How does the empty set differ from a set containing the empty set?

In set theory, the empty set and a set containing the empty set are fundamentally different objects because they possess a different number of elements.

• The Empty Set ($\varnothing$): This is a set that contains no elements at all. By definition, its cardinality is 0.12
• A Set Containing the Empty Set ($\{\varnothing \}$): This is a set that contains exactly one element: the empty set itself. Because it contains one member, its cardinality is 1.12

To distinguish them, consider their membership: the empty set has no members, whereas the set $\{\varnothing \}$ has one member, which is the empty set. While the empty set is a subset of every set, it is only a member of sets that specifically include it as an element.12