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

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

newbie
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    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. – Scott - Слава Україні Feb 04 '16 at 18:25
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    In addition to the excellent answers below, look at http://math.stackexchange.com/questions/1281436/confusion-about-the-null-empty-set-being-contained-in-other-sets -- especially part 3 of the question. – David K Feb 04 '16 at 22:41
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    $\in\ne\subset$. – Martín-Blas Pérez Pinilla Feb 05 '16 at 07:26
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    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. – Steve Jessop Feb 05 '16 at 11:05

3 Answers3

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

Onil90
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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 $\emptyset$.

EDIT:

In one sentence, (Thanks to @Henry)

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

GoodDeeds
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    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 $\emptyset$. – Henry Feb 04 '16 at 22:49
  • @Henry That's a nice way of saying it, have added it to the answer. Thank you. – GoodDeeds Feb 05 '16 at 10:43
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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, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.

Deusovi
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