Math 2.0

Mathematical logic and foundations Definition

Identity function

Builds on Function

Definition

Let AA be a set. The identity function of AA is the function idA ⁣:A→A\mathrm{id}_A \colon A \to A with idA(a)=a\mathrm{id}_A(a) = a for each a∈Aa \in A.

Remarks

For each function f ⁣:A→Bf \colon A \to B, f∘idA=ff \circ \mathrm{id}_A = f and idB∘f=f\mathrm{id}_B \circ f = f. See function composition. The identity function is a bijection, and it is its own inverse. Its graph is the diagonal {(a,a)∣a∈A}\{(a, a) \mid a \in A\}.

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 levelFunctiondefinitionIdentity function, definition, no evidence levelIdentity 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.
  • no evidence level
  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. Identity function definition

Where this leads

5 pages build on this one.

Further reading

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