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:

  1. Reflexivity: $aRa, \forall a \in X$
  2. Transitivity: $aRb, bRc \Rightarrow aRc$
  3. 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