Math 2.0

Mathematical logic and foundations Proposition

Surjective iff it has a right inverse

Builds on Surjective functionRight inverseAxiom of choice

Statement

Let f ⁣:A→Bf \colon A \to B be a function. Assume the axiom of choice. Then ff is surjective if and only if ff has a right inverse.

Remarks

If f∘h=idBf \circ h = \mathrm{id}_B, then each bb equals f(h(b))f(h(b)), so ff is surjective. This direction does not use choice. For the other direction, choose for each b∈Bb \in B an element h(b)h(b) of the nonempty set f−1({b})f^{-1}(\{b\}). The statement "each surjection has a right inverse" is equivalent to the axiom of choice.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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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 levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionIdentity function, definition, no evidence levelIdentity functiondefinitionChoice function, definition, no evidence levelChoice functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementRight inverse, definition, no evidence levelRight inversedefinitionSurjective iff it has a right inverse, statement, no evidence levelSurjective iff it has…statement
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. 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. Codomain definition
  19. Function composition definition
  20. Identity function definition
  21. Choice function definition
  22. Surjective function definition
  23. Axiom of choice statement
  24. Right inverse definition
  25. Surjective iff it has a right inverse statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Right inverses
About this page
Tag0TYQNHVB4C
URIm2://0TYQNHVB4C
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 0TYQNHVB4C.
UsesSurjective function, Right inverse, Axiom of choice

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