Math 2.0

Mathematical logic and foundations Definition

Bijective function

Builds on Injective functionSurjective function

Definition

Let f ⁣:A→Bf \colon A \to B be a function. The function ff is bijective when it is injective and surjective. Equivalently, for each b∈Bb \in B, there is exactly one a∈Aa \in A with f(a)=bf(a) = b.

Remarks

Other names are bijection and one-to-one correspondence. The function x↦2x+1x \mapsto 2x + 1 on ℝ\mathbb{R} is bijective. A function is bijective if and only if it has an inverse function. See bijective iff it has a two-sided inverse. Two finite sets have the same number of elements exactly when there is a bijection between them.

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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 levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinition
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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. Injective function definition
  16. Surjective function definition
  17. Bijective function definition

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111 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bijection, lead section
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