Math 2.0

Mathematical logic and foundations Proposition

Injective iff it has a left inverse

Builds on Injective functionLeft inverse

Statement

Let AA be a nonempty set, let BB be a set and let f ⁣:A→Bf \colon A \to B be a function. Then ff is injective if and only if ff has a left inverse.

Remarks

The hypothesis that AA is nonempty is necessary. The empty function from ∅\varnothing to a nonempty set is injective, but there is no function from a nonempty set to ∅\varnothing. To build a left inverse, fix a0∈Aa_0 \in A. Send each b=f(a)b = f(a) to aa, and send each b∉f(A)b \notin f(A) to a0a_0.

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To understand this page from the ground up, read these 18 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 levelFunctiondefinitionFunction composition, definition, no evidence levelFunction compositiondefinitionIdentity function, definition, no evidence levelIdentity functiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionLeft inverse, definition, no evidence levelLeft inversedefinitionInjective iff it has a left inverse, statement, no evidence levelInjective 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. Function composition definition
  15. Identity function definition
  16. Injective function definition
  17. Left inverse definition
  18. Injective iff it has a left inverse statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Left inverses
About this page
Tag0HYFEEZAEM
URIm2://0HYFEEZAEM
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 0HYFEEZAEM.
UsesInjective function, Left inverse

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