Math 2.0

Mathematical logic and foundations Definition

Well-founded relation

Builds on Binary relationMinimal element

Definition

Let RR be a binary relation on a set AA. The relation RR is well-founded when each nonempty subset SS of AA has an element mm such that no s∈Ss \in S satisfies sRms \mathrel{R} m.

Remarks

The relation << on ℕ\mathbb{N} is well-founded. The relation << on ℤ\mathbb{Z} is not, because ℤ\mathbb{Z} itself has no minimal element. If RR is well-founded, there is no sequence x0,x1,x2,…x_0, x_1, x_2, \dots with xn+1Rxnx_{n+1} \mathrel{R} x_n for each nn. The converse needs the axiom of dependent choice. Well-founded relations allow proofs by transfinite induction.

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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 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 relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionMinimal element, definition, no evidence levelMinimal elementdefinitionWell-founded relation, definition, no evidence levelWell-founded relationdefinition
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. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Minimal element definition
  17. Well-founded relation definition

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Well-founded_relation, lead section
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DefinesWell-founded relation
UsesBinary relation, Minimal element

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