Math 2.0

Mathematical logic and foundations Definition

Equinumerous sets

Builds on Bijective function

Definition

Let AA and BB be sets. The sets AA and BB are equinumerous, written A≈BA \approx B, when there is a bijective function f ⁣:A→Bf \colon A \to B.

Remarks

Equinumerosity is reflexive, symmetric and transitive. The identity, the inverse and the composition of bijections give the three properties. The set ℕ\mathbb{N} is equinumerous with the set of even natural numbers, by n↦2nn \mapsto 2n. So an infinite set can be equinumerous with a proper subset. Other names are "equipotent" and "have the same cardinality".

Reading path

To understand this page from the ground up, read these 18 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 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 functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinition
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. Equinumerous sets definition

Where this leads

60 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinality, Equinumerosity
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