Why is $\{\emptyset\}$ not a subset of $\{\{\emptyset\}\}$?
It contains this element, but why is it not a subset?
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7$\begingroup$ 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 - Слава УкраїніFeb 4, 2016 at 18:25
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2$\begingroup$ In addition to the excellent answers below, look at math.stackexchange.com/questions/1281436/… -- especially part 3 of the question. $\endgroup$– David KFeb 4, 2016 at 22:41
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7$\begingroup$ $\in\ne\subset$. $\endgroup$– Martín-Blas Pérez PinillaFeb 5, 2016 at 7:26
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1$\begingroup$ 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 JessopFeb 5, 2016 at 11:05
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3 Answers
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It is not a subset because its element $\emptyset$ does not belong to the set $\{\{\emptyset\}\}$.
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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 $∅$.
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1$\begingroup$ 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 $\emptyset$. $\endgroup$– HenryFeb 4, 2016 at 22:49
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$\begingroup$ @Henry That's a nice way of saying it, have added it to the answer. Thank you. $\endgroup$ Feb 5, 2016 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, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.
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$\begingroup$ You didn't define B, which makes this answer a bit hard to follow. $\endgroup$– djechlinFeb 4, 2016 at 22:08
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$\begingroup$ sure you can, just open the bag and take the plastic bag with nothing inside out. $\endgroup$– djechlinFeb 4, 2016 at 22:42
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5$\begingroup$ @153330 Then you have fun with whatever visualization or lack of visualization you want. $\endgroup$– DeusoviFeb 5, 2016 at 3:02
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1$\begingroup$ The bag layers' count would be more obvious if you replace $\emptyset$ with$\{\,\}$ after the first use. $\endgroup$– CiaPanFeb 5, 2016 at 10:16