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I have came across this in a textbook:

$\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$

I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset?

Mauro ALLEGRANZA
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PotWashMike
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    I think that I wrote an answer to this about 30 times now. I'll go fetch a duplicate. Hang on. – Asaf Karagila Jan 14 '14 at 21:06
  • The subsets of $A={{2}}$ are elements in $\mathcal{P}({{2}})={\emptyset,A}$ – Listing Jan 14 '14 at 21:07
  • @AsafKaragila sorry – PotWashMike Jan 14 '14 at 21:08
  • @Listing I think I get it actually. Are you saying that as a subset is a set of the elements in another set, as {2} is an element of the other set it cannot essentially exist as its own in this context, it must be contained within a set i.e. {{2}} thus making it equal and not a subset? (I'm not a mathematician, sorry about the informal logic) – PotWashMike Jan 14 '14 at 21:10
  • Not word for word, but surely close enough to help you: http://math.stackexchange.com/questions/169198/element-of-a-set/ and http://math.stackexchange.com/questions/135218/set-theory-subset-of-set/ (And I am far more than certain there are a couple more!) – Asaf Karagila Jan 14 '14 at 21:12
  • @AsafKaragila Thank you for taking the time to find those links and to edit my question. – PotWashMike Jan 14 '14 at 21:15
  • Try to recall the definition of a subset. It helps you to understand why {2}⊈{{2}}. – Dave Clifford Jan 14 '14 at 21:29

2 Answers2

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The assertion $\{ 2 \} \subseteq \{ \{ 2 \} \}$ states that every element of $\{2\}$ is an element of $\{\{2\}\}$. But $2$ is the only element of $\{2\}$, and $2$ is not an element of $\{\{2\}\}$ because the only element of this latter set is $\{2\}$, and $2 \ne \{2\}$!

Just because something is a set and it's an element of another set doesn't mean it's a subset of it.

Clive Newstead
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  • Took me a lot of times to read through that to fully understand it. But I couldn't have asked for a better explanation. Thank you. I get it now! – PotWashMike Jan 14 '14 at 21:12
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To phrase it differently:

  1. $\{\{2\}\}$ is a set containing a set
  2. $\{2\}$ is a set containing a number

Since they are sets containing different kind of things, they cannot have a superset-subset relation to each other...

String
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  • Thank you for your answer. – PotWashMike Jan 14 '14 at 21:14
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    (Depends on the definitions, ${2}$ is also a set containing a set.) – Asaf Karagila Jan 14 '14 at 21:14
  • @AsafKaragila: I think I know what you mean, but that would make ${{2}}$ a set containing a set containing a set... Still I do not see why this would be necessary to think about in the current situation. It rather complicates things. – String Jan 14 '14 at 21:19
  • Is the set it contains, the empty set? – PotWashMike Jan 14 '14 at 21:21
  • http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers – String Jan 14 '14 at 21:23
  • But it is really irrelevant as it confuses the point... – String Jan 14 '14 at 21:23
  • String: I used to think so myself. But I have to say that pointing out to students in the first class of the semester/year/degree, that sets can be (and usually are) members of other sets is important. The point is that at least in our common-sense interpretation of things, $2\neq{2}$, and therefore ${2}$ and ${{2}}$ are different sets with different elements. – Asaf Karagila Jan 14 '14 at 21:37
  • @AsafKaragila: Ok, you are probably right. I just though that the current case was hard enough... – String Jan 14 '14 at 21:58