Math 2.0

Mathematical logic and foundations Proposition

Image of an intersection

Builds on Image of a setIntersection

Statement

Let f ⁣:A→Bf \colon A \to B be a function and let SS and TT be subsets of AA. Then the image of S∩TS \cap T is a subset of the intersection of the images: f(S∩T)⊆f(S)∩f(T).f(S \cap T) \subseteq f(S) \cap f(T).

Remarks

The inclusion can be strict. For the constant function f ⁣:{1,2}→{0}f \colon \{1, 2\} \to \{0\}, S={1}S = \{1\} and T={2}T = \{2\}, the left side is ∅\varnothing and the right side is {0}\{0\}. Equality holds for all SS and TT if and only if ff is injective. For unions, equality always holds: f(S∪T)=f(S)∪f(T)f(S \cup T) = f(S) \cup f(T).

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionExistential 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 setdefinitionImage of an intersection, statement, no evidence levelImage of an intersect…statement
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.
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Implication definition
  6. Predicate definition
  7. Ordered pair definition
  8. Intersection definition
  9. Existential quantifier definition
  10. Universal quantifier definition
  11. Cartesian product definition
  12. Subset definition
  13. Uniqueness quantifier definition
  14. Binary relation definition
  15. Function definition
  16. Image of a set definition
  17. Image of an intersection statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Image_(mathematics), Properties
About this page
Tag0RX7DDBYCK
URIm2://0RX7DDBYCK
Kindstatement, proposition
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 0RX7DDBYCK.
UsesImage of a set, Intersection

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