# Definition:Well-Founded Relation/Definition 1

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## 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**.

## Sources

- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $1$