# Category:Set Intersection

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This category contains results about **Set Intersection**.

Definitions specific to this category can be found in Definitions/Set Intersection.

Let $S$ and $T$ be sets.

The **(set) intersection** of $S$ and $T$ is written $S \cap T$.

It means the set which consists of all the elements which are contained in both of $S$ and $T$:

- $x \in S \cap T \iff x \in S \land x \in T$

## Also see

## Subcategories

This category has the following 43 subcategories, out of 43 total.

### A

- Absorption Laws (12 P)

### C

### D

### E

- Examples of Set Intersection (21 P)

### I

- Intersection is Associative (5 P)
- Intersection is Commutative (5 P)
- Intersection of Empty Set (3 P)
- Intersections of Families (3 P)

### S

### U

- Union of Intersections (3 P)

## Pages in category "Set Intersection"

The following 157 pages are in this category, out of 157 total.

### A

### C

- Cartesian Product Distributes over Intersection
- Cartesian Product of Intersections
- Centralizer in Subgroup is Intersection
- Characteristic Function of Intersection
- Closure of Intersection and Symmetric Difference imply Closure of Set Difference
- Closure of Intersection and Symmetric Difference imply Closure of Union
- Closure of Intersection is Subset of Intersection of Closures
- Closure of Intersection may not equal Intersection of Closures

### D

### E

- Empty Intersection iff Subset of Complement
- Empty Intersection iff Subset of Complement/Corollary
- Empty Intersection iff Subset of Relative Complement
- Equal Set Differences iff Equal Intersections
- Equivalence of Definitions of Symmetric Difference
- Existence of Set with Singleton Intersections with Disjoint Collection

### F

### I

- Identity of Power Set with Intersection
- Image of Intersection under Injection
- Image of Intersection under Mapping
- Image of Intersection under One-to-Many Relation
- Image of Intersection under Relation
- Index of Intersection of Subgroups
- Infinite Intersection of Open Sets of Metric Space may not be Open
- Interior of Finite Intersection equals Intersection of Interiors
- Interior of Intersection may not equal Intersection of Interiors
- Intersection Complement of Set with Itself is Complement
- Intersection Distributes over Symmetric Difference
- Intersection Distributes over Union
- Intersection is Associative
- Intersection is Commutative
- Intersection is Decreasing
- Intersection is Empty and Union is Universe if Sets are Complementary
- Intersection is Empty Implies Intersection of Subsets is Empty
- Intersection is Largest Subset
- Intersection is Largest Subset/Family of Sets
- Intersection is Largest Subset/General Result
- Intersection is Subset
- Intersection is Subset of Union
- Intersection is Subset of Union of Intersections with Complements
- Intersection is Subset/Family of Sets
- Intersection is Subset/General Result
- Intersection of All Ring Ideals Containing Subset is Smallest
- Intersection of All Subrings Containing Subset is Smallest
- Intersection of Antisymmetric Relations is Antisymmetric
- Intersection of Closed Sets is Closed
- Intersection of Closed Sets is Closed/Closure Operator
- Intersection of Congruence Classes
- Intersection of Elements of Power Set
- Intersection of Empty Set
- Intersection of Equivalences
- Intersection of Family is Subset of Intersection of Subset of Family
- Intersection of Horizontal Sections is Horizontal Section of Intersection
- Intersection of Image with Subset of Codomain
- Intersection of Inductive Sets
- Intersection of Injective Image with Relative Complement
- Intersection of Interiors contains Interior of Intersection
- Intersection of Normal Subgroup with Sylow P-Subgroup
- Intersection of Normal Subgroups is Normal
- Intersection of Open Set with Closure of Set is Subset of Closure of Intersection
- Intersection of Orderings is Ordering
- Intersection of Ordinals is Ordinal
- Intersection of Power Sets
- Intersection of Primitive Recursive Sets
- Intersection of Reflexive Relations is Reflexive
- Intersection of Regular Closed Sets is not necessarily Regular Closed
- Intersection of Relation with Inverse is Symmetric Relation
- Intersection of Ring Ideals is Ideal
- Intersection of Ring Ideals is Largest Ideal Contained in all Ideals
- Intersection of Semilattice Ideals is Ideal/Set of Sets
- Intersection of Set whose Every Element is Closed under Chain Unions is also Closed under Chain Unions
- Intersection of Set whose Every Element is Closed under Mapping is also Closed under Mapping
- Intersection of Singleton
- Intersection of Subfields Containing Subset is Smallest
- Intersection of Subfields is Largest Subfield Contained in all Subfields
- Intersection of Subfields is Subfield
- Intersection of Subgroups is Subgroup
- Intersection of Subgroups is Subgroup/General Result
- Intersection of Submagmas is Largest Submagma
- Intersection of Subrings is Largest Subring Contained in all Subrings
- Intersection of Subrings is Subring
- Intersection of Subsemigroups
- Intersection of Subsets is Subset/Set of Sets
- Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup
- Intersection of Symmetric Relations is Symmetric
- Intersection of Transitive Relations is Transitive
- Intersection of Transitive Relations is Transitive/General Result
- Intersection of Two Ordinals is Ordinal
- Intersection of Unions with Complements is Subset of Union
- Intersection of Vertical Sections is Vertical Section of Intersection
- Intersection with Complement
- Intersection with Complement is Empty iff Subset
- Intersection with Empty Set
- Intersection with Relative Complement is Empty
- Intersection with Set Difference is Set Difference with Intersection
- Intersection with Subset is Subset
- Intersection with Universe

