# Definition:Strictly Well-Founded Relation/Definition 1

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## Definition

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

$\RR$ is a **strictly well-founded relation on $S$** if and only if every non-empty subset of $S$ has a strictly minimal element under $\RR$.

## Also known as

A **strictly well-founded relation** is also known in the literature as a **foundational relation**.

It is commonplace in the literature and on the internet to use the term **well-founded relation** for **strictly well-founded relation**.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more cumbersome and arguably more precise **strictly well-founded relation** in preference to all others.

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

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.21$