Math 2.0

Mathematical logic and foundations Definition

Graph of a function

Builds on FunctionCartesian product

Definition

Let f ⁣:A→Bf \colon A \to B be a function. The graph of ff is the set Γ(f)={(a,f(a))∣a∈A}⊆A×B.\Gamma(f) = \{(a, f(a)) \mid a \in A\} \subseteq A \times B.

Remarks

For a real function f ⁣:ℝ→ℝf \colon \mathbb{R} \to \mathbb{R}, the graph is a curve in the plane ℝ2\mathbb{R}^2. Let G⊆A×BG \subseteq A \times B. GG is the graph of a function from AA to BB if and only if each a∈Aa \in A is the first entry of exactly one pair. In the set-theoretic definition of a function, the function and its graph are the same set.

Reading path

To understand this page from the ground up, read these 14 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 levelFunctiondefinitionGraph of a function, definition, no evidence levelGraph of a functiondefinition
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.
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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. Graph of a function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Graph_of_a_function, lead section
About this page
Tag0Y0KFHF47G
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Kinddefinition
WritingW1 stated
MSC03E20
DefinesGraph of a function
UsesFunction, Cartesian product

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