Math 2.0

Mathematical logic and foundations Proposition

Cardinality of the power set

Builds on Power setIndicator functionCardinality

Statement

Let AA be a set. Then the power set of AA is equinumerous with the set {0,1}A\{0, 1\}^A of functions from AA to {0,1}\{0, 1\}, so ∣𝒫(A)∣=2∣A∣.|\mathcal{P}(A)| = 2^{|A|}.

Remarks

The bijection sends a subset SS to its indicator function 𝟏S\mathbf{1}_S. For a finite set with nn elements, the power set has 2n2^n elements. For A=ℕA = \mathbb{N}, ∣𝒫(ℕ)∣=2ℵ0|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}, the cardinality of the continuum.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionPower set, definition, no evidence levelPower setdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionIndicator function, definition, no evidence levelIndicator functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCardinality of the power set, statement, no evidence levelCardinality of the po…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Power set definition
  14. Function definition
  15. Codomain definition
  16. Injective function definition
  17. Indicator function definition
  18. Surjective function definition
  19. Bijective function definition
  20. Equinumerous sets definition
  21. Cardinality definition
  22. Cardinality of the power set statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Power_set, Representing subsets as functions
About this page
Tag0GPFKS96T7
URIm2://0GPFKS96T7
Kindstatement, proposition
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0GPFKS96T7.
UsesPower set, Indicator function, Cardinality

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