In set theory, what does it mean for a set to be transitive, and what does it have to do with transitivity of a relation?
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$\begingroup$A set $X$ is transitive means that $Y \in X$ implies $Y \subset X$. In other words, a set $X$ is transitive whenever $Y \in X$ and $Z \in Y$ implies $Z \in X$. Thus, the $\in$ relation is transitive in this case.
$\endgroup$ 3 $\begingroup$From what I read on wikipedia
A transitive set is one in which inclusion "$\in$" is transitive.
So $A$ is transitive, if whenever for sets $X$ and $Y$ if $Y \in X$ and $X \in A$ then $Y \in A$.
This statement is equivalent to and can be restated as:
$A$ is transitive, if whenever the set $X \in A$, it follows that $X \subset A$.
Notice that this refers to elements that are sets, and not urelements (elements that are not sets).
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Also, I believe, I could be wrong, that if $A$ is transitive and $X \in A|X \subset A$, that $X$ need not be transitive. A counter example could be $X=${{1,{2,3}}, 1,{2,3},2,3}.
$X$ has two set elements, {1,{2,3}} and {2,3} which are both subsets of $X$. {2,3} is vacuously transitive as it has not set elements (only urelements). But {1,{2,3}} is not transitive as {2,3} $\in$ {1,{2,3}} but {2,3} $\not \subset$ {1,{2,3}}.
I'd appreciate it if someone would correct me if I am wrong and if the definition of transitive would require $X \in A$ be that $X$ is also transitive. (And if so was wikipedia wrong?). So to make my set "regressively" transitive I'd need $X=$ {{1,{2,3},2,3},1,{2,3},2,3}.
$A$ is transitive if for any set $x$ that is an element of $A$ ($x \in A$) then $x \subset A$.
This impies that if $y \in x$, $x \in A$ and $y$ is a set, then $y \in A$. Hence the name "transitive". Inclusion is transitive.
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