Let $f: X \rightarrow Y$ be a function. Define a relation $R$ in $X$ given by $R = \{(a, b) : f(a) = f(b)\}$. Examine if $R$ is an equivalence relation.

(A) To determine if $R$ is an equivalence relation,we must check if it is reflexive,symmetric,and transitive.
$1$. Reflexive: For every $a \in X$,we have $f(a) = f(a)$,which implies $(a, a) \in R$. Thus,$R$ is reflexive.
$2$. Symmetric: Let $(a, b) \in R$. Then $f(a) = f(b)$,which implies $f(b) = f(a)$. Therefore,$(b, a) \in R$. Thus,$R$ is symmetric.
$3$. Transitive: Let $(a, b) \in R$ and $(b, c) \in R$. Then $f(a) = f(b)$ and $f(b) = f(c)$. By the transitive property of equality,$f(a) = f(c)$,which implies $(a, c) \in R$. Thus,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.