The relation


The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is
(a) symmetric only
(b) reflexive only
(c) an equivalence relation
(d) transitive only


(c) an equivalence relation

$R=\{(a, b): a=b$ and $a, b \in A\}$

Reflexivity: Let $a \in A$



$\Rightarrow(a, a) \in R$ for all $a \in A$

So, $R$ is reflexive on $A$.

Symmetry: Let $a, b \in A$ such that $(a, b) \in R$. Then,

$(a, b) \in R$

$\Rightarrow a=b$

$\Rightarrow b=a$

$\Rightarrow(b, a) \in R$ for all $a \in A$

So, $R$ is symmetric on $A$.

Transitive: Let $a, b, c \in A$ such that $(a, b) \in R$ and $(b, c) \in R$. Then,

$(a, b) \in R \Rightarrow a=b$

and $(b, c) \in R \Rightarrow b=c$

$\Rightarrow a=c$

$\Rightarrow(a, c) \in R$ for all $a \in A$

So, $R$ is transitive on $A$.

Hence, R is an equivalence relation on A.

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