Math 2.0

Mathematical logic and foundations Definition

Injective function

Builds on Function

Definition

Let f ⁣:A→Bf \colon A \to B be a function. The function ff is injective when, for all a,a′∈Aa, a' \in A, f(a)=f(a′)f(a) = f(a') implies a=a′a = a'.

Remarks

Other names are one-to-one function and injection. The function x↦2xx \mapsto 2x on ℝ\mathbb{R} is injective, and x↦x2x \mapsto x^2 on ℝ\mathbb{R} is not, because f(1)=f(−1)f(1) = f(-1). An injective function with a nonempty domain has a left inverse. See injective iff it has a left inverse.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
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 levelFunctiondefinitionInjective function, definition, no evidence levelInjective 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. Injective function definition

Where this leads

345 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Injective_function, lead section
About this page
Tag0PVG1CRAJN
URIm2://0PVG1CRAJN
Kinddefinition
WritingW1 stated
MSC03E20
DefinesInjective function
UsesFunction

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by