1. Equality Question

Why is it the case that $\emptyset=\{\emptyset\}$,

and also that

$\{1,2,3\}=\{\{1\},\{2\},\{3\}\}$ ?

I'm having trouble proving these statements from the definition.

Let $x\in\emptyset$... OK, now I'm lost.

2. Both of these statements are false. Sets are equal iff they have the same elements. The empty set $\emptyset$ does not have any elements whereas $\{\emptyset\}$ has one element $\emptyset$. Similarly, $1\ne\{1\}$, $1\ne\{2\}$, $1\ne\{3\}$, so $1\in\{1,2,3\}$ but $1\notin\{\{1\},\{2\},\{3\}\}$.

Maybe there is something in the context that would make sense of these equalities?

3. Thanks for the reply. I should have specified that the problems were to determine whether or not the sets were equal. Sorry for the confusion.