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transitive closure (Definition)

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)$

$\displaystyle \operatorname{tc}(X)=\bigcup_{n<\omega} f(n)$



"transitive closure" is owned by Henry.
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See Also: transitive
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Cross-references: function, transitive set
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This is version 1 of transitive closure, born on 2002-09-28.
Object id is 3486, canonical name is TransitiveClosure.
Accessed 5212 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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