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well-founded relation (Definition)

A binary relation $ R$ on a class $ X$ is well-founded if and only if

  • each nonempty subclass of $ X$ contains an $ R$-minimal element and,
  • for each $ x \in X$, $ \lbrace y \mid y\,R\,x \rbrace$ is a set.

The notion of a well-founded relation is a generalization of that of a well-ordering relation: proof by induction and definition by recursion may be carried out over well-founded relations.



"well-founded relation" is owned by ratboy. [ full author list (2) ]
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See Also: relation, R-minimal element

Also defines:  well-founded
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Cross-references: induction, relation, well-ordering, contains, subclass, binary relation
There are 8 references to this entry.

This is version 6 of well-founded relation, born on 2007-07-18, modified 2008-04-01.
Object id is 9781, canonical name is WellFoundedRelation.
Accessed 686 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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