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

Homeomorphism

Isometry

Equivalent metrics

🏷 Category=Analysis