Math 2.0

Mathematical logic and foundations Definition

Inverse function

Builds on Bijective function

Definition

Let f ⁣:A→Bf \colon A \to B be a bijective function. The inverse function of ff is the function f−1 ⁣:B→Af^{-1} \colon B \to A such that, for each b∈Bb \in B, f−1(b)f^{-1}(b) is the unique a∈Aa \in A with f(a)=bf(a) = b.

Remarks

Then f−1∘f=idAf^{-1} \circ f = \mathrm{id}_A and f∘f−1=idBf \circ f^{-1} = \mathrm{id}_B. See bijective iff it has a two-sided inverse. The inverse f−1f^{-1} is a bijection, and its inverse is ff. Example: the inverse of x↦2x+1x \mapsto 2x + 1 on ℝ\mathbb{R} is y↦(y−1)/2y \mapsto (y - 1)/2. Do not confuse f−1(b)f^{-1}(b) with the preimage f−1({b})f^{-1}(\{b\}), which exists for every function.

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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 levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionInverse function, definition, no evidence levelInverse 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. Injective function definition
  16. Surjective function definition
  17. Bijective function definition
  18. Inverse function definition

Where this leads

35 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function, lead section
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MSC03E20
DefinesInverse function
UsesBijective function

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