GATE CS 1998 Q2.3 — Sets, Relations and Functions

To determine the properties of the binary relation $R$ on the set $A = \{1, 2, 3, 4\}$, let us analyze each property step-by-step:

1. Reflexivity: A relation $R$ on set $A$ is reflexive if $(a, a) \in R$ for all $a \in A$. Here, $(4, 4) \notin R$. Thus, $R$ is not reflexive.
2. Irreflexivity: A relation $R$ on set $A$ is irreflexive if $(a, a) \notin R$ for all $a \in A$. Here, $(1, 1), (2, 2), (3, 3) \in R$. Thus, $R$ is not irreflexive.
3. Transitivity: A relation $R$ is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$. Let us verify the non-trivial compositions:
   • For $(3, 2) \in R$ and $(2, y) \in R$:
     • $(3, 2)$ and $(2, 1) \implies (3, 1) \in R$
     • $(3, 2)$ and $(2, 2) \implies (3, 2) \in R$
     • $(3, 2)$ and $(2, 3) \implies (3, 3) \in R$
     • $(3, 2)$ and $(2, 4) \implies (3, 4) \in R$
   • For $(2, 3) \in R$ and $(3, y) \in R$:
     • $(2, 3)$ and $(3, 1) \implies (2, 1) \in R$
     • $(2, 3)$ and $(3, 2) \implies (2, 2) \in R$
     • $(2, 3)$ and $(3, 3) \implies (2, 3) \in R$
     • $(2, 3)$ and $(3, 4) \implies (2, 4) \in R$
   All other combinations also satisfy the condition. Thus, $R$ is transitive.

Since $R$ is neither reflexive nor irreflexive, but is transitive, the correct option is B.