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The following definition is a generalization of the concept of minimal elements in a set and of a well-order.

## Definition: Well-founded Relation

Let $V$ be a set, and let $R\subseteq V\times V$ be a binary relation. The relation $R$ is called well-founded if every non-empty subset $S\subseteq V$ contains a minimal element $m$ with respect to $R$. In other words, no element $x\in S$ is in the left-sided relation $xRm.$ Formally:

$$\forall S\subseteq V\; \exists x\in S \Rightarrow \exists m\in S \quad \forall x\in S \quad (x,m)\not\in R.$$

| | | | | created: 2019-02-01 00:04:53 | modified: 2019-02-17 07:48:40 | by: bookofproofs | references: [8055]

## 1.Example: Examples of Well-founded Relations

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### Bibliography (further reading)

[8055] Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018