The empty set versus a set containing only the empty set

Question

Please help me unpack the following claim:

The empty set is different from a set containing only the empty set.
∅ ≠ {∅}

Answer 1

Intuitively, the empty set contains nothing. The set containing only the empty set is a set maden of one element, which in this case is the empty set. Hence it contains something, is just a set maden by another set.

Another way to see at this is by looking at the number of elements in the sets: by definition, the empty set has zero elements, while the set containing the empty set has one element.

An obvious necessary condition for set equality is that the sets we are equaling have the same number of elements, which is not the case here. Hence the two sets are different.