### P

- Power Set is Closed under Intersection
- Power Set with Intersection and Subset Relation is Ordered Semigroup
- Power Set with Intersection and Superset Relation is Ordered Semigroup
- Power Set with Intersection is Commutative Monoid
- Powerset of Subset is Closed under Intersection
- Preimage of Intersection under Mapping
- Preimage of Intersection under Relation
- Preimage of Intersection under Relation/Family of Sets
- Preimage of Intersection under Relation/General Result
- Product of Subset with Intersection

### S

- Set Difference and Intersection are Disjoint
- Set Difference and Intersection form Partition
- Set Difference and Intersection form Partition/Corollary 1
- Set Difference as Intersection with Complement
- Set Difference as Intersection with Relative Complement
- Set Difference as Symmetric Difference with Intersection
- Set Difference Equals First Set iff Empty Intersection
- Set Difference Intersection with First Set is Set Difference
- Set Difference Intersection with Second Set is Empty Set
- Set Difference is Disjoint with Reverse
- Set Difference is Right Distributive over Set Intersection
- Set Difference of Intersection with Set is Empty Set
- Set Difference Union Intersection
- Set Difference with Intersection
- Set Difference with Intersection is Difference
- Set Difference with Set Difference
- Set Difference with Set Difference is Union of Set Difference with Intersection
- Set Differences and Intersection form Partition of Union
- Set Equation: Intersection
- Set Intersection Distributes over Set Difference
- Set Intersection expressed as Intersection Complement
- Set Intersection is Idempotent
- Set Intersection is Idempotent/Indexed Family
- Set Intersection is not Cancellable
- Set Intersection is Self-Distributive
- Set Intersection Not Cancellable
- Set Intersection Preserves Subsets
- Set is Subset of Intersection of Supersets
- Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection
- Set System Closed under Intersection is Commutative Semigroup
- Set Union expressed as Intersection Complement
- Symmetric Difference with Intersection forms Boolean Ring
- Symmetric Difference with Intersection forms Ring

### U

- Union as Symmetric Difference with Intersection
- Union Distributes over Intersection
- Union equals Intersection iff Sets are Equal
- Union minus Symmetric Difference equals Intersection
- Union of Intersections
- Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3
- Union with Intersection equals Intersection with Union iff Subset