Math 2.0

Mathematical logic and foundations Proof

Proof that a composition of injections is injective

Builds on Injective functionFunction composition

Proof

We check the definition of injectivity directly.

  1. Let f ⁣:A→Bf \colon A \to B and g ⁣:B→Cg \colon B \to C be injective functions.
  2. Let a,a′∈Aa, a' \in A with (g∘f)(a)=(g∘f)(a′)(g \circ f)(a) = (g \circ f)(a').
  3. By the definition of composition, step 2 says g(f(a))=g(f(a′))g(f(a)) = g(f(a')).
  4. Since gg is injective and f(a),f(a′)∈Bf(a), f(a') \in B, step 3 gives f(a)=f(a′)f(a) = f(a').
  5. Since ff is injective, step 4 gives a=a′a = a'.
  6. So (g∘f)(a)=(g∘f)(a′)(g \circ f)(a) = (g \circ f)(a') implies a=a′a = a' for all a,a′∈Aa, a' \in A. Hence g∘fg \circ f is injective.

Idea

To go back from an equal output of g∘fg \circ f to equal inputs, peel off one function at a time. Injectivity of gg removes gg, and then injectivity of ff removes ff.

Reading path

To understand this page from the ground up, read these 16 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 compositiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionProof that a composition of injections is injective, proof, no evidence levelProof that a composit…proof
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. Injective function definition
  16. Proof that a composition of injections is injective proof

Where this leads

122 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Composite_of_Injections_is_Injection
About this page
Tag6ZEQS0E0MA
URIm2://6ZEQS0E0MA
Kindproof
WritingW0 stub
MSC03E20
ProvesComposition of injections is injective
UsesInjective function, Function composition
Changed4 November 2026

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