# transitive closure

The 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 $f(n+1)=\bigcup f(n)$

 $\operatorname{tc}(X)=\bigcup_{n<\omega}f(n)$
Title transitive closure TransitiveClosure 2013-03-22 13:04:33 2013-03-22 13:04:33 Henry (455) Henry (455) 4 Henry (455) Definition msc 03E20 Transitive