# Equivalence Relation

An Equivalence Relation is a binary relation that is a reflexive relation, a symmetric relation, and a transitive relation.

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/equivalence_relation Retrieved:2015-6-2.
• In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition. The intersection of any two different cells is empty; the union of all the cells equals the original set. These cells are formally called equivalence classes.
• Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are "a ~ b" and "ab”, which are used when R is the obvious relation being referenced, and variations of "a ~R b”, "aR b”, or "aRb" otherwise.
• A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in X:
• X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as $[a] = \{b\in X \mid a\sim b\}$ .