Math 2.0

Mathematical logic and foundations Definition

Right inverse

Builds on Function compositionIdentity function

Definition

Let f ⁣:A→Bf \colon A \to B be a function. A right inverse of ff is a function h ⁣:B→Ah \colon B \to A such that f∘h=idBf \circ h = \mathrm{id}_B, that is, f(h(b))=bf(h(b)) = b for each b∈Bb \in B.

Remarks

A function has a right inverse exactly when it is surjective. The proof uses the axiom of choice. See surjective iff it has a right inverse. A right inverse is in general not unique. A right inverse is injective.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

Show the path as a graph
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 functiondefinitionRight inverse, definition, no evidence levelRight inversedefinition
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. Right inverse definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, Right inverses
About this page
Tag0DDPZAG5ER
URIm2://0DDPZAG5ER
Kinddefinition
WritingW0 stub
MSC03E20
DefinesRight inverse
UsesFunction composition, Identity function

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by