Solving $ \{ \varnothing, \{ x\} \} = \{ y, \{ \varnothing \} \}$.

Well, you have to look at several cases:

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

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

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

4. 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 $x=\varnothing$.

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 $$x=\emptyset=y$$