# 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 $\mathcal R$ 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 13 subcategories, out of 13 total.

### C

### E

### F

### M

### O

### Q

### T

### U

## Pages in category "Equivalence Relations"

The following 91 pages are in this category, out of 91 total.

### A

- Absolute Value Function on Integers induces Equivalence Relation
- Addition of Cross-Relation Equivalence Classes on Natural Numbers is Cancellable
- Addition of Cross-Relation Equivalence Classes on Natural Numbers is Well-Defined
- Arc-Connectedness is Equivalence Relation
- Associatehood is Equivalence Relation

### C

- Cardinality of Set of Induced Equivalence Classes of Injection
- Cardinality of Set of Induced Equivalence Classes of Surjection
- Ceiling defines Equivalence Relation
- Composite of Inverse of Mapping with Mapping
- Composite of Mapping with Inverse
- Composite of Quotient Mappings
- Congruence Modulo Real Number is Equivalence Relation
- Connected Equivalence Relation is Trivial
- Connectedness Between Two Points is an Equivalence Relation
- Connectedness of Points is Equivalence Relation
- Construction of Inverse Completion/Equivalence Relation
- Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements
- Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes
- Construction of Inverse Completion/Quotient Mapping is Injective
- Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements
- Construction of Inverse Completion/Quotient Structure
- Construction of Inverse Completion/Quotient Structure is Commutative Semigroup
- Correspondence Theorem (Set Theory)
- Cross-Relation Equivalence Classes on Natural Numbers are Cancellable for Addition
- Cross-Relation on Natural Numbers is Equivalence Relation
- Cross-Relation on Real Numbers is Equivalence Relation

### 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
- Equivalence Relation is Congruence for Left Operation
- Equivalence Relation is Congruence for Right Operation
- Equivalent Characterizations of Finer Equivalence Relation

### H

### I

### M

### P

### 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
- 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
- Row Equivalence is Equivalence Relation

### S

- Saturation Under Equivalence Relation in Terms of Graph
- Set Equality is Equivalence Relation
- Set Equivalence is Equivalence Relation
- Set Equivalence Less One Element
- Set of Mappings which map to Same Element induces Equivalence Relation
- Strict Weak Ordering Induces Partition
- Symmetric Preordering is Equivalence Relation
- Symmetric Transitive and Serial Relation is Reflexive