Math 2.0

Mathematical logic and foundations Definition

Choice function

Builds on FunctionUnion of a family of sets

Definition

Let XX be a set of nonempty sets. A choice function on XX is a function c ⁣:X→⋃Xc \colon X \to \bigcup X such that c(S)∈Sc(S) \in S for each S∈XS \in X.

Remarks

For a finite set XX, a choice function exists without any axiom of choice. For the set of nonempty subsets of ℕ\mathbb{N}, the function that sends each subset to its least element is a choice function. The axiom of choice says that each set of nonempty sets has a choice function.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionChoice function, definition, no evidence levelChoice functiondefinition
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.
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Disjunction definition
  5. Implication definition
  6. Predicate definition
  7. Family of sets definition
  8. Ordered pair definition
  9. Union definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Union of a family of sets definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Choice function definition

Where this leads

32 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Choice_function, lead section
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WritingW1 stated
MSC03E25
DefinesChoice function
UsesFunction, Union of a family of sets

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