Why is it that given $E=\{ 1, \{2,3\}, \{2,4\} \}$ we have $1\in E$ but $2\not\in E$? I am reading "Book of proof" of Richard Hammard and it says that given the set $E=\{ 1, \{2,3\}, \{2,4\} \}$, we have that $2\not\in E$ . I do not understand why is it so. I understand that the set $E$ is a collection of the elements "$1$", "$\{2,3\}$" and "$\{2,4\}$". But I do not understand why I could not go further and deduce that since $2\in\{2,3\}$ and $\{2,3\}$ is an element of $E$ then $2$ must be an element of $E$.
Sorry if this is a very basic question but I do not even seem to know how to ask for this clarification. 
 A: Let's denote the set of all the basketball teams in the NBA by $E$.
Denote the Lakers by $u$ (set of players).
Denote Michael Jordan by $x$ (singleton on a player).
Obviously the Lakers is a basketball team in the NBA, and therefore $u\in E$.
Michael Jordan was a player in the Lakers, and therefore $x\in u$
But Michael Jordan is obviously not an NBA basketball team, and therefore $x\notin E$ 
A: I think this is a deliberately provocative example, chosen by the author to illustrate the thought, that the relation, $\in$ is 'blind' to whatever structure the elements may have. The elements of a set are just anonymous 'lumps' - labels, if you will - and we don't care about what these labels mean, if indeed they mean anything at all.
Surprisingly, this is, in my experience, often one of the more difficult aspects of mathematics: you sometimes have to deliberately ignore any further knowledge and limit yourself to just the particular aspect that you are studying.
Sets theory is perhaps 'worse' than other areas of maths - a general set has no structure; the only trait that all sets have, is cardinality: loosely speaking how many elements the set contains. From that point of view, the set $\{ apple, pear, orange \}$ is exactly the same as $\{1, \{2,3 \}, \{2,4\} \} $; they are isomorphic to use the modern (category theoretical) term, which simply means they are indistinguishable given the tools of, here, set theory.
A: The set $E$ has exactly three elements: 1, {2,3} and {2,4}. If 2 were an element of $E$, then it would be a fourth element, and we would write $E=\{1,2,\{2,3\}, \{2,4\}\}$. You need to make a distinction between an element such as $2$ and a set that contains exactly one element, such as $\{2\}$.  As another example, $\emptyset$ is the empty set, but $\{\emptyset\}$ is a set that contains exactly one element, and that element is the empty set.
