Let `f : X->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.

Explanation

To determine whether the relation \( R \) defined by \( R = \{(a, b) : f(a) = f(b)\} \) is an equivalence relation, we need to check if it satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step 1: Check Reflexivity
A relation \( R \) is reflexive if for every element \( a \in X \), the pair \( (a, a) \) is in \( R \).

- For \( (a, a) \) to be in \( R \), we need \( f(a) = f(a) \).
- Since \( f(a) \) is always equal to itself, \( (a, a) \) is indeed in \( R \) for all \( a \in X \).

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