275: Transitive Closure of Subset

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A definition of transitive closure of subset

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of set.
• The reader admits the transfinite recursion theorem.

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Target Context

• The reader will have a definition of transitive closure of subset.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

For any set, $S$, the natural numbers set, $N$, the formula, $\varphi (x,y)$, where $x$ is any function from any subset of $N$ such that $y=S\cup \cup \cup imax$ where $ima\bullet$ denotes the image of the argument, $\bar{S}:=\cup imaf$ where $f$ is the function constructed by the transfinite recursion theorem

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2: Note

As the name suggests, any transitive closure of any subset is a transitive set that contains the subset, as is proved by a proposition.

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References

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