Math 2.0

Mathematical logic and foundations Proposition

Product of countable sets is countable

Builds on Countable setCartesian productCantor pairing function

Statement

Let AA and BB be countable sets. Then the Cartesian product A×BA \times B is countable.

Remarks

Proof: take injections f ⁣:A→ℕf \colon A \to \mathbb{N} and g ⁣:B→ℕg \colon B \to \mathbb{N}. Then (a,b)↦π(f(a),g(b))(a, b) \mapsto \pi(f(a), g(b)) is an injection to ℕ\mathbb{N}, where π\pi is the Cantor pairing function. By induction, a finite product A1×⋯×AnA_1 \times \dots \times A_n of countable sets is countable. An infinite product need not be countable: the set of sequences of 00 and 11 is uncountable.

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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 relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionCountable set, definition, no evidence levelCountable setdefinitionBijective function, definition, no evidence levelBijective functiondefinitionCantor pairing function, example, no evidence levelCantor pairing functi…exampleProduct of countable sets is countable, statement, no evidence levelProduct of countable …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. Function definition
  14. Codomain definition
  15. Injective function definition
  16. Surjective function definition
  17. Countable set definition
  18. Bijective function definition
  19. Cantor pairing function example
  20. Product of countable sets is countable statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Countable_set, Formal overview
About this page
Tag08G4XH86JC
URIm2://08G4XH86JC
Kindstatement, proposition
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 08G4XH86JC.
UsesCountable set, Cartesian product, Cantor pairing function

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