Math 2.0

Mathematical logic and foundations Definition

Indicator function

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Definition

Let XX be a set and let AA be a subset of XX. The indicator function of AA is the function 𝟏A ⁣:X→{0,1}\mathbf{1}_A \colon X \to \{0, 1\} with 𝟏A(x)={1if x∈A,0if x∉A.\mathbf{1}_A(x) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}

Remarks

Other names are characteristic function and the notation χA\chi_A. For subsets AA and BB of XX, 𝟏A∩B=𝟏A𝟏B\mathbf{1}_{A \cap B} = \mathbf{1}_A \mathbf{1}_B and 𝟏Ac=1−𝟏A\mathbf{1}_{A^c} = 1 - \mathbf{1}_A. The map A↦𝟏AA \mapsto \mathbf{1}_A is a bijection from the power set of XX to the set of functions from XX to {0,1}\{0, 1\}.

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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 levelFunctiondefinitionIndicator function, definition, no evidence levelIndicator 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. Indicator function definition

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52 pages build on this one.

Further reading

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