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Concept
Transitive set
Summary
In set theory, a branch of mathematics, a set is called transitive if either of the following equivalent conditions hold:
whenever , and , then .
whenever , and is not an urelement, then is a subset of .
Similarly, a class is transitive if every element of is a subset of .
Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class.
Any of the stages and leading to the construction of the von Neumann universe and Gödel's constructible universe are transitive sets. The universes and themselves are transitive classes.
This is a complete list of all finite transitive sets with up to 20 brackets:
A set is transitive if and only if , where is the union of all elements of that are sets, .
If is transitive, then is transitive.
If and are transitive, then and are transitive. In general, if is a class all of whose elements are transitive sets, then and are transitive. (The first sentence in this paragraph is the case of .)
A set that does not contain urelements is transitive if and only if it is a subset of its own power set, The power set of a transitive set without urelements is transitive.
The transitive closure of a set is the smallest (with respect to inclusion) transitive set that includes (i.e. ). Suppose one is given a set , then the transitive closure of is
Proof. Denote and . Then we claim that the set
is transitive, and whenever is a transitive set including then .
Assume . Then for some and so . Since , . Thus is transitive.
Now let be as above. We prove by induction that for all , thus proving that : The base case holds since . Now assume . Then . But is transitive so , hence . This completes the proof.
Note that this is the set of all of the objects related to by the transitive closure of the membership relation, since the union of a set can be expressed in terms of the relative product of the membership relation with itself.
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