Math 2.0

Combinatorics Proof

Proof of the pigeonhole principle

Builds on Injective functionBijective functionImage of a setBijection principleComplement principle

Proof

We argue by contradiction.

  1. Suppose that f:A→Bf : A \to B is injective, and let C=f(A)C = f(A) be the image of AA.
  2. Then ff is a bijection from AA onto CC: it is injective, and every element of CC is a value of ff.
  3. Composing a numbering of AA with ff numbers CC. So CC is finite and ∣C∣=∣A∣|C| = |A|, as in the bijection principle.
  4. The set CC is a subset of BB. By the complement principle, ∣B∖C∣=∣B∣−∣C∣=∣B∣−∣A∣|B \setminus C| = |B| - |C| = |B| - |A|.
  5. The left side is a natural number, but ∣B∣−∣A∣<0|B| - |A| < 0 because ∣A∣>∣B∣|A| > |B|. This is a contradiction.
  6. Hence ff is not injective. So there are a≠a′a \ne a' in AA with f(a)=f(a′)f(a) = f(a').

Idea

An injective map copies AA onto a subset of BB of the same size. A subset of BB cannot be larger than BB, so this is impossible when AA has more elements than BB.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first. The path takes the proofs Proof that a composition of injections is injective, Proof that a function is bijective iff it has a two-sided inverse, Proof of the complement principle, Proof that a composition of surjections is surjective, Proof of the rule of sum, Proof of the bijection principle.

Show the path as a graph
Cardinality of a finite set, definition, no evidence levelCardinality of a fini…definitionElement, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionSubset, 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 compositiondefinitionIdentity function, definition, no evidence levelIdentity functiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionImage of a set, definition, no evidence levelImage of a setdefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionLeft inverse, definition, no evidence levelLeft inversedefinitionRight inverse, definition, no evidence levelRight inversedefinitionProof that a composition of injections is injective, proof, no evidence levelProof that a composit…proofProof that a composition of surjections is surjective, proof, no evidence levelProof that a composit…proofBijective function, definition, no evidence levelBijective functiondefinitionComposition of injections is injective, statement, no evidence levelComposition of inject…statementComposition of surjections is surjective, statement, no evidence levelComposition of surjec…statementProof of the rule of sum, proof, no evidence levelProof of the rule of …proofProof that a function is bijective iff it has a two-sided inverse, proof, no evidence levelProof that a function…proofRule of sum, statement, no evidence levelRule of sumstatementBijective iff it has a two-sided inverse, statement, no evidence levelBijective iff it has …statementProof of the complement principle, proof, no evidence levelProof of the compleme…proofProof of the bijection principle, proof, no evidence levelProof of the bijectio…proofComplement principle, statement, no evidence levelComplement principlestatementBijection principle, statement, no evidence levelBijection principlestatementProof of the pigeonhole principle, proof, no evidence levelProof of the pigeonho…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. Cardinality of a finite set definition
  2. Element definition
  3. Proposition concept
  4. Set definition
  5. Conjunction definition
  6. Implication definition
  7. Predicate definition
  8. Empty set object
  9. Ordered pair definition
  10. Intersection definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Disjoint sets definition
  15. Subset definition
  16. Uniqueness quantifier definition
  17. Binary relation definition
  18. Function definition
  19. Codomain definition
  20. Function composition definition
  21. Identity function definition
  22. Injective function definition
  23. Image of a set definition
  24. Surjective function definition
  25. Left inverse definition
  26. Right inverse definition
  27. Proof that a composition of injections is injective proof
  28. Proof that a composition of surjections is surjective proof
  29. Bijective function definition
  30. Composition of injections is injective statement
  31. Composition of surjections is surjective statement
  32. Proof of the rule of sum proof
  33. Proof that a function is bijective iff it has a two-sided inverse proof
  34. Rule of sum statement
  35. Bijective iff it has a two-sided inverse statement
  36. Proof of the complement principle proof
  37. Proof of the bijection principle proof
  38. Complement principle statement
  39. Bijection principle statement
  40. Proof of the pigeonhole principle proof

Where this leads

39 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Pigeonhole_Principle
About this page
TagE0R6P2FMCM
URIm2://E0R6P2FMCM
Kindproof
WritingW0 stub
MSC05A05
ProvesPigeonhole principle
UsesInjective function, Bijective function, Image of a set, Bijection principle, Complement principle
MethodsProof by contradiction
Changed4 November 2026

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