Let $f: \mathrm{X} \rightarrow \mathrm{Y}$ be a function. Define a relation $\mathrm{R}$ in $\mathrm{X}$ given by $\mathrm{R}=\{(a, b): f(a)=f(b)\}$. Examine whether $\mathrm{R}$ is an equivalence relation or not.

Step 1: Check for Reflexivity

For a relation to be reflexive, every element must be related to itself. This means for any element $a$ in set $X$, the pair $(a, a)$ must be in $R$. Since $f(a)$ is always equal to $f(a)$, the condition $f(a) = f(a)$ is always true for any $a \in X$.

Step 2: Conclusion on Reflexivity

Since $f(a) = f(a)$ is always true, it implies that $(a, a) \in R$ for all $a \in X$. Therefore, the relation $R$ is reflexive.

Step 3: Check for Symmetry

For a relation to be symmetric, if $(a, b)$ is in $R$, then $(b, a)$ must also be in $R$. If $(a, b) \in R$, it means $f(a) = f(b)$. Since equality is symmetric, $f(b)$ must also be equal to $f(a)$. This implies that $(b, a) \in R$.

Step 4: Conclusion on Symmetry

Since $f(a) = f(b)$ implies $f(b) = f(a)$, the relation $R$ is symmetric.

Step 5: Check for Transitivity

For a relation to be transitive, if $(a, b)$ is in $R$ and $(b, c)$ is in $R$, then $(a, c)$ must also be in $R$. If $(a, b) \in R$, then $f(a) = f(b)$. If $(b, c) \in R$, then $f(b) = f(c)$. By the transitive property of equality, if $f(a) = f(b)$ and $f(b) = f(c)$, then $f(a) = f(c)$. This implies that $(a, c) \in R$.

Step 6: Conclusion on Transitivity

Since $f(a) = f(b)$ and $f(b) = f(c)$ implies $f(a) = f(c)$, the relation $R$ is transitive.

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