Math 2.0

Mathematical logic and foundations Proposition

Bijective iff it has a two-sided inverse

Builds on Bijective functionLeft inverseRight inverse

Statement

Let f ⁣:A→Bf \colon A \to B be a function. Then ff is bijective if and only if there is a function g ⁣:B→Ag \colon B \to A with g∘f=idAg \circ f = \mathrm{id}_A and f∘g=idBf \circ g = \mathrm{id}_B.

Remarks

The function gg is then unique. It is the inverse function f−1f^{-1}. This statement does not need the axiom of choice: for a bijection, g(b)g(b) is the unique aa with f(a)=bf(a) = b. If ff has a left inverse gg and a right inverse hh, then g=hg = h.

Reading path

To understand this page from the ground up, read these 22 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 levelCodomaindefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionIdentity function, definition, no evidence levelIdentity functiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionLeft inverse, definition, no evidence levelLeft inversedefinitionRight inverse, definition, no evidence levelRight inversedefinitionBijective function, definition, no evidence levelBijective functiondefinitionBijective iff it has a two-sided inverse, statement, no evidence levelBijective 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. 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. Function composition definition
  16. Identity function definition
  17. Injective function definition
  18. Surjective function definition
  19. Left inverse definition
  20. Right inverse definition
  21. Bijective function definition
  22. Bijective iff it has a two-sided inverse statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Two-sided inverses
About this page
Tag01VG2A0BHW
URIm2://01VG2A0BHW
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 01VG2A0BHW.
UsesBijective function, Left inverse, Right inverse

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