# Category:Power Set

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This category contains results about **power sets**.

The **power set** of a set $S$ is the set defined and denoted as:

- $\powerset S := \set {T: T \subseteq S}$

That is, the set whose elements are all of the subsets of $S$.

## Subcategories

This category has the following 13 subcategories, out of 13 total.

### A

- Axiom of Powers (1 P)

### B

### C

- Cantor's Theorem (9 P)
- Cardinality of Power Set (5 P)

### E

- Examples of Power Sets (7 P)

### I

### N

### P

## Pages in category "Power Set"

The following 91 pages are in this category, out of 91 total.

### B

### C

- Cantor's Theorem
- Cantor's Theorem (Strong Version)
- Cardinality of Power Set is Invariant
- Cardinality of Power Set of Finite Set
- Cardinality of Power Set of Natural Numbers Equals Cardinality of Real Numbers
- Cardinality of Set less than Cardinality of Power Set
- Choice Function for Power Set implies Choice Function for Set
- Compact Closure is Set of Finite Subsets in Lattice of Power Set
- Condition for Power Set to be Totally Ordered
- Cowen's Theorem/Lemma 1

### E

### F

### I

### P

- Power Set and Two-Valued Functions are Isomorphic Boolean Rings
- Power Set can be Derived using Axiom of Abstraction
- Power Set Exists and is Unique
- Power Set is Algebra of Sets
- Power Set is Boolean Ring
- Power Set is Closed under Complement
- Power Set is Closed under Countable Unions
- Power Set is Closed under Intersection
- Power Set is Closed under Set Complement
- Power Set is Closed under Set Difference
- Power Set is Closed under Symmetric Difference
- Power Set is Closed under Union
- Power Set is Complete Lattice
- Power Set is Filter in Lattice of Power Set
- Power Set is Lattice
- Power Set is Nonempty
- Power Set less Empty Set has no Smallest Element iff not Singleton
- Power Set of Doubleton
- Power Set of Empty Set
- Power Set of Finite Set is Finite
- Power Set of Natural Numbers has Cardinality of Continuum
- Power Set of Natural Numbers is Uncountable
- Power Set of Singleton
- Power Set of Subset
- Power Set of Transitive Set is Transitive
- 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
- Power Set with Union and Intersection forms Boolean Algebra
- 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
- Power Structure of Subset is Closed iff Subset is Closed
- Powerset is not Subset of its Set
- Powerset of Subset is Closed under Intersection
- Powerset of Subset is Closed under Symmetric Difference
- Powerset of Subset is Closed under Union

### S

- Set equals Union of Power Set
- Set is Element of its Power Set
- Set is Equivalent to Proper Subset of Power Set
- Set is Subset of Power Set of Union
- Set is Transitive iff Subset of Power Set
- Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection
- Singleton of Power Set less Empty Set is Minimal Subset
- Singleton of Set is Filter in Lattice of Power Set
- Singleton of Subset is Element of Powerset of Powerset
- Subset is Element of Power Set
- Subset Relation on Power Set is Partial Ordering
- Supremum of Power Set
- Symmetric Difference on Power Set forms Abelian Group
- Symmetric Difference with Intersection forms Boolean Ring
- Symmetric Difference with Intersection forms Ring
- Symmetric Difference with Union does not form Ring