Let X, Y be non empty sets and f: X \rightarrow Y a mapping and R_f a relation wit R_f := {(x,y) \in X \times X | f(x) = f(y)}, Show that R_f is an equivalence relation and that for every x \in X the equivalence class is defined by [x] := {x \in X | f(x) = y} for all y \in f(X).

To demonstrate that \( R_f \) is an equivalence relation and to describe the equivalence classes, we proceed with the following steps:

### Part 1: Show that \( R_f \) is an equivalence relation

An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. Let’s check these properties for \( R_f \).

#### Reflexivity:
A relation \( R \) on a set \( X \) is reflexive if, for all \( x \in X \), \( (x, x) \in R \). We need to show that for every \( x \in X \), \( (x, x) \in R_f \). 

By the definition of \( R_f \),
\[
(x, x) \in R_f \iff f(x) = f(x),
\]
which is true because a function is always equal to itself for the same input. Therefore, \( R_f \) is reflexive.

#### Symmetry:
A relation \( R \) on a set \( X \) is symmetric if, for all \( x, y \in X \), \( (x, y) \in R \) implies \( (y, x) \in R \). We need to show that if \( (x, y) \in R_f \), then \( (y, x) \in R_f \).

By the definition of \( R_f \),
\[
(x, y) \in R_f \iff f(x) = f(y).
\]
Since equality of functions is symmetric, \( f(x) = f(y) \iff f(y) = f(x) \), we have that \( (y, x) \in R_f \). Therefore, \( R_f \) is symmetric.

#### Transitivity:
A relation \( R \) on a set \( X \) is transitive if, for all \( x, y, z \in X \), \( (x, y) \in R \) and \( (y, z) \in R \) implies \( (x, z) \in R \). We need to show that if \( (x, y) \in R_f \) and \( (y, z) \in R_f \), then \( (x, z) \in R_f \).

By the definition of \( R_f \),
\[
(x, y) \in R_f \iff f(x) = f(y),
\]
and
\[
(y, z) \in R_f \iff f(y) = f(z).
\]
If \( f(x) = f(y) \) and \( f(y) = f(z) \), then \( f(x) = f(z) \), so \( (x, z) \in R_f \). Therefore, \( R_f \) is transitive.

Since \( R_f \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Part 2: Define the equivalence classes

Given that \( R_f \) is an equivalence relation, for each \( x \in X \), the equivalence class of \( x \), denoted by \( [x] \), consists of all elements in \( X \) that are related to \( x \) under \( R_f \). This means:
\[
[x] = \{ y \in X \mid (x, y) \in R_f \}.
\]
By the definition of \( R_f \),
\[
(x, y) \in R_f \iff f(x) = f(y).
\]
Thus, the equivalence class of \( x \) is:
\[
[x] = \{ y \in X \mid f(x) = f(y) \}.
\]
In other words, the equivalence class of \( x \) contains all elements of \( X \) that are mapped to the same value as \( x \) under the function \( f \).

Alternatively, for all \( x \in X \) and for every \( y \in f(X) \) (where \( f(X) \) is the image of \( f \)), the equivalence class of \( x \) can be described as:
\[
[x] = \{ x' \in X \mid f(x') = f(x) \}.
\]

### Conclusion:
We have shown that \( R_f \) is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity. Moreover, the equivalence class of an element \( x \in X \) under \( R_f \) is the set of all elements in \( X \) that are mapped to the same value as \( x \) by the function \( f \).

### Do you need further clarification, or would you like more details on any of these points?

Here are five follow-up questions to deepen understanding:
1. How does the definition of an equivalence relation change if we modify the function \( f \)?
2. Can \( R_f \) fail to be an equivalence relation for certain types of functions?
3. How do equivalence classes partition the set \( X \)?
4. How would the equivalence classes look if \( f \) is injective?
5. How do the equivalence classes change if \( f \) is a constant function?

**Tip:** When dealing with equivalence relations, remember that they naturally induce a partition of the set into disjoint equivalence classes.

Let X, Y be non empty sets and f: X → Y a mapping and R_f a relation with R_f := {(x,y) ∈ X × X | f(x) = f(y)}. Show that R_f is an equivalence relation and that for every x ∈ X the equivalence class is defined by [x] := {x ∈ X | f(x) = y} for all y ∈ f(X).

Learn how to prove that the relation R_f derived from a function f: X → Y is an equivalence relation. This step-by-step guide covers reflexivity, symmetry, and transitivity, and explores how equivalence classes are defined by the function f. Suitable for undergraduate students studying set theory and equivalence relations.

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