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I just started set theory (ncert maths 11th grade) and it had a question:

Let $A = \{1 , 2 , \{3,4\} , 5 , 6\}$

Is $\{3,4\}$ a subset of A?

The book said no. I think the answer should be yes, since $A$ contains a set which contains the elements $3,4$ hence $A$ contains $3,4$ and by the definition of subset $\{3,4\}$ should be a subset of $A$.

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BatMandor
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  • While the set in question is different, the principle is the same. – Asaf Karagila May 24 '18 at 17:28
  • I was just about to post this as an answer before the question was closed, but I'll include it anyway since using OP's example may help understanding:

    The book is correct: $A$ doesn't contain $3$, nor does it contain $4$ even though it does contain a set containing $3$ and $4$. It's important to distinguish between these notions! It is the case that ${1,2}$ is a subset of $A$, since all the elements of ${1,2}$ are also elements of $A$, and it is the case that ${3,4}$ is itself an element of $A$, but ${3,4}$ is not a subset of $A$.

     – B. Mehta May 24 '18 at 17:29
  • @B.Mehta: As a principle, always assume that all the basic questions are duplicates. At least until you put some effort into finding a suitable duplicate, or the lack thereof. – Asaf Karagila May 24 '18 at 17:30
  • @AsafKaragila Yes, I'd expected this to be a duplicate but I couldn't find a duplicate in the related list which addressed this concern, nor on the list of common questions so I started writing an answer anyway. – B. Mehta May 24 '18 at 17:33
  • @AsafKaragila In fact it may be worth adding the duplicate target to that list, it does seem like a very common question. – B. Mehta May 24 '18 at 17:34
  • @B.Mehta: If you have some suggestions, I'd be happy to add them. – Asaf Karagila May 24 '18 at 17:34
  • @AsafKaragila I'm sure you'll know better than I, but this and this probably ought to be on that list, under the same bullet point. – B. Mehta May 24 '18 at 17:37
  • I am sorry for inconvenience. – BatMandor May 24 '18 at 17:38
  • "since A contains a set which contains the elements 3,4 hence A contains 3,4" That is a fundamental misconception. A does not contain 3 or 4, it contains a set that contains 3 and 4. A set does not contain the stuff inside it's elements. Think of it like a phone book. A phonebook contains people. My library contains a phone book. But my library does not contain people. (That would be slavery and that is morally wrong.) – fleablood May 24 '18 at 17:40
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    "I am sorry for inconvenience" You should not be sorry for asking a legitimate question. And this is a very legitimate question. Search this site for answers can be difficult so no one can blame you asking a question that has already been answered. – fleablood May 24 '18 at 17:43
  • Someone might argue that a phonebook does not contain people but references to people. Okay, but what about a cafeteria that sells bag lunches containing bananas but does not sell bananas? Or... I hate this analogy, but eveyone else finds it useful... think of sets as "bags" holding things. If you have a bag with a 1, a 2 and another bag inside it that has a 3 and a 4, the outer bag doesn't actually have the stuff in the inner bag as its contents. It just has the bag itself. – fleablood May 24 '18 at 17:47
  • @fleablood This is where I think I am wrong: If B is a subset of A, then every element in B is an element in A. But in here, 3 and 4 are not elements of A, but they are elements of an element in A. So basically when we say a /epsilon A, then we mean that a is an element of A, a does not belong to A. About your phonebook example, phonebook contains people's number not people either, hence the library contains people's number, which makes sense, to me. – BatMandor May 24 '18 at 17:49
  • "a does not belong to A". The meaning of "belong to" is not clear. $a \in A$ means $a$ is an element of $A$. $B\subset A$ means be is a set and that all the elements that are in $B$ are elements of $A$. Member ship is not transitive. If $a \in B$ and $B \in A$ that does not mean $a \in A$. I am member The AlphaFinger Society. And the AlphaFinger Society is a collection of people. And the AlphaFinger Society is a member The Associated Coalition of Societies which is a collection of societies. But i'm not a member of the ACS which has societies, not people, as members. – fleablood May 24 '18 at 17:57
  • @fleablood, ohk so in set theory, belongs to means it is an element of the set and not like, it is inside it (like a ball in box) – BatMandor May 24 '18 at 18:06
  • That's strange way of looking at it but if it works for you... I personally do think of elements as being inside the set but not nested layers deep inside the set. I'd say $A={a,b,{c,d}}$ has three things inside it; the $a$, $b$ and ${c,d}$. the stuff that are nested inside ${c,d}$ are not directly inside $A$ but nested too deep. But "inside" is just a ... words. The elements of $A$ are $a$, $b$ and ${c,d}$. That's it. The elements within elements are not elements. A set is a list (where order doesn't matter) and $c$ and $d$ are simply not on the list. – fleablood May 24 '18 at 18:53
  • ... and corporations are NOT people! – fleablood May 24 '18 at 18:53

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