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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.

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    $\begingroup$ $E$ contains three elements: $1, \{2,3\},$ and $\{2,4\}$. None of these elements is $2$. In set theory, unlike in real life, if $A$ contains $B$ and $B$ contains $C$ we can't deduce that $A$ contains $C$: a set contains its elements and nothing else. $\endgroup$ – TonyK Jan 17 '20 at 11:44
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    $\begingroup$ As you said $E$ has three elements, all of which are different from $2$. $2$ is an element of an element of $E$, but being an element is not transitive. (It is however true that $2$ belongs to the transitive closure of $E$) $\endgroup$ – Alessandro Codenotti Jan 17 '20 at 11:45
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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.

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  • $\begingroup$ What would be a good source to understand more precisely the definition of the ∈ relationship? Would you recommend any book or some other material? $\endgroup$ – César D. Vázquez Jan 17 '20 at 12:30
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    $\begingroup$ My absolute favourite is 'Naive Set Theory' by P R Halmos, which to my mind is a very good read. I don't recall if he actually goes very much into the $\in$ relation; though. Or, if you want a more rigorous treatment, try youtube.com/watch?v=V49i_LM8B0E - the first lecture in a series by Frederic Schuller: "Lectures on the Geometric Anatomy of Theoretical Physics" - he starts right from the bedrock with propositional logic. It isn't easy, but he is a breathtakingly brilliant lecturer and well worth listening to. Plus, you end up understanding fundamental set theory. $\endgroup$ – j4nd3r53n Jan 17 '20 at 12:53
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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 $x\notin E$

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  • $\begingroup$ What would be a good source to understand more precisely the definition of the ∈ relationship? Would you recommend any book or some other material? $\endgroup$ – César D. Vázquez Jan 17 '20 at 12:28
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    $\begingroup$ @CésarD.Vázquez Unfortunately I could not find a book I like. I recommend those lecture notes by Itay Kaplan, In case the link will die. Although it is not easy to read, this is my reccomendation. $\endgroup$ – TheHolyJoker Jan 17 '20 at 12:33
  • $\begingroup$ Haha they don't seem very user friendly. I'll give them a try! thanks $\endgroup$ – César D. Vázquez Jan 17 '20 at 12:38
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    $\begingroup$ @CésarD.Vázquez on second thought "Topology" by James Munkres is one of the best well-written text-book I read, I highly recommend it, although it is not a classic book for set theory. $\endgroup$ – TheHolyJoker Jan 17 '20 at 12:40
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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.

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