmetric.

Therefore, $R$ is Not reflexive but symmetric, making C the correct option.