Math 2.0

Mathematical logic and foundations Definition

Intersection of a family of sets

Builds on Family of setsUniversal quantifierIntersection

Definition

Let (Ai)i∈I(A_i)_{i \in I} be a family of sets with a nonempty index set II. The intersection of the family is the set ⋂i∈IAi={x∣∀i∈I, x∈Ai}.\bigcap_{i \in I} A_i = \{x \mid \forall i \in I, \, x \in A_i\}. So xx is an element of the intersection when x∈Aix \in A_i for each i∈Ii \in I.

Remarks

For I={1,2}I = \{1, 2\}, this is the intersection A1∩A2A_1 \cap A_2. For an empty index set, the condition is vacuously true for each xx. So the intersection is defined only relative to a universal set UU, and it is UU. Example: ⋂n≥1(−1/n,1/n)={0}\bigcap_{n \ge 1} (-1/n, 1/n) = \{0\} in ℝ\mathbb{R}.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definition
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Predicate definition
  6. Family of sets definition
  7. Intersection definition
  8. Universal quantifier definition
  9. Intersection of a family of sets definition

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47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Intersection_(set_theory), Arbitrary intersections
About this page
Tag0839XNQXDK
URIm2://0839XNQXDK
Aliasesindexed-intersection, arbitrary-intersection
Kinddefinition
WritingW1 stated
MSC03E20
DefinesIntersection
UsesFamily of sets, Universal quantifier, Intersection

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