Question

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is
$\left( A \right)$ Symmetric only
$\left( B \right)$ Reflexive only
$\left( C \right)$ An equivalence relation
$\left( D \right)$ Transitive relation

Answer 1

Hint: In this particular question use the concept if a given set satisfies all the relations (i.e. symmetric, reflexive and transitive) then this relation is called as an equivalence relation so use this concept to reach the solution of the question.

Complete step-by-step answer:
Given set, A = {1, 2, 3}
Given relation, R = {(1, 1), (2, 2), (3, 3)}
Reflexive relation
If all the ordered pair elements of set A are in R then R is known to be called a reflexive relation.
Therefore, (a, a) $ \in $R, for all a$ \in $A
Where, a = a
So in set (A) all ordered pairs are (1, 1), (2, 2) and (3, 3) so all these ordered pairs are in set R so R is a reflexive relation.
Symmetric relation
If a, b $ \in $A such that (a, b) $ \in $ R then (b, a) $ \in $ R so this is called a symmetric relation.
In this case if a = b then this condition is also satisfied.
$ \Rightarrow a = b$
$ \Rightarrow b = a$
$ \Rightarrow \left( {b,a} \right) \in R$ For all a, b$ \in $A
So, R is also a symmetric relation.
Transitive relation
If a, b, c $ \in $A such that (a, b) $ \in $ R and (b, c) $ \in $ R then (a, c) $ \in $ R so this is called a transitive relation.
In this case if a = b = c then this condition is also satisfied.
$ \Rightarrow a = b,b = c$
$ \Rightarrow a = c$
$ \Rightarrow \left( {a,c} \right) \in R$ For all a, b, c$ \in $A
So, R is also a transitive relation.
Now as we all know if any relation R satisfying reflexive, symmetric and transitive then it is called as an equivalence relation.
So relation R is an equivalence relation.
Hence option (C) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember that for a given relation check all the possible conditions it satisfies then decide which relation it follows i.e. if we check whether it is symmetric or not and it satisfies then do not mark the answer check all the given options then mark the correct one as above doing this we will get the correct answer.