# Category:Equivalence Relations

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This category contains results about Equivalence Relations.

Definitions specific to this category can be found in Definitions/Equivalence Relations.

Let $\RR$ be a relation on a set $S$.

Let $\RR$ be:

- $(1): \quad$ reflexive
- $(2): \quad$ symmetric
- $(3): \quad$ transitive

Then $\RR$ is an **equivalence relation** on $S$.

## Subcategories

This category has the following 17 subcategories, out of 17 total.

### C

- Congruence Modulo Subgroup (8 P)

### E

### F

### Q

### R

### T

### U

- Union of Equivalences (3 P)

### W

- Well-Defined Mappings (4 P)
- Well-Defined Relations (2 P)

## Pages in category "Equivalence Relations"

The following 52 pages are in this category, out of 52 total.

### C

- Cardinality of Set of Induced Equivalence Classes of Injection
- Cardinality of Set of Induced Equivalence Classes of Surjection
- Cauchy Sequences in Vector Spaces with Equivalent Norms Coincide
- Composite of Inverse of Mapping with Mapping
- Composite of Mapping with Inverse
- Composite of Quotient Mappings
- Connected Equivalence Relation is Trivial
- Construction of Inverse Completion
- Convergent Sequences in Vector Spaces with Equivalent Norms Coincide
- Correspondence Theorem (Set Theory)
- Cross-Relation Equivalence Classes on Natural Numbers are Cancellable for Addition

### D

### E

- Equivalence iff Diagonal and Inverse Composite
- Equivalence Induced on Preordering defines Ordering
- Equivalence of Definitions of Equivalence Relation
- Equivalence of Definitions of Saturated Set Under Equivalence Relation
- Equivalence of Definitions of Saturation Under Equivalence Relation
- Equivalent Characterizations of Finer Equivalence Relation

### R

- Reflexive Circular Relation is Equivalence
- Reflexive Euclidean Relation is Equivalence
- Relation Induced by Mapping is Equivalence Relation
- Relation Induced by Partition is Equivalence
- Relation Induced by Quotient Set is Equivalence
- Relation Isomorphism is Equivalence Relation
- Relation Isomorphism Preserves Equivalence Relations
- Relation on Empty Set is Equivalence
- Relation Partitions Set iff Equivalence
- Relation Partitions Set iff Equivalence/Proof
- Renaming Mapping from Set of Mappings on Single Element
- Renaming Mapping is Bijection
- Renaming Mapping is Well-Defined
- Restriction of Equivalence Relation is Equivalence
- Restriction of Ordering is Ordering