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?

It might be a good idea to start with something maybe slightly more basic:
What is the difference between these two things: $\emptyset$ and $\{\emptyset\}$.
The first is just the empty set (which has no elements)
The second is a set with exactly one element - which is the empty set.
Think about this for a while, and it should become clearer.
