Math 2.0

Mathematical logic and foundations Definition

Transitive closure

Builds on Binary relationTransitive relation

Definition

Let RR be a binary relation on a set AA. The transitive closure of RR is the relation R+R^{+} on AA such that aR+ba \mathrel{R^{+}} b when there are n≥1n \ge 1 and x0,…,xn∈Ax_0, \dots, x_n \in A with a=x0Rx1Rx2⋯xn−1Rxn=b.a = x_0 \mathrel{R} x_1 \mathrel{R} x_2 \cdots x_{n-1} \mathrel{R} x_n = b.

Remarks

The relation R+R^{+} is transitive, it contains RR, and it is a subset of each transitive relation that contains RR. It is also the intersection of all transitive relations on AA that contain RR. Example: the transitive closure of "is a parent of" is "is an ancestor of".

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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 pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionTransitive closure, definition, no evidence levelTransitive closuredefinition
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. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Transitive relation definition
  12. Transitive closure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Transitive_closure, lead section
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Kinddefinition
WritingW1 stated
MSC03E20
DefinesTransitive closure
UsesBinary relation, Transitive relation

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