# Category:Equivalence Classes

This category contains results about Equivalence Classes.

Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$.

Let $x \in S$.

Then the **equivalence class of $x$ under $\mathcal R$** is the set:

- $\eqclass x {\mathcal R} = \set {y \in S: \tuple {x, y} \in \mathcal R}$

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Equivalence Classes"

The following 15 pages are in this category, out of 15 total.

### E

- Element in its own Equivalence Class
- Equivalence Class Equivalent Statements
- Equivalence Class holds Equivalent Elements
- Equivalence Class is not Empty
- Equivalence Class is Unique
- Equivalence Class of Element is Subset
- Equivalence Class of Equal Elements of Cross-Relation
- Equivalence Class of Fixed Element
- Equivalence Class of Fixed Element/Corollary
- Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms
- Equivalence Classes are Disjoint
- Equivalence Classes induced by Derivative Function on Set of Functions
- Equivalence Classes of Cross-Relation on Natural Numbers