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 $x\in X$, $\{y\mid yRx\}$ 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
Canonical name	WellfoundedRelation
Date of creation	2013-03-22 17:24:31
Last modified on	2013-03-22 17:24:31
Owner	ratboy (4018)
Last modified by	ratboy (4018)
Numerical id	9
Author	ratboy (4018)
Entry type	Definition
Classification	msc 03E20
Related topic	Relation
Related topic	RMinimalElement
Defines	well-founded