## Equivalence Class

An **equivalence class** is a subset of a space that contains all the elements equivalent to a specific element, under some equivalence relation.

Symbolically, equivalence class of $a$ is $[a] = \{x \in X | xRa \}$, where $R$ is an equivalence relation defined on space $X$.
Element $a$ can be called a **class representative** of this equivalence class.

### Equivalence relation

An **equivalence relation** is a reflexive, transitive and symmetric bi-variate relation on some space.

Symbolically, $R$ is an equivalence relation on space $X$ if:

- Reflexivity: $aRa, \forall a \in X$
- Transitivity: $aRb, bRc \Rightarrow aRc$
- Symmetry: $aRb \Rightarrow bRa$

### Equivalence relation partitions a space into equivalence classes.

It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of the underlying space.

A **set of class representatives** is a subset of the underlying space which contains exactly one element from each equivalence class.

### Examples

Isomorphism

Homeomorphism

Isometry

Equivalent metrics

🏷 Category=Analysis