Math 2.0

Mathematical logic and foundations Definition

Composition of relations

Builds on Binary relationExistential quantifier

Definition

Let AA, BB and CC be sets, let R⊆A×BR \subseteq A \times B and let S⊆B×CS \subseteq B \times C be binary relations. The composition of RR and SS is the relation from AA to CC R;S={(a,c)∈A×C∣∃b∈B, aRb and bSc}.R \mathbin{;} S = \{(a, c) \in A \times C \mid \exists b \in B, \, a \mathrel{R} b \text{ and } b \mathrel{S} c\}.

Remarks

Many authors write S∘RS \circ R for R;SR \mathbin{;} S. This order agrees with function composition. Composition of relations is associative. A relation RR on a set is transitive exactly when R;R⊆RR \mathbin{;} R \subseteq R.

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To understand this page from the ground up, read these 12 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 levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionComposition of relations, definition, no evidence levelComposition of relati…definition
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. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Binary relation definition
  12. Composition of relations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_of_relations, lead section
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DefinesComposition of relations
UsesBinary relation, Existential quantifier

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