[Math] Subtracting empty set from another

Question

I am considering two sets $A = \{\emptyset, \{\emptyset\}\}\setminus\{\emptyset\}$, $B = \{\emptyset, \{\emptyset\}\}\setminus\{\{\emptyset\}\}$. From the definition of the set difference, I have to consider only those elements of the first set that don't belong to the second one. So my guess is $B = \{ \emptyset \}$. $A$ is more problematic – the empty set doesn't have any elements. My guess is there are no elements to remove. Then, is $A = \{\emptyset, \{\emptyset\}\}$ the correct result?

Answer 1

You are not removing the empty set. You are removing a set whose only element is the empty set, as $$\emptyset\ne\{\emptyset\}$$

So, $$A\setminus\{\emptyset\}=\{\{\emptyset\}\}$$

Your claim would be true if the set being removed were indeed empty, i.e., $$A\setminus\emptyset=A$$