Equivalence relation

equivalenceequivalentmoduloequivalence relationsequivalence classesequivalence kernelequivalencesequivalencyFine (mathematics)finer
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.wikipedia
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Transitive relation

transitivetransitivitytransitive property
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
. Transitivity is a key property of both partial orders and equivalence relations.

Equivalence class

quotient setequivalence classesquotient
As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes.
have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set

Setoid

Bishop setconstructive setoidExtensional set
X together with the relation ~ is called a setoid.
In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~.

Modular arithmetic

modulomodcongruent
The congruence relation satisfies all the conditions of an equivalence relation:

Reflexive relation

reflexivereflexivityirreflexive
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.

Symmetric relation

symmetricsymmetryasymmetric
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

Congruence relation

congruencecongruentcongruences
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements.

Preorder

preordered setpreordered setsquasiorder
Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder.

Invariant (mathematics)

invariantinvariantsinvariance
See also invariant.
More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.

Equality (mathematics)

equalityequalequalities
These three properties make equality an equivalence relation.

Partial equivalence relation

If R is also reflexive, then R is an equivalence relation.

Quotient space (topology)

quotient spacequotient topologyquotient
If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient space for the details.
The points to be identified are specified by an equivalence relation.

Bell number

Bell numbersall partitions of a setexpression
Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B n :
The nth of these numbers, B n, counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it.

Apartness relation

apart
# The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity.

Integer

integersintegralZ
. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation

Congruence (geometry)

congruentcongruencecongruent triangles
Congruence is an equivalence relation.

Serial relation

totaltotal relationserial
In this case the relation is an equivalence relation.

Dependency relation

Independency relationDependenceDependency
In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation by discarding transitivity.

Cardinality

cardinalitiesnumber of elementssize
The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.

Groupoid

groupoidsBrandt groupoidgroupoid and double groupoid representations
The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

Homeomorphism

homeomorphichomeomorphismstopologically equivalent
"Being homeomorphic" is an equivalence relation on topological spaces.

Order theory

orderorder relationordering
Much of mathematics is grounded in the study of equivalences, and order relations.
It is also the only relation that is both a partial order and an equivalence relation.

Kernel (algebra)

kernelkernelsKernel (mathematics)
Less formally, the equivalence relation ker on X, takes each function f: X→X to its kernel ker f.
It turns out that ker f is an equivalence relation on M, and in fact a congruence relation.

Partition of a set

partitionpartitionspartitioned
As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes.
Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation.

Binary relation

relationrelationsidentity relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.