Solving $ \{ \varnothing, \{ x\} \} = \{ y, \{ \varnothing \} \}$. The equation is:
$$ \{ \varnothing, \{ x\} \} = \{ y, \{ \varnothing \} \}$$
where $ \varnothing $ denotes empty set.
I guess I need to break this into cases for each side. Help me get started.
 A: Well, you have to look at several cases:


*

*Since $\varnothing$ is a member of the left-hand side, then it is also a member of the right-hand side.

*Since the members of the right-hand side are $y$ and $\left\{\varnothing\right\}$, then $\varnothing=\left\{\varnothing\right\}$ or $\varnothing=y$.

*However, $\varnothing\in\left\{\varnothing\right\}$, whereas $\varnothing\not\in\varnothing$. Therefore $\varnothing\neq\left\{\varnothing\right\}$.

*By 2. and 3., the only possibility is $\varnothing=y$.
Now do a similar analysis for $\left\{x\right\}$, knowing that $y=\varnothing$, and conclude that $x=\varnothing$.
A: Two sets are equal if they have exactly the same elements.
so, $\emptyset$ must be in $\{y,\{\emptyset\}\}$
but, it is known that
$\emptyset \ne \{\emptyset\}$
thus
$$\emptyset=y$$
and
$$\{x\}=\{\emptyset\}$$
which means that
$$x=\emptyset=y$$
A: Break things into cases!
Two sets are equal iff they have the same elements. In particular, we have $$\{a,b\}=\{u,v\}\quad\iff\quad \mbox{EITHER ($a=u$ and $b=v$) OR ($a=v$ and $b=u$).}$$ 
(NOTE: this isn't an exclusive "or," since we could have $a=b=u=v$.)
In this case, this tells us that in order to have the desired equality we must have one of the following:


*

*$\emptyset=y$, $\{x\}=\{\emptyset\}$.

*$\color{red}{\emptyset=\{\emptyset\}}$, $y=\{x\}$.
Note the red clause: $\emptyset=\{\emptyset\}$ is not true (this is important to understand), so that second bulletpoint can't hold.
So our only option is the first bulletpoint. That is, $$x=y=\emptyset.$$
A: If two sets are equal, then they have exactly the same elements in each. Since $\emptyset$ is an element of the LHS, it is an element of the RHS. It cannot be $\{\emptyset\}$, since that is not empty, so $y=\emptyset$. It follows that $\{x\}=\{\emptyset\}$, which implies that $x=\emptyset$ also.
