Math 2.0

Mathematical logic and foundations Proposition

Composition of injections is injective

Builds on Injective functionFunction composition

Statement

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

Remarks

Proof: if g(f(a))=g(f(a′))g(f(a)) = g(f(a')), then f(a)=f(a′)f(a) = f(a') because gg is injective, and a=a′a = a' because ff is injective. A partial converse holds: if g∘fg \circ f is injective, then ff is injective. The function gg need not be injective.

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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 compositiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionComposition of injections is injective, statement, no evidence levelComposition of inject…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. Function composition definition
  15. Injective function definition
  16. Composition of injections is injective statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Injective_function, Other properties
About this page
Tag01FPXMTBX1
URIm2://01FPXMTBX1
Aliasesinjective-comp
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 01FPXMTBX1.
UsesInjective function, Function composition

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