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True or False. if $X ∈ Y$ and $Y$ is transitive, then $X$ is transitive. So far I have: To show that $X$ is transitive, we argue $A ∈ B ∈ X$ implies $A ∈ X$. Suppose $A ∈ B ∈ X$. Since $X ∈ Y$, then $A ∈ B ∈ X ∈ Y$. Since $Y$ is transitive, then $A ∈ B ∈ X ⊆ Y$. Since $Y$ is transitive, then $A ∈ X ∈ Y$. Is this right so far? Or am I doing it wrong? Also if I am heading in the right direction can someone point out to me where to go from here? Thank you very much.

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Its false: think about the cumulative hierarchy. Assuming regularity, every set $x$ is an element of $V_\alpha$ for some ordinal $\alpha$. But $V_\alpha$ is transitive, so if the conjecture were true, then every set would be transitive. –  goblin Dec 6 '13 at 12:59

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The statement is false. In fact any set $X$, transitive or not, can be an element of an appropriately chosen transitive set $Y$. For instance, the set $X = \{\{\emptyset\}\}$ is not transitive. Try to find a set $Y$ such that $X \in Y$ and $Y$ is transitive.

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