What does it mean for a number to be in a set? Frustratingly my book gives me several examples of a number in a set but offers no explanation at all.
Anyways what is going on here? According to the book $2$ is not an element of these sets:
$$\{\{2\},\{\{2\}\}\}$$
$$\{\{2\},\{2,\{2\}\}\}$$
$$\{\{\{2\}\}\}$$
What is going on? Clearly $2$ is in all of those sets. Or are they saying that $2$ isn't in any of these sets but a set is in all these sets and in that set is $2$? Which really seems like a logical fallacy because $2$ is in those sets contained in a set means the set has $2$ even if it is behind a layer of sets.
For example you wouldn't say that there are no cars in a neighborhood if all the cars in in a garage, so why does math take this approach?

When we say "$2$ is in a set," we mean that $2$ is an element of the set. In the first case, there are exactly two elements of $\{\{2\}, \{\{2\}\}\}$, namely
$$\{2\}$$ and $$\{\{2\}\}$$
In general, one can find the elements of a given set by erasing the outermost braces.
Neither of these are the number $2$; the first one is a set containing $2$ as an element, and the other is a set containing the set $\{2\}$ as an element.
If we wanted to have $2$ be an element of the set, we'd need either $2 = \{2\}$ or $2 = \{\{2\}\}$, neither of which is true.
