A definition of transitive closure of subset
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader admits the transfinite recursion theorem.
Target Context
- The reader will have a definition of transitive closure of subset.
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.
Main Body
1: Definition
For any set, \(S\), the natural numbers set, \(N\), the formula, \(\phi (x, y)\), where \(x\) is any function from any subset of \(N\) such that \(y = S \cup \cup \cup ima \text{ }x\) where \(ima \text{ }\bullet\) denotes the image of the argument, \(\overline S := \cup ima \text{ }f\) where \(f\) is the function constructed by the transfinite recursion theorem
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.
