# Category:Relation Theory

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This category contains results about Relation Theory.

Definitions specific to this category can be found in Definitions/Relation Theory.

**Relation theory** is the subfield of set theory concerned with the properties of relations and relational structures.

As a relation has the same conceptual definition as a (graph-theoretical) graph, it follows that there is considerable overlap between the fields of relation theory and graph theory.

## Subcategories

This category has the following 52 subcategories, out of 52 total.

### A

- Approximating Relations (10 P)
- Auxiliary Relations (21 P)

### C

- Circular Relations (1 P)
- Codomains (Relation Theory) (empty)

### E

- Euclidean Relations (1 P)

### G

### I

- Inverses of Mappings (empty)

### L

- Left-Total Relations (5 P)

### M

- Many-to-One Relations (1 P)
- Membership Relation (3 P)

### O

### P

### R

- Relation Isomorphisms (14 P)
- Relational Closures (6 P)
- Relational Structures (empty)
- Right-Total Relations (5 P)

### S

- Schröder Rule (3 P)
- Serial Relations (6 P)
- Symmetric Closures (6 P)

### T

- Trichotomies (2 P)

### W

- Well-Defined Relations (2 P)

## Pages in category "Relation Theory"

The following 46 pages are in this category, out of 46 total.

### C

### E

### I

- Image is Subset of Codomain
- Image is Subset of Codomain/Corollary 1
- Image of Domain of Relation is Image Set
- Image of Element is Subset
- Image of Empty Set is Empty Set
- Image of Intersection under Relation
- Image of Relation is Domain of Inverse Relation
- Image of Set Difference under Relation
- Image of Singleton under Relation
- Image of Subset under Relation equals Union of Images of Elements
- Image of Subset under Relation is Subset of Image
- Image of Subset under Relation is Subset of Image/Corollary 1
- Image of Union under Relation
- Image Preserves Subsets
- Image under Subset of Relation is Subset of Image under Relation
- Intersection of Relations is Relation
- Inverse Image under Embedding of Image under Relation of Image of Point

### M

### P

### R

### U

- Union of Left-Total Relations is Left-Total
- Union of Many-to-One Relations with Disjoint Domains is Many-to-One
- Union of One-to-Many Relations with Disjoint Images is One-to-Many
- Union of Relations is Relation
- Union of Right-Total Relations is Right-Total
- Union of Union of Relation is Union of Domain with Image
- Uniqueness Condition for Relation Value