well-founded relationWellFoundedRelation2007-07-18 23:10:112008-04-01 23:44:44Definitionwell-founded% this is the default PlanetMath preamble. as your knowledge
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A binary relation $R$ on a \PMlinkname{class}{Class} $X$ is \emph{well-founded} if and only if
\begin{itemize}
\item each nonempty subclass of $X$ contains an $R$-minimal element and,
\item for each $x \in X$, $\lbrace y \mid y\,R\,x \rbrace$ is a set.
\end{itemize}
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.