Some clarifications on my understanding of sets Suppose I have $X = \{\varnothing, \mathbb N, \mathbb Z \}$. Are all nonempty subsets of $\mathbb N$ and $\mathbb Z$ automatically elements of $X$? In particular, is $\mathbb N \cap \mathbb Z$ in $X$? I'm asking because I read somewhere that $\varnothing$ and $\{ \varnothing\}$ are not the same thing. Does it also apply for example to $\mathbb R$ and $\{ \mathbb R\}$?
 A: If you have $X=\{\emptyset, \mathbb N, \mathbb Z\}$, then that means $X$ is a set with exactly three elements: the empty set, the naturals and the integers. The statement $a\in X$ is therefore true if and only if one of the following three statements is true:

*

*$a=\emptyset$.

*$a=\mathbb N$.

*$a=\mathbb Z$.

If the three statements above are all false, then $a$ is not an element of $X$.
In particular:

Are all nonempty subsets of $\mathbb N$ and $\mathbb Z$ automatically elements of $X$?

No. Not all. In fact, only $\mathbb N$ and $\mathbb Z$ are elemnents of $X$. No other nonempty subsets of $\mathbb Z$ or $\mathbb N$ are elements of $X$.

In particular, is $\mathbb N \cap \mathbb Z$ in $X$?

Well, yes, but only because $\mathbb N\cap \mathbb Z = \mathbb N$, and we already know that $\mathbb N\in X$.

Also:

I'm asking because I read somewhere that $\varnothing$ and $\{ \varnothing\}$ are not the same thing.

Well, of course they are not the same thing. $\varnothing$ has zero elements, and $\{\varnothing\}$ has one element, so they cannot be the same set. In fact, it is always true that if $A$ is a set, then $A$ and $\{A\}$ will be two different sets.
