Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?

It contains this element, but why is it not a subset?

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    $\begingroup$ You're trying to interpret the word "contain" too broadly. For example, a library may be said to contain letters, words, sentences, paragraphs, and chapters; but when we think of a library as a set, we generally think of it as a set of books, so only the books are members of the set, and not the piecemeal contents/components of the books. $\endgroup$ – Scott Feb 4 '16 at 18:25
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    $\begingroup$ In addition to the excellent answers below, look at math.stackexchange.com/questions/1281436/… -- especially part 3 of the question. $\endgroup$ – David K Feb 4 '16 at 22:41
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    $\begingroup$ $\in\ne\subset$. $\endgroup$ – Martín-Blas Pérez Pinilla Feb 5 '16 at 7:26
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    $\begingroup$ Incidentally, there is a word for a set of which every element is also a subset, and that word is "transitive". $\{\{\emptyset\}\}$ is not a transitive set, but $\{\{\emptyset\},\emptyset\}$ is. $\endgroup$ – Steve Jessop Feb 5 '16 at 11:05

It is not a subset because its element $\emptyset$ does not belong to the set $\{\{\emptyset\}\}$.


The element $\{\emptyset\}$ and the element $\emptyset$ are different.

$\{\emptyset\}$ is an element of $\{\{\emptyset\}\}$, whereas $\emptyset$ is not.

A subset is a set whose every element is also a part of the given set.

Thus, the subsets of $\{\{\emptyset\}\}$ are $\{\{\emptyset\}\}$ and the empty set $\{\}$, also denoted by $\emptyset$.


In one sentence, (Thanks to @Henry)

$\{\{∅\}\}$ has a single element $\{∅\}$ and two subsets $\{\{∅\}\}$ and $∅$, while $\{∅\}$ has a single element $∅$ and two subsets $\{∅\}$ and $∅$.

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    $\begingroup$ You could say that $\{\{\emptyset\}\}$ has a single element $\{\emptyset\}$ and two subsets $\{\{\emptyset\}\}$ and $\emptyset$. Meanwhile $\{\emptyset\}$ has a single element $\emptyset$ and two subsets $\{\emptyset\}$ and $\emptyset$. $\endgroup$ – Henry Feb 4 '16 at 22:49
  • $\begingroup$ @Henry That's a nice way of saying it, have added it to the answer. Thank you. $\endgroup$ – GoodDeeds Feb 5 '16 at 10:43

You can think of sets like plastic bags if you want; the empty set is just a plastic bag with nothing in it, $\{\emptyset\}$ is a plastic bag with another plastic bag in it, and $\{\{\emptyset\}\}$ is three layers of plastic bags.

The element relation $A\in B$ means that you could open up bag $B$ and take out $A$.

The subset relation $A\subset B$ means that every object that you could directly take out of $A$ can also be directly taken out of $B$.

So, look at $\{\emptyset\}$. You can "open it up and" take out $\emptyset$, but you can't do that with $\{\{\emptyset\}\}$. Therefore, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.

  • $\begingroup$ You didn't define B, which makes this answer a bit hard to follow. $\endgroup$ – djechlin Feb 4 '16 at 22:08
  • $\begingroup$ @djechlin Fixed, thanks! $\endgroup$ – Deusovi Feb 4 '16 at 22:24
  • $\begingroup$ sure you can, just open the bag and take the plastic bag with nothing inside out. $\endgroup$ – djechlin Feb 4 '16 at 22:42
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    $\begingroup$ @153330 Then you have fun with whatever visualization or lack of visualization you want. $\endgroup$ – Deusovi Feb 5 '16 at 3:02
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    $\begingroup$ The bag layers' count would be more obvious if you replace $\emptyset$ with$\{\,\}$ after the first use. $\endgroup$ – CiaPan Feb 5 '16 at 10:16

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