Math 2.0

Mathematical logic and foundations Definition

Image of a set

Builds on FunctionSubset

Definition

Let f ⁣:A→Bf \colon A \to B be a function and let SS be a subset of AA. The image of SS under ff is the set f(S)={f(s)∣s∈S}={b∈B∣∃s∈S, f(s)=b}.f(S) = \{f(s) \mid s \in S\} = \{b \in B \mid \exists s \in S, \, f(s) = b\}.

Remarks

Another notation is f[S]f[S]. It avoids a confusion with f(a)f(a) for an element aa. The image f(A)f(A) of the whole domain is the range of ff. Example: for f(x)=x2f(x) = x^2 on ℝ\mathbb{R}, f([−1,2])=[0,4]f([-1, 2]) = [0, 4]. The image of a union is the union of the images, but the image of an intersection can be smaller. See image of an intersection.

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 levelFunctiondefinitionImage of a set, definition, no evidence levelImage of a setdefinition
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. Image of a set definition

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Image_(mathematics), Image of a subset
About this page
Tag0Q3F20XB0A
URIm2://0Q3F20XB0A
Kinddefinition
WritingW0 stub
MSC03E20
DefinesImage of a set
UsesFunction, Subset

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