Well-founded relation

Type of binary relation

In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset (or subclass) S ⊆ X has a minimal element with respect to R; that is, there exists an m ∈ S such that for every s ∈ S, one does not have s R m. More formally, a relation is well-founded if: ( ∀ S ⊆ X) [ S ≠ ∅ ⟹ ( ∃ m ∈ S) ( ∀ s ∈ S) ¬ ( s R m) ] . {\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].} Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.

From Wikipedia, under CC BY-SA. More on occurri.