Definition: The power set of a set \(A\) is the set of all subsets of \(A\). The power set of \(A\) is written \(\mathcal{P}(A).\)

For example, let \(A = \{a, b, c\}\). The power set of \(A\) is \[\mathcal{P}(A) = \{\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\}\] Notice that the power set is a set of sets. So, while \(\{a\} \in \mathcal{P}(A)\), \(a \not\in \mathcal{P}(A)\) since \(a\) is not a set. Also, notice that \(\emptyset \in \mathcal{P}(A)\) and \(A \in \mathcal{P}(A)\). The empty set belongs to every power set, and every set belongs to its own power set.