Why is $\big\{X:X\subseteq\{3,2,a\}\text{ and }|X|=4\big\}$ empty? Write out the following set by listing its elements between braces:
$$\big\{X:X\subseteq\{3,2,a\}\text{ and }|X|=4\big\}=\emptyset$$
The above answer is correct, however can someone explain why this is so? If the size of $X$ must be $4$, why can the only element be none?
 A: On the left we have a set of all sets $X$, that are subsets of set $\{3,2,a\}$ and have $4$ elements. But the subset of set with $3$ elements can have at most $3$ elements, so there aren't any such sets $X$. That's why on the right side we have an empty set.
A: Try creating a list of every subset of $\{3,2,a\}$. You get the following:
$$\newcommand{\set}[1]{\left\{ #1 \right\}}\begin{align*}
&\varnothing \\
&\set{3} \\
&\set{2} \\
&\set{3, 2} \\
&\set{a} \\
&\set{3, a} \\
&\set{2, a} \\
&\set{3, 2, a}
\end{align*}$$
Remember that, for finite sets, $|X|$ gives you the number of elements in $X$.
Do any of the above sets $X$ have $|X|=4$? Clearly, not.
Hence, the collection of subsets of $\set{3,2,a}$ with precisely $4$ elements -- that is, the set
$$\set{X \mid X \subseteq \set{3,2,a} \text{ and } |X| = 4}$$
must be empty: no set satisfies those conditions.

Remember, you are being asked to give a list of the sets $X$ which meet the conditions

*

*$X \subseteq \set{3,2,a}$

*$|X|=4$
a "set of sets," if you will. You are not being asked to list the members of such a set, but the sets that meet the conditions.
