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 $a \in A$
Here,
$a=a$
$\Rightarrow(a, a) \in R$ 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 $A$.
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.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.