4
$\begingroup$

What is the difference between

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

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

and

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

where $\varnothing$ is the empty set?

$\endgroup$
  • 6
    $\begingroup$ 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 $\{\emptyset\}$? $\endgroup$ – Arturo Magidin Sep 18 '11 at 21:20
  • $\begingroup$ that's what I thought, but I wasn't sure, it sounded like a trick question to start with. $\endgroup$ – Caleb Jares Sep 18 '11 at 21:49
12
$\begingroup$

$\varnothing$ 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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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