transitive closureTransitiveClosure2002-09-28 19:33:302002-09-28 19:33:30Definition% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}The \emph{transitive closure} of a set $X$ is the smallest transitive set $\operatorname{tc}(X)$ such that $X\subseteq \operatorname{tc}(X)$.
The transitive closure of a set can be constructed as follows:
Define a function $f$ on $\omega$ by $f(0)=X$ and $f(n+1)=\bigcup f(n)$
$$\operatorname{tc}(X)=\bigcup_{n<\omega} f(n)$$