If R is a relation on the set
Question:

If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) all the three options

Solution:

(d) all the three options

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

Reflexivity: Let $a \in A$. Then,

$a=a$

$\Rightarrow(a, a) \in R$ 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 $A$.

Transitivity: 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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