Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question already has an answer here:

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

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

share|cite|improve this question

marked as duplicate by Rory Daulton, Claude Leibovici, Eric Wofsey, Joel Reyes Noche, Jack's wasted life Feb 5 at 12:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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 Feb 4 at 18:25
In addition to the excellent answers below, look at… -- especially part 3 of the question. – David K Feb 4 at 22:41
$\in\ne\subset$. – Martín-Blas Pérez Pinilla Feb 5 at 7:26
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 5 at 11:05

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

share|cite|improve this answer

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 $∅$.

share|cite|improve this answer
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 4 at 22:49
@Henry That's a nice way of saying it, have added it to the answer. Thank you. – GoodDeeds Feb 5 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, $\{\emptyset\}\not\subset \{\{\emptyset\}\}$.

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.