# 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**

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.