Let $x\in y$ and $y\in z$. Does this imply that $x\in z$? For example: Let $y=\{A,B\}$ and $z=\{\{A,B\},C\}$. If $x=A$, then $x\in y$. My understanding, however, is that $x$ is not an element of $z$ since $A$ is not an element of $z$.

  • 5
    $\begingroup$ You are correct. $z$ has a set that holds $A$, but $z$ does not hold $A$. $\endgroup$ Sep 20, 2014 at 5:13
  • $\begingroup$ The really tough question is whether $x$ can be an element of $x$ ... $\endgroup$
    – orangeskid
    Sep 20, 2014 at 5:29

2 Answers 2


That's correct, though it can still potentially be true: \begin{align} x&=A\\ y&=\{A,B\}\\ z&=\{\{A,B\},A\}.\end{align} So we have to know the sets completely to be sure.

  • $\begingroup$ Sure, by modifying the example we can make anything become true :) $\endgroup$
    – Ja͢ck
    Sep 23, 2014 at 11:02

You have described the transitive property of $z$.

$z$ is transitive if and only if:

$y \in z \land x \in y \implies x \in z$

In your example, $z$ is therefore not transitive.


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