# Definition:Well-Founded Relation/Definition 1

## Definition

Let $\struct {S, \RR}$ be a relational structure.

$\RR$ is a well-founded relation on $S$ if and only if:

$\forall T \subseteq S: T \ne \O: \exists z \in T: \forall y \in T \setminus \set z: \tuple {y, z} \notin \RR$

where $\O$ is the empty set.

That is, $\RR$ is a strictly well-founded relation on $S$ if and only if:

for every non-empty subset $T$ of $S$, there exists an element $z$ in $T$ such that for all $y \in T \setminus \set z$, it is not the case that $y \mathrel \RR z$.

## Also known as

The term well-founded relation is often used the literature for what on $\mathsf{Pr} \infty \mathsf{fWiki}$ we call a strictly well-founded relation.

In order to emphasise the differences between the two, at some point a $\mathsf{Pr} \infty \mathsf{fWiki}$ editor coined the term strongly well-founded relation.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the less unwieldy term well-founded relation in preference to others.

Some sources do not hyphenate, and present the name as wellfounded relation.

## Also see

• Results about well-founded relations can be found here.