Math 2.0

Mathematical logic and foundations Theorem

Distributive laws for sets

Builds on UnionIntersection

Statement

Let AA, BB and CC be sets. Then intersection distributes over union, and union distributes over intersection: A∩(B∪C)=(A∩B)∪(A∩C),A \cap (B \cup C) = (A \cap B) \cup (A \cap C), A∪(B∩C)=(A∪B)∩(A∪C).A \cup (B \cap C) = (A \cup B) \cap (A \cup C).

Remarks

The laws follow from the distributive laws of logic for ∧\land and ∨\lor. The second law has no analogue for numbers: a+bca + bc is not (a+b)(a+c)(a + b)(a + c) in general.

Reading path

To understand this page from the ground up, read these 8 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 levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionDistributive laws for sets, statement, no evidence levelDistributive laws for…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.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Conjunction definition
  5. Disjunction definition
  6. Intersection definition
  7. Union definition
  8. Distributive laws for sets statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebra_of_sets, lead section
About this page
Tag08N6EVA7DV
URIm2://08N6EVA7DV
Aliasesdistributivity-of-union-and-intersection
Kindstatement, theorem
WritingW0 stub
MSC03E20
StatusThis copy of the site has no build data. Run m2 status 08N6EVA7DV.
UsesUnion, Intersection

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