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

Question

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

Accepted Answer

> [!TLDR]
> The relation $R$ defined by $f(a) = f(b)$ is always an equivalence relation on $X$ because it is reflexive, symmetric, and transitive — the three properties follow directly from the equality of function values.

The core idea here is simple: we are grouping elements of $X$ based on whether they map to the same output in $Y$ under $f$. This is exactly the "kernel" of the function — the relation that identifies elements that are indistinguishable from the function's perspective. Proving it's an equivalence relation means checking three properties, each of which flows naturally from the fact that equality itself is an equivalence relation.

Let's walk through it step by step.

1. **Reflexive**: For any $a \in X$, we need $(a, a) \in R$.  
   Since $f(a) = f(a)$ always (a value is equal to itself), the condition $f(a) = f(b)$ holds when $b = a$. So $(a, a) \in R$ for every $a \in X$.  
   *Reasoning*: This works because equality is reflexive — nothing deeper.

2. **Symmetric**: If $(a, b) \in R$, then $f(a) = f(b)$. But equality is symmetric: if $f(a) = f(b)$, then $f(b) = f(a)$. So $(b, a) \in R$ as well.  
   *Reasoning*: The condition is an equation; swapping the two sides doesn't change the truth.

3. **Transitive**: If $(a, b) \in R$ and $(b, c) \in R$, then $f(a) = f(b)$ and $f(b) = f(c)$. By the transitivity of equality, $f(a) = f(c)$. Hence $(a, c) \in R$.  
   *Reasoning*: Equality chains together — if two values both equal the same third value, they equal each other. …

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

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Method: Checking Equivalence for a Relation Built From a Function

Steps

Common Mistakes

Q.Let f:X→Yf: X \to Yf:X→Y be a function. Define a relation RRR in XXX given by R={(a,b):f(a)=f(b)}R = \{(a, b): f(a) = f(b)\}R={(a,b):f(a)=f(b)}. Examine whether RRR is an equivalence relation or not.

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

Unlock everything free for 14 days