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
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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