# Category:Set Union

This category contains results about Set Union.

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

Let $S$ and $T$ be sets.

The **(set) union** of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$:

- $x \in S \cup T \iff x \in S \lor x \in T$

## Subcategories

This category has the following 20 subcategories, out of 20 total.

### A

### D

### E

### F

### I

### P

### S

### U

## Pages in category "Set Union"

The following 155 pages are in this category, out of 155 total.

### C

- Cardinality of Set Union
- Cardinality of Set Union/Corollary
- Cardinality of Set Union/General Case
- Cardinality of Union not greater than Product
- Cartesian Product Distributes over Union
- Cartesian Product of Unions
- Cartesian Product of Unions/Corollary
- Cartesian Product of Unions/General Result
- Characteristic Function of Union
- Characteristic Function of Union/Variant 1
- Characteristic Function of Union/Variant 2
- Characteristic Function of Union/Variant 3
- Closure of Finite Union equals Union of Closures
- Closure of Infinite Union may not equal Union of Closures
- Closure of Intersection and Symmetric Difference imply Closure of Union
- Closure of Union and Complement imply Closure of Set Difference
- Closure of Union contains Union of Closures
- Complement Union with Superset is Universe
- Complement Union with Superset is Universe/Corollary
- Condition for Agreement of Family of Mappings
- Countable Union of Countable Sets is Countable
- Countable Union of Finite Sets is Countable

### D

### E

### F

### I

- Identity of Power Set with Union
- Image of Union under Mapping
- Image of Union under Mapping/Family of Sets
- Image of Union under Mapping/General Result
- Image of Union under Relation
- Image of Union under Relation/Family of Sets
- Image of Union under Relation/General Result
- Inclusion-Exclusion Principle
- Inclusion-Exclusion Principle/Corollary
- Increasing Union of Ideals is Ideal/Sequence
- Increasing Union of Sequence of Ideals is Ideal
- Increasing Union of Subrings is Subring
- Indexed Union Equality
- Indexed Union Subset
- Infinite Union of Closed Sets of Metric Space may not be Closed
- Interior of Union is not necessarily Union of Interiors
- Intersection Distributes over Union
- Intersection is Subset of Union
- Intersection is Subset of Union of Intersections with Complements
- Intersection of Unions with Complements is Subset of Union
- Intersection Operation on Supersets of Subset is Closed

### P

- Power Set is Closed under Union
- Power Set with Union is Commutative Monoid
- Preimage of Subset under Mapping equals Union of Preimages of Elements
- Preimage of Union under Mapping
- Preimage of Union under Mapping/Family of Sets
- Preimage of Union under Mapping/Family of Sets/Proof 1
- Preimage of Union under Mapping/Family of Sets/Proof 2
- Preimage of Union under Mapping/General Result
- Preimage of Union under Relation
- Preimage of Union under Relation/Family of Sets
- Preimage of Union under Relation/General Result
- Product of Subset with Union

### S

- Set Difference and Intersection form Partition
- Set Difference and Intersection form Partition/Corollary
- Set Difference is Right Distributive over Union
- Set Difference is Subset of Union of Differences
- Set Difference Union First Set is First Set
- Set Difference Union Intersection
- Set Difference Union Second Set is Union
- Set Difference with Set Difference is Union of Set Difference with Intersection
- Set Difference with Union
- Set Difference with Union is Set Difference
- Set Equation: Union
- Set is Subset of Union
- Set is Subset of Union/Family of Sets
- Set is Subset of Union/General Result
- Set is Subset of Union/Set of Sets
- Set System Closed under Union is Commutative Semigroup
- Set Theory/Examples/Unions and Intersections
- Set Union is not Cancellable
- Set Union Preserves Subsets
- Set Union Preserves Subsets/Corollary
- Set Union Preserves Subsets/Corollary/Proof 1
- Set Union Preserves Subsets/Corollary/Proof 2
- Set Union Preserves Subsets/Families of Sets
- Set Union Preserves Subsets/General Result
- Set Union Preserves Subsets/Proof 1
- Set Union Preserves Subsets/Proof 2
- Subsets in Increasing Union
- Symmetric Difference is Subset of Union
- Symmetric Difference is Subset of Union of Symmetric Differences
- Symmetric Difference of Unions
- Symmetric Difference of Unions is Subset of Union of Symmetric Differences

### U

- Union as Symmetric Difference with Intersection
- Union Distributes over Intersection
- Union Distributes over Union
- Union equals Intersection iff Sets are Equal
- Union is Associative
- Union is Associative/Family of Sets
- Union is Commutative
- Union is Empty iff Sets are Empty
- Union is Idempotent
- Union is Increasing
- Union is Smallest Superset
- Union is Smallest Superset/Family of Sets
- Union is Smallest Superset/General Result
- Union is Smallest Superset/Set of Sets
- Union of Closure with Closure of Complement is Whole Space
- Union of Countable Sets of Sets
- Union of Disjoint Singletons is Doubleton
- Union of Doubleton
- Union of Elements of Power Set
- Union of Empty Set
- Union of Equivalences
- Union of Finite Sets is Finite
- Union of Finite Sets is Finite/Proof 1
- Union of Finite Sets is Finite/Proof 2
- Union of Intersections
- Union of Inverses of Mappings is Inverse of Union of Mappings
- Union of Mappings with Disjoint Domains is Mapping
- Union of Non-Disjoint Convex Sets is Convex Set
- Union of Open Sets of Metric Space is Open
- Union of Power Sets
- Union of Power Sets not always Equal to Powerset of Union
- Union of Primitive Recursive Sets
- Union of Reflexive Relations is Reflexive
- Union of Relative Complements of Nested Subsets
- Union of Set of Sets when a Set Intersects All
- Union of Set of Singletons
- Union of Singleton
- Union of Small Classes is Small
- Union of Subgroups
- Union of Subset of Family is Subset of Union of Family
- Union of Subsets is Subset
- Union of Subsets is Subset/Family of Sets
- Union of Subsets is Subset/Set of Sets
- Union of Subsets is Subset/Subset of Power Set
- Union of Symmetric Differences
- Union of Symmetric Relations is Symmetric
- Union of Transitive Relations Not Always Transitive
- Union Operation on Supersets of Subset is Closed
- Union with Complement
- Union with Empty Set
- Union with Relative Complement
- Union with Set Difference
- Union with Superset is Superset
- Union with Universe