Math 2.0

Mathematical logic and foundations Definition

Preimage of a set

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Definition

Let f ⁣:A→Bf \colon A \to B be a function and let TT be a subset of BB. The preimage of TT under ff is the set f−1(T)={a∈A∣f(a)∈T}.f^{-1}(T) = \{a \in A \mid f(a) \in T\}.

Remarks

Another name is the inverse image. The notation f−1(T)f^{-1}(T) makes sense for every ff, also when ff has no inverse function. Preimages respect all set operations: f−1(T1∩T2)=f−1(T1)∩f−1(T2)f^{-1}(T_1 \cap T_2) = f^{-1}(T_1) \cap f^{-1}(T_2), and the same holds for unions and complements. Example: for f(x)=x2f(x) = x^2 on ℝ\mathbb{R}, f−1([1,4])=[−2,−1]∪[1,2]f^{-1}([1, 4]) = [-2, -1] \cup [1, 2].

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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 levelFunctiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinition
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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. Preimage of a set definition

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429 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Image_(mathematics), Inverse image
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