# Equivalence Relation

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

**AKA:**Equivalence Order, ≡, ~.**Context:**- It can range from being a Partial Equivalence Relation to being a Full Equivalence Relation.
- It can range from being a Singleton Equivalence Relation to being a Set Equivalence Relation.
- It can define an Equivalence Class Set.

**Example(s):**- ≡(2,2) ⇒ True.
- ≡(
*x*,*x*) ⇒ True. - ≡({2},{1,2,3}) ⇒ False.
- ≡({1,2,3},{1,2,3}) ⇒ True.
- a Semantic Equivalence Relation.
- an Equivalence Premise.

**Counter-Example(s):****See:**Semantic Equivalence, Identity Relation, if And Only if, Set (Mathematics), Partition of a Set, Empty Set, Clustering Task.

## 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 "*a*≡*b*”, which are used when R*is the obvious relation being referenced, and variations of "*a*~*R_{}*a*≡_{R}*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*:**a*~*a*. (Reflexivity)- if
*a*~*b*then*b*~*a*. (Symmetry) - if
*a*~*b*and*b*~*c*then*a*~*c*. (Transitivity)

*X*together with the relation ~ is called a setoid. The equivalence class of a*under ~, denoted [*a*], is defined as [math] [a] = \{b\in X \mid a\sim b\} [/math] .*

- In mathematics, an

### 2005

- (ANSI Z39.19, 2005) ⇒ ANSI. (2005). “ANSI/NISO Z39.19 - Guidelines for the Construction, Format, and Management of Monolingual Controlled Vocabularies." ANSI.
- QUOTE: "
*equivalence relationship A relationship between or among terms in a controlled vocabulary that leads to one or more terms that are to be used instead of the term from which the cross-reference is made; begins with the word SEE or USE.*

- QUOTE: "