Math 2.0

Mathematical logic and foundations Definition

Cardinality

Builds on Equinumerous sets

Definition

Let AA be a set. The cardinality of AA, written ∣A∣|A|, is an object assigned to AA such that, for all sets AA and BB, ∣A∣=∣B∣  ⟺  A and B are [equinumerous](m2node:08EJHHMB5R).|A| = |B| \iff A \text{ and } B \text{ are } \text{[equinumerous](m2node:08EJHHMB5R)}. For a finite set AA, ∣A∣|A| is the number nn of its elements, that is, the unique n∈ℕn \in \mathbb{N} with A≈{1,…,n}A \approx \{1, \dots, n\}. For an infinite set, ∣A∣|A| is a cardinal number.

Remarks

Other notations are #A\#A, card⁡(A)\operatorname{card}(A) and A‾‾\overline{\overline{A}}. With the axiom of choice, one can take ∣A∣|A| to be the least ordinal equinumerous with AA. ∣A∣≤∣B∣|A| \le |B| means that there is an injection from AA to BB. See order of cardinals.

Reading path

To understand this page from the ground up, read these 19 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 levelCardinalitydefinition
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

Where this leads

33 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinality, lead section
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DefinesCardinality
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