Math 2.0

Mathematical logic and foundations Proposition

Composition of surjections is surjective

Builds on Surjective functionFunction composition

Statement

Let f ⁣:A→Bf \colon A \to B and g ⁣:B→Cg \colon B \to C be surjective functions. Then the composition g∘f ⁣:A→Cg \circ f \colon A \to C is surjective.

Remarks

Proof: let c∈Cc \in C. Since gg is surjective, c=g(b)c = g(b) for some b∈Bb \in B. Since ff is surjective, b=f(a)b = f(a) for some a∈Aa \in A. Then c=(g∘f)(a)c = (g \circ f)(a). A partial converse holds: if g∘fg \circ f is surjective, then gg is surjective. The function ff need not be surjective. Together with the result for injections, this shows that a composition of bijections is a bijection.

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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 compositiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionComposition of surjections is surjective, statement, no evidence levelComposition of surjec…statement
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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. Codomain definition
  15. Function composition definition
  16. Surjective function definition
  17. Composition of surjections is surjective statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Surjective_function, Composition and decomposition
About this page
Tag07EH4R3RVQ
URIm2://07EH4R3RVQ
Aliasessurjective-comp
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 07EH4R3RVQ.
UsesSurjective function, Function composition

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