GATE CS 2015 Set 3 Q41 — Sets, Relations and Functions

Let the relation $R$ be defined on the set of ordered pairs of positive integers $\mathbb{Z}^+ \times \mathbb{Z}^+$ by:

Let's analyze the properties of reflexivity and symmetry for $R$:

1. Reflexivity

For $R$ to be reflexive, every pair $((p,q), (p,q))$ must belong to $R$ for all $p, q \in \mathbb{Z}^+$.

Applying the definition of the relation:

This condition only holds when $p = q$. For any pair where $p \neq q$ (for example, $(1, 2)$):

Since $-1 \neq 1$, the pair $((1,2), (1,2)) \notin R$. Therefore, $R$ is not reflexive.

2. Symmetry

For $R$ to be symmetric, if $((p,q), (r,s)) \in R$, then $((r,s), (p,q)) \in R$ must also hold.

Assume $((p,q), (r,s)) \in R$:

Multiplying both sides of the equation by $-1$:

This is precisely the condition for $((r,s), (p,q)) \in R$. Therefore, $R$ is symmetric.

Thus, the relation $R$ is not reflexive but symmetric.