# Category:Restrictions

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This category contains results about **Restrictions**.

Definitions specific to this category can be found in **Definitions/Restrictions**.

Let $\RR$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

The **restriction of $\RR$ to $X \times Y$** is the relation on $X \times Y$ defined as:

- $\RR {\restriction_{X \times Y} }: = R \cap \paren {X \times Y}$

where $R \subseteq S \times T$ is the subset of the Cartesian product of $S$ and $T$ which defines the relation $\RR$.

If $Y = T$, then we simply call this the **restriction of $\RR$ to $X$**, and denote it as $\RR {\restriction_X}$.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Restrictions"

The following 41 pages are in this category, out of 41 total.

### C

### I

### R

- Restriction is Subset of Relation
- Restriction of Antireflexive Relation is Antireflexive
- Restriction of Antisymmetric Relation is Antisymmetric
- Restriction of Antitransitive Relation is Antitransitive
- Restriction of Asymmetric Relation is Asymmetric
- Restriction of Congruence Relation is Congruence
- Restriction of Connected Relation is Connected
- Restriction of Equivalence Relation is Equivalence
- Restriction of Homomorphism is Homomorphism
- Restriction of Inverse is Inverse of Restriction
- Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image
- Restriction of Mapping is Mapping
- Restriction of Mapping is Mapping (General Result)
- Restriction of Mapping is Mapping/General Result
- Restriction of Mapping to Image is Surjection
- Restriction of Non-Connected Relation is Not Necessarily Non-Connected
- Restriction of Non-Reflexive Relation is Not Necessarily Non-Reflexive
- Restriction of Non-Symmetric Relation is Not Necessarily Non-Symmetric
- Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive
- Restriction of Ordering is Ordering
- Restriction of Reflexive Relation is Reflexive
- Restriction of Restriction is Restriction
- Restriction of Serial Relation is Not Necessarily Serial
- Restriction of Symmetric Relation is Symmetric
- Restriction of Transitive Relation is Transitive
- Restriction of Union of Mappings which Agree Equals Mapping