Editing Equivalence relation (section)

== Connections to other relations ==

* A [[partial order]] is a relation that is reflexive, {{em|[[Antisymmetric relation|antisymmetric]]}}, and transitive.
* [[Equality (mathematics)|Equality]] is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In [[algebraic expression]]s, equal variables may be [[Substitution (algebra)|substituted]] for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class.
* A [[strict partial order]] is irreflexive, transitive, and [[asymmetric relation|asymmetric]].
* A [[partial equivalence relation]] is transitive and symmetric. Such a relation is reflexive [[if and only if]] it is [[total relation|total]], that is, if for all <math>a,</math> there exists some <math>b \text{ such that } a \sim b.</math><ref group="proof">''If:'' Given <math>a,</math> let <math>a \sim b</math> hold using totality, then <math>b \sim a</math> by symmetry, hence <math>a \sim a</math> by transitivity. &mdash; ''Only if:'' Given <math>a,</math> choose <math>b = a,</math> then <math>a \sim b</math> by reflexivity.</ref> Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and total relation.
* A [[ternary equivalence relation]] is a ternary analogue to the usual (binary) equivalence relation.
* A reflexive and symmetric relation is a [[dependency relation]] (if finite), and a [[tolerance relation]] if infinite.
* A [[preorder]] is reflexive and transitive.
* A [[congruence relation]] is an equivalence relation whose domain <math>X</math> is also the underlying set for an [[algebraic structure]], and which respects the additional structure. In general, congruence relations play the role of [[Kernel (algebra)|kernels]] of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the [[normal subgroup]]s).
* Any equivalence relation is the negation of an [[apartness relation]], though the converse statement only holds in classical mathematics (as opposed to [[constructive mathematics]]), since it is equivalent to the [[law of excluded middle]].
* Each relation that is both reflexive and left (or right) [[Euclidean relation|Euclidean]] is also an equivalence relation.