Math 2.0

Mathematical logic and foundations Definition

Restriction of a function

Builds on FunctionSubset

Definition

Let f ⁣:A→Bf \colon A \to B be a function and let SS be a subset of AA. The restriction of ff to SS is the function f∣S ⁣:S→Bf|_S \colon S \to B with f∣S(s)=f(s)f|_S(s) = f(s) for each s∈Ss \in S.

Remarks

The graph of f∣Sf|_S is {(s,f(s))∣s∈S}\{(s, f(s)) \mid s \in S\}, a subset of the graph of ff. The function ff is an extension of f∣Sf|_S. Example: x↦x2x \mapsto x^2 on ℝ\mathbb{R} is not injective, but its restriction to [0,∞)[0, \infty) is injective.

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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 levelFunctiondefinitionRestriction of a function, definition, no evidence levelRestriction of a func…definition
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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. Restriction of a function definition

Where this leads

21 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Restriction_(mathematics), lead section
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MSC03E20
DefinesRestriction of a function
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