Is X a subset of the {X}? I'm new to this website. I'm trying to understand the precise definition of a subset.
In particular, this problem is tripping me up: 
True or False? 
$\{\emptyset, \{\emptyset\}\} \subseteq \{\{\emptyset, \{\emptyset\}\}\}$
I know that any set X is a subset of itself. I'm taking a guess however, that the answer to this is false. Is this because $\{\{\emptyset, \{\emptyset\}\}\}$ contains only one element which is $\{\emptyset, \{\emptyset\}\}$?

You are correct; the particular relation of $X$ to $\{X\}$ is that $X\in \{X\}$ rather than that $X\subseteq \{X\}$ (which is generally false). In the given question of whether $\{\emptyset, \{\emptyset\}\} \subseteq \{\{\emptyset, \{\emptyset\}\}\}$, one can test to see if every element of the left hand side is in the right; in particular, the left contains $\emptyset$ while the right does not, so the statement is false. (In more generality $X\subseteq \{X\}$ holds only when $X=\emptyset$ in ZFC)
