Math 2.0

Mathematical logic and foundations Definition

Countably infinite set

Builds on Bijective function

Definition

A set AA is countably infinite when there is a bijective function f ⁣:ℕ→Af \colon \mathbb{N} \to A.

Remarks

Another name is denumerable. Its cardinality is aleph-null. Examples are ℕ\mathbb{N}, ℤ\mathbb{Z} and ℚ\mathbb{Q}, and also the set of primes. A set is countably infinite exactly when it is countable and infinite.

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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 functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionCountably infinite set, definition, no evidence levelCountably infinite setdefinition
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. Bijective function definition
  18. Countably infinite set definition

Where this leads

21 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Countable_set, Definition
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WritingW1 stated
MSC03E10
DefinesCountably infinite set
UsesBijective function

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  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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