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A) $\{x \in \mathbb{Z} \mid 4 < x < 10\}$

B) $\{x \in \mathbb{Z} \mid x \text{ is the square of an integer}\}$

C) $\{4, \{4\}\}$

D) $\{\{4\},\{\{4\}\}\}$

E) $\{\{\{4\}\}\}$

Here are the answers I came up with. I am pretty confident about A-C, but I'm not sure about D and E.

A: $4$ is not an element because $x < 4$. False

B: This can be true, only if the integer being squared is $2$. True

C: This contained $4$ and the subset $\{4\}$, so True.

D: Both of these are subsets, so False.

E: This is only a subset, which is not the same as $4$. False.

Could someone confirm my logic on this is correct?

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    $\begingroup$ Your reasoning about A) should be: $4$ is not an element of that set because $4<4<10$ is not true. $\endgroup$ – drhab Feb 15 '16 at 15:48
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Your answers to all of them are correct, but $D$ and $E$ are poorly expressed. For $D$ you might say $\{4\}$ and $\{\{4\}\}$ are elements but $4$ is not.

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    $\begingroup$ I see. Thank you! @RossMillikan $\endgroup$ – CSstudent Feb 15 '16 at 15:41
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All of your answers are absolutely correct. {4} but not 4, is an element of {{4}}. Similarly, {{4}} is an element of {{{4}}}.

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    $\begingroup$ Excellent. Thank you! $\endgroup$ – CSstudent Feb 15 '16 at 15:39
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    $\begingroup$ In D, $\{4\}$ is an element of the set $\{\{4\}, \{\{4\}\}$ rather than a subset. A similar comment applies to E. $\endgroup$ – N. F. Taussig Feb 15 '16 at 15:45
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    $\begingroup$ @N.F.Taussig You are right thank you. $\endgroup$ – Win Vineeth Feb 15 '16 at 15:46

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