# Category:Set Union

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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 $T$:

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

## Also see

## Subcategories

This category has the following 44 subcategories, out of 44 total.

### A

- Absorption Laws (13 P)

### B

### C

- Cartesian Product of Unions (4 P)

### D

### E

- Examples of Set Union (19 P)

### I

- Image of Union under Mapping (5 P)

### P

- Product of Subset with Union (3 P)

### S

- Set Difference with Union (2 P)
- Set is Subset of Union (4 P)
- Set Union Preserves Subsets (8 P)

### U

- Union is Associative (3 P)
- Union is Commutative (3 P)
- Union Mappings (5 P)
- Union of Equivalences (3 P)
- Union of Intersections (3 P)
- Union of Subgroups (5 P)
- Union of Subsets is Subset (5 P)
- Union with Empty Set (3 P)
- Unions of Families (3 P)

## Pages in category "Set Union"

The following 189 pages are in this category, out of 189 total.

### B

### C

- Cardinality of Set Union
- Cardinality of Union not greater than Product
- Cartesian Product Distributes over Union
- Cartesian Product of Unions
- Characteristic Function of Disjoint Union
- Characteristic Function of Union
- Characterization of Set Equals Union of Sets
- Choice Function Exists for Well-Orderable Union of Sets
- Choice Function for Set does not imply Choice Function for Union of Set
- 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
- Composition of Symmetric Relation with Itself is Union of Products of Images
- Condition for Agreement of Family of Mappings
- Countable Union of Countable Sets is Countable
- Countable Union of Finite Sets is Countable
- Countable Union of Meager Sets is Meager

### D

### E

### F

### I

- Identity of Power Set with Union
- Image of Subset under Mapping equals Union of Images of Elements
- Image of Subset under Relation equals Union of Images of Elements
- Image of Union under Mapping
- Image of Union under Relation
- 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
- Infimum of Union of Bounded Below Sets of Real Numbers
- 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 Empty and Union is Universe if Sets are Complementary
- 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 Countable Unions
- Power Set is Closed under Union
- Power Set with Union and Subset Relation is Ordered Semigroup
- Power Set with Union and Superset Relation is Ordered Semigroup
- Power Set with Union is Commutative Monoid
- Powerset of Subset is Closed under Union
- Preimage of Subset under Mapping equals Union of Preimages of Elements
- Preimage of Union under Mapping
- Preimage of Union under Relation
- Product of Subset with Union

### R

### S

- Set Difference and Intersection form Partition/Corollary 1
- 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 Set Difference is Union of Set Difference with Intersection/Corollary
- Set Difference with Union
- Set Difference with Union is Set Difference
- Set Differences and Intersection form Partition of Union
- Set equals Union of Power Set
- Set Equation: Union
- Set is Subset of Power Set of Union
- Set is Subset of Union
- Set is Subset of Union of Family
- 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 Union can be Derived using Axiom of Abstraction
- Set Union expressed as Intersection Complement
- Set Union is Idempotent
- Set Union is not Cancellable
- Set Union is Self-Distributive
- Set Union Preserves Subsets
- Subsets in Increasing Union
- Sum of Union of Subsets of Vector Space and Subset
- Supremum of Union of Bounded Above Sets of Real Numbers
- 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
- Symmetric Difference with Union does not form Ring

### U

- Union as Symmetric Difference with Intersection
- Union Distributes over Intersection
- Union equals Intersection iff Sets are Equal
- Union is Associative
- Union is Commutative
- Union is Empty iff Sets are Empty
- Union is Increasing
- Union is Increasing Sequence of Sets
- 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 minus Symmetric Difference equals Intersection
- Union of Antisymmetric Relation with Inverse is Antisymmetric iff Diagonal
- Union of Balanced Sets in Vector Space is Balanced
- Union of Bijections with Disjoint Domains and Codomains is Bijection
- Union of Bounded Above Real Subsets is Bounded Above
- Union of Bounded Below Real Subsets is Bounded Below
- Union of Closure with Closure of Complement is Whole Space
- Union of Countable Sets of Sets
- Union of Disjoint Singletons is Doubleton
- Union of Elements of Power Set
- Union of Empty Set
- Union of Equivalences
- Union of Event with Complement is Certainty
- Union of Family of Subsets is Subset
- Union of Finite Sets is Finite
- Union of Horizontal Sections is Horizontal Section of Union
- Union of Interiors is Subset of Interior of Union
- Union of Intersections
- Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3
- Union of Inverse of Relations is Inverse of their Union
- Union of Inverses of Mappings is Inverse of Union of Mappings
- Union of Mappings which Agree is Mapping
- Union of Mappings which Agree is Mapping/Family of Mappings
- Union of Mappings with Disjoint Domains is Mapping
- Union of Meager Sets is Meager Set
- Union of Non-Disjoint Convex Sets is Convex Set
- Union of Open Sets is Open
- Union of Open Sets of Metric Space is Open
- Union of Open Sets of Normed Vector Space is Open
- Union of Orderings is not necessarily Ordering
- Union of Ordinal is Subset of Itself
- 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 Regular Open Sets is not necessarily Regular Open
- Union of Relation with Inverse is Symmetric Relation
- Union of Relative Complements of Nested Subsets
- Union of Set of Ordinals is Ordinal
- Union of Set of Sets is Empty iff Sets are Empty
- Union of Set of Sets is Non-empty iff some Set is Non-empty
- 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/Subset of Power Set
- Union of Symmetric Differences
- Union of Symmetric Relations is Symmetric
- Union of Topologies on Singleton or Doubleton is Topology
- Union of Transitive Relations Not Always Transitive
- Union of Two Compact Sets is Compact
- Union of Union of Cartesian Product
- Union of Union of Cartesian Product with Empty Factor
- Union of Upper Sections is Upper
- Union of Vertical Sections is Vertical Section of Union
- Union Operation on Supersets of Subset is Closed
- Union with Complement
- Union with Empty Set
- Union with Intersection equals Intersection with Union iff Subset
- Union with Relative Complement
- Union with Set Difference
- Union with Superset is Superset
- Union with Universe