# Category:Subsets

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This category contains results about Subsets.

Definitions specific to this category can be found in Definitions/Subsets.

Let $S$ and $T$ be sets.

$S$ is a **subset** of a set $T$ if and only if all of the elements of $S$ are also elements of $T$.

This is denoted:

- $S \subseteq T$

That is:

- $S \subseteq T \iff \forall x: \paren {x \in S \implies x \in T}$

If the elements of $S$ are not all also elements of $T$, then $S$ is not a **subset** of $T$:

- $S \nsubseteq T$ means $\neg \paren {S \subseteq T}$

## Subcategories

This category has the following 25 subcategories, out of 25 total.

### C

- Cartesian Product of Subsets (6 P)

### E

- Examples of Subsets (11 P)

### I

- Image of Subset under Mapping (empty)

### P

### R

### S

- Set Difference is Subset (3 P)
- Set Union Preserves Subsets (8 P)

### U

- Union of Subsets is Subset (5 P)
- Union with Empty Set (3 P)

## Pages in category "Subsets"

The following 108 pages are in this category, out of 108 total.

### C

- Cardinality of Proper Subset of Finite Set
- Cardinality of Subset of Finite Set
- Cartesian Product of Family of Subsets
- Cartesian Product of Subsets
- Complement of Relative Complement is Union with Complement
- Complement Relative to Subset is Subset of Complement Relative to Superset
- Complement Union with Superset is Universe
- Complement Union with Superset is Universe/Corollary
- Cycle of Subsets implies Set Equality

### E

### I

- Image of Subset under Mapping is Subset of Image
- Image of Subset under Relation equals Union of Images of Elements
- Image of Subset under Relation is Subset of Image
- Image of Subset under Relation is Subset of Image/Corollary 1
- Image of Subset under Relation is Subset of Image/Corollary 3
- Image Preserves Subsets
- Infinite Set has Countably Infinite Subset
- Infinite Set is Equivalent to Proper Subset
- Interior of Subset
- Intersection is Largest Subset
- Intersection is Largest Subset/Family of Sets
- Intersection is Largest Subset/General Result
- Intersection is Subset
- Intersection is Subset/Family of Sets
- Intersection is Subset/General Result
- Intersection of Family is Subset of Intersection of Subset of Family
- Intersection of Subsets is Subset/Set of Sets
- Intersection with Complement is Empty iff Subset
- Intersection with Subset is Subset
- Inverse of Subset of Relation is Subset of Inverse

### P

- Power Set of Subset
- Powerset of Subset is Closed under Intersection
- Powerset of Subset is Closed under Symmetric Difference
- Powerset of Subset is Closed under Union
- Preimage of Subset is Subset of Preimage
- Preimage of Subset under Inclusion Mapping
- Preimage of Subset under Mapping equals Union of Preimages of Elements
- Product Space of Subspaces is Subspace of Product Space
- Pullback of Subset Inclusion

### S

- Set Complement inverts Subsets
- Set Complement inverts Subsets/Corollary
- Set Difference is Subset
- Set Difference with Proper Subset
- Set Difference with Proper Subset is Proper Subset
- Set Difference with Superset is Empty Set
- Set Inequality
- Set Intersection Preserves Subsets
- Set is Equivalent to Proper Subset of Power Set
- Set is Subset of Intersection of Supersets
- Set is Subset of Itself
- 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 of Finite Subsets of Countable Set is Countable
- Set Union Preserves Subsets
- Singleton Equality
- Singleton of Element is Subset
- Singleton of Subset is Element of Powerset of Powerset
- Subset equals Preimage of Image iff Mapping is Injection
- Subset equals Preimage of Image implies Injection
- Subset Equivalences
- Subset in Subsets
- Subset Maps to Subset
- Subset of Countable Set is Countable
- Subset of Countably Infinite Set is Countable
- Subset of Finite Set is Finite
- Subset of Naturals is Finite iff Bounded
- Subset of Preimage under Relation is Preimage of Subset
- Subset of Preimage under Relation is Preimage of Subset/Corollary
- Subset of Set with Propositional Function
- Subset Relation is Antisymmetric
- Subset Relation is Ordering
- Subset Relation is Ordering/General Result
- Subset Relation is Transitive
- Subset Relation on Power Set is Partial Ordering
- Subsets in Increasing Union
- Subsets of Disjoint Sets are Disjoint

### U

- 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 Power Sets
- Union of Power Sets not always Equal to Powerset of Union
- 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 with Empty Set
- Union with Intersection equals Intersection with Union iff Subset
- Union with Superset is Superset