**Question:**

Mark the tick against the correct answer in the following:

Let $A=\{a, b, c\}$ and let $R=\{(a, a),(a, b),(b, a)\} .$ Then, $R$ is

A. reflexive and symmetric but not transitive

B. reflexive and transitive but not symmetric

C. symmetric and transitive but not reflexive

D. an equivalence relation

**Solution:**

Given set $A=\{a, b, c\}$

And $R=\{(a, a),(a, b),(b, a)\}$

Formula

For a relation $R$ in set $A$

Reflexive

The relation is reflexive if $(a, a) \in R$ for every $a \in A$

Symmetric

The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$

Transitive

Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$

Equivalence

If the relation is reflexive, symmetric and transitive, it is an equivalence relation.

Check for reflexive

Since, $(b, b) \notin R$ and $(c, c) \notin R$

Therefore, $R$ is not reflexive ....... (1)

Check for symmetric

Since, $(a, b) \in R$ and $(b, a) \in R$

Therefore, $R$ is symmetric ....... (2)

Check for transitive

Here, $(a, b) \in R$ and $(b, a) \in R$ and $(a, a) \in R$

Therefore, $\mathrm{R}$ is transitive ....... (3)

Now, according to the equations (1), (2), (3)

Correct option will be (C)

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