Math 2.0

Mathematical logic and foundations Definition

Surjective function

Builds on FunctionCodomain

Definition

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

Remarks

Other names are onto function and surjection. Equivalently, the range of ff is the whole codomain BB. The function x↦x3x \mapsto x^3 on ℝ\mathbb{R} is surjective, and x↦x2x \mapsto x^2 on ℝ\mathbb{R} is not, because −1-1 is not a value.

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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 levelCodomaindefinitionSurjective function, definition, no evidence levelSurjective 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. Surjective function definition

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116 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Surjective_function, lead section
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MSC03E20
DefinesSurjective function
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