Which of the following relations on the set A ={1, 2, 3} are equivalence ?

(A) R = {(1, 1), (2, 2), (1, 2), (2, 1)}

(B) R = {(1, 1), (2, 2), (3, 3)}

(C) R = {(1, 1), (1, 2), (2, 1)}

(D) R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}

(E) R = {(1, 1), (2, 2), (3, 3), (1, 2)}

Choose the correct answer from the options given below :

(B) and (D) only

The correct answer is Option (1) → (B) and (D) only ##

To determine if a relation $R$ on a set $A = \{1, 2, 3\}$ is an $\textbf{Equivalence Relation}$, it must satisfy three specific properties: $\textbf{Reflexive}, \textbf{Symmetric}$, and $\textbf{Transitive}$.

$\textbf{Reflexive:}$ Every element must be related to itself. For set $A$, $R$ must contain $(1, 1), (2, 2)$, and $(3, 3)$.

$\textbf{Symmetric:}$ If $(a, b) \in R$, then $(b, a)$ must also be in $R$.

$\textbf{Transitive:}$ If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$.

$\textbf{Analyzing the Relations}$

$\textbf{(A): Not Equivalence.}$ It is missing $(3, 3)$. Therefore, it fails the $\textbf{Reflexive}$ property.

$\textbf{(B): Equivalence.}$ This is the $\textbf{Identity Relation}$. It is reflexive $(1, 1, 2, 2, 3, 3)$, vacuously symmetric, and vacuously transitive.

$\textbf{(C): Not Equivalence.}$ Like (A), it is missing $(2, 2)$ and $(3, 3)$. It fails the \textbf{Reflexive} property.

$\textbf{(D): Equivalence.}$ This is the $\textbf{Universal Relation}$ ($A \times A$). Every possible pair is present, so it automatically satisfies all three properties.

$\textbf{(E): Not Equivalence.}$ While it is reflexive $(1, 1, 2, 2, 3, 3)$, it is $\textbf{not symmetric}$ because $(1, 2)$ is present but $(2, 1)$ is missing.