When a set has an empty set as an element, e.g.$ \{\emptyset, a, b \}$. What is the powerset?
Is it: $$ \{ \emptyset, \{ \emptyset \}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
Or
$$ \{ \emptyset, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
Or
$$ \{ \{\emptyset\}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
The confusion arises for me because, the powerset of every non-empty set has an empty set. Well the original set already has the empty set. So we don't need a subset with an empty set.
Somehow, the first one seems correct. Yet, I can't seem to accept it.
Best Answer
The first one is correct.
This is because $\emptyset$ and $\{\emptyset\}$ are different. The first is an empty set whereas the second is a set whose only element is the empty set.
Both are subsets of the given set. This is because the $\emptyset$ is the subset of every set, and as it happens to be an element of the given set, the set containing it as its element is also its subset.