# well-founded relation

A binary relation $R$ on a class (http://planetmath.org/Class) $X$ is well-founded if and only if

• each nonempty subclass of $X$ contains an $R$-minimal element and,

• for each $x\in X$, $\{y\mid y\,R\,x\}$ 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.

Title well-founded relation WellfoundedRelation 2013-03-22 17:24:31 2013-03-22 17:24:31 ratboy (4018) ratboy (4018) 9 ratboy (4018) Definition msc 03E20 Relation RMinimalElement well-founded