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- Well-founded_relation abstract "In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S⊆X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than") for the rest of the s ∈ S.(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.)Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x0, x1, x2, ... of elements of X such that xn+1 R xn for every natural number n.In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order.In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded.A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R-1 is well-founded on X. In this case R is also said to satisfy the ascending chain condition.".
- Well-founded_relation wikiPageID "319712".
- Well-founded_relation wikiPageRevisionID "599890411".
- Well-founded_relation hasPhotoCollection Well-founded_relation.
- Well-founded_relation subject Category:Mathematical_relations.
- Well-founded_relation subject Category:Wellfoundedness.
- Well-founded_relation comment "In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S⊆X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than") for the rest of the s ∈ S.(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.)Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x0, x1, x2, ... ".
- Well-founded_relation label "Fundierte Menge".
- Well-founded_relation label "Relación bien fundada".
- Well-founded_relation label "Relacja dobrze ufundowana".
- Well-founded_relation label "Relation bien fondée".
- Well-founded_relation label "Relação bem-fundada".
- Well-founded_relation label "Welgefundeerde relatie".
- Well-founded_relation label "Well-founded relation".
- Well-founded_relation label "Фундированное множество".
- Well-founded_relation label "整礎関係".
- Well-founded_relation label "良基关系".
- Well-founded_relation sameAs Fundovaná_relace.
- Well-founded_relation sameAs Fundierte_Menge.
- Well-founded_relation sameAs Relación_bien_fundada.
- Well-founded_relation sameAs Relation_bien_fondée.
- Well-founded_relation sameAs 整礎関係.
- Well-founded_relation sameAs Welgefundeerde_relatie.
- Well-founded_relation sameAs Relacja_dobrze_ufundowana.
- Well-founded_relation sameAs Relação_bem-fundada.
- Well-founded_relation sameAs m.01vb_2.
- Well-founded_relation sameAs Q338021.
- Well-founded_relation sameAs Q338021.
- Well-founded_relation wasDerivedFrom Well-founded_relation?oldid=599890411.
- Well-founded_relation isPrimaryTopicOf Well-founded_relation.