# Category:Set Intersection

This category contains results about Set Intersection.

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

Let $S$ and $T$ be any two 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 29 subcategories, out of 29 total.

### A

### C

### D

### E

### I

### P

### S

### U

## Pages in category "Set Intersection"

The following 142 pages are in this category, out of 142 total.

### C

- Cartesian Product Distributes over Intersection
- Cartesian Product of Intersections
- Centralizer in Subgroup is Intersection
- Characteristic Function of Intersection
- Characteristic Function of Intersection/Variant 1
- Characteristic Function of Intersection/Variant 2
- 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

### 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

### I

- Identity of Power Set with Intersection
- Image of Intersection under Injection
- Image of Intersection under Mapping
- Image of Intersection under Mapping/Family of Sets
- Image of Intersection under Mapping/General Result
- Image of Intersection under One-to-Many Relation
- Image of Intersection under One-to-Many Relation/Family of Sets
- Image of Intersection under One-to-Many Relation/General Result
- Image of Intersection under Relation
- Image of Intersection under Relation/Family of Sets
- Image of Intersection under Relation/General Result
- 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 Intersection
- 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 Idempotent
- Intersection is Idempotent/Indexed Family
- 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 Closed Sets is Closed/Closure Operator
- Intersection of Congruence Classes
- Intersection of Elements of Power Set
- Intersection of Empty Set
- Intersection of Empty Set/Paradoxical Implications
- Intersection of Equivalences
- Intersection of Family is Subset of Intersection of Subset of Family
- Intersection of Image with Subset of Codomain
- Intersection of Injective Image with Relative Complement
- Intersection of Interiors contains Interior of Intersection
- Intersection of Normal Subgroup with Sylow P-Subgroup
- Intersection of Power Sets
- Intersection of Primitive Recursive Sets
- Intersection of Reflexive Relations is Reflexive
- Intersection of Relation with Inverse is Symmetric Relation
- Intersection of Ring Ideals Containing Subset is Smallest
- 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 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 Subrings Containing Subset is Smallest
- Intersection of Subrings is Largest Subring Contained in all Subrings
- Intersection of Subrings is Subring
- Intersection of Subsemigroups
- Intersection of Subsemigroups/General Result
- 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 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 is Commutative Monoid
- 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 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 Disjoint with Reverse
- 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 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 Equation: Intersection
- Set Intersection Distributes over Set Difference
- Set Intersection expressed as Intersection Complement
- Set Intersection is not Cancellable
- Set Intersection Not Cancellable
- Set Intersection Preserves Subsets
- Set is Subset of Intersection of Supersets
- Set System Closed under Intersection is Commutative Semigroup
- Set Union expressed as Intersection Complement
- Symmetric Difference with Intersection forms Ring