Math 2.0

Mathematical logic and foundations Definition

Left inverse

Builds on Function compositionIdentity function

Definition

Let f ⁣:A→Bf \colon A \to B be a function. A left inverse of ff is a function g ⁣:B→Ag \colon B \to A such that g∘f=idAg \circ f = \mathrm{id}_A, that is, g(f(a))=ag(f(a)) = a for each a∈Aa \in A.

Remarks

A function with a nonempty domain has a left inverse exactly when it is injective. See injective iff it has a left inverse. A left inverse is in general not unique. Its values on B∖f(A)B \setminus f(A) can be any elements of AA. If gg is a left inverse of ff, then ff is a right inverse of gg.

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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 levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionIdentity function, definition, no evidence levelIdentity functiondefinitionLeft inverse, definition, no evidence levelLeft inversedefinition
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  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. Function composition definition
  15. Identity function definition
  16. Left inverse definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Left inverses
About this page
Tag02486SD328
URIm2://02486SD328
Kinddefinition
WritingW0 stub
MSC03E20
DefinesLeft inverse
UsesFunction composition, Identity function

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