# Differences in $\varnothing$, {$\varnothing$}, and $\subseteq$

What is the difference between

{$\varnothing$} $\subseteq$ {$\varnothing$, {$\varnothing$}},

{{$\varnothing$}} $\subseteq$ {$\varnothing$, {$\varnothing$}},

and

{{$\varnothing$}} $\subseteq$ {{$\varnothing$}, {$\varnothing$}},

where $\varnothing$ is the empty set?

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They are all true statements, but about different things. For the first to be true, you need $\emptyset$ to be an element of $\{\emptyset,\{\emptyset\}\}$, which it is. The second is true because $\{\emptyset\}$ is an element of $\{\emptyset,\{\emptyset\}\}$. The third is true as well, though the right hand side is just equal to $\{\{\emptyset\}\}$; the repetition is immaterial. Is your question really "what is the difference between $\emptyset$ and $\{\emptyset\}$? – Arturo Magidin Sep 18 '11 at 21:20
that's what I thought, but I wasn't sure, it sounded like a trick question to start with. – Caleb Jares Sep 18 '11 at 21:49

$\varnothing$ is the empty set; it has no elements.

$\{\varnothing\}$ is a set that has exactly one element; that element is $\varnothing$ (a bag that contains an empty bag is not itself empty).

$\bigl\{\{\varnothing\}\bigr\}$ is a set whose only element is the set whose only element is $\varnothing$. It is different from $\varnothing$ (which has no elements), and from $\{\varnothing\}$ (which has a single element which has no elements, whereas the single element of $\bigl\{\{\varnothing\}\bigr\}$ does have elements).

And $\bigl\{ \{\varnothing\},\{\varnothing\}\bigr\} = \bigl\{ \{\varnothing\}\bigr\}$, by the Axiom of Extensionality: two sets $A$ and $B$ are equal if and only if for every $x$, $(x\in A\leftrightarrow x\in B)$ which holds here.

So the first statement says that $\{\varnothing\}$ is a subset of $\bigl\{\varnothing,\{\varnothing\}\bigr\}$; essentially, that $\varnothing$ is an element of the set on the right; true.

The second statement says that $\bigl\{\{\varnothing\}\bigr\}$ is a subset of $\bigl\{\varnothing,\{\varnothing\}\bigr\}$; again, essentially that $\{\varnothing\}$ is an element of the set on the right; also true, but a different statement from the first (it refers to different elements).

The third statement says that $\bigl\{\{\varnothing\}\bigr\}$ is a subset of $\bigl\{\{\varnothing\},\{\varnothing\}\bigr\} = \bigl\{ \{\varnothing\}\bigr\}$; that is, that $\{\varnothing\}$ is an element of $\bigl\{\{\varnothing\}\bigr\}$. Also true.

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