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.
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.
For every $a \in X ,(a, a) \in R ,$ since $f(a)=f(a),$ showing that $R$ is reflexive. Similarly, $(a, b) \in R \Rightarrow f(a)=f(b)$ $ \Rightarrow f(b)=f(a)$ $ \Rightarrow(b, a) \in$ $R$. Therefore, $R$ is symmetric. Further, $(a, b) \in R$ and $(b, c) \in R \Rightarrow$ $f(a)=f(b)$ and $f(b)=f(c) \Rightarrow f(a)$ $=f(c) \Rightarrow(a, c) \in R ,$ which implies that $R$ is transitive. Hence, $R$ is an equivalence relation.
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