Math 2.0

Mathematical logic and foundations Definition

Order of cardinals

Builds on CardinalityInjective function

Definition

Let AA and BB be sets. The cardinality of AA is at most the cardinality of BB, written ∣A∣≤∣B∣|A| \le |B|, when there is an injective function f ⁣:A→Bf \colon A \to B. Write ∣A∣<∣B∣|A| < |B| when ∣A∣≤∣B∣|A| \le |B| and ∣A∣≠∣B∣|A| \neq |B|.

Remarks

The relation does not depend on the choice of AA and BB in their classes of equinumerous sets. It is reflexive and transitive. It is antisymmetric by the Schröder–Bernstein theorem. With the axiom of choice, any two cardinals are comparable. See comparability of cardinals.

Reading path

To understand this page from the ground up, read these 20 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 setsdefinitionCardinality, definition, no evidence levelCardinalitydefinitionOrder of cardinals, definition, no evidence levelOrder of cardinalsdefinition
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
  19. Cardinality definition
  20. Order of cardinals definition

Where this leads

9 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinal_number, Cardinal comparison
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MSC03E10
DefinesOrder of cardinals
UsesCardinality, Injective function

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