# 4 is an element in the following set:

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?

• Your reasoning about A) should be: $4$ is not an element of that set because $4<4<10$ is not true. – drhab Feb 15 '16 at 15:48

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.
• In D, $\{4\}$ is an element of the set $\{\{4\}, \{\{4\}\}$ rather than a subset. A similar comment applies to E. – N. F. Taussig Feb 15 '16 at 15:45