4 Power set

The power set of $S$ , denoted as $\mathcal{P}(S)$ , is the set of all possible subsets of $S$ . Yes, an element in a set can be a set! Here is an example:

$\mathcal{P}(\rm {PrimaryColors}) = \{ \{\}, \{\rm {red}\}, \{\rm {blue}\}, \{\... ...en}\}, \{\rm {red}, \rm {green}\},\cdots\{\rm {red},\rm {green}, \rm {blue}\}\}$

In this example, there should be $2^{\vert S\vert}=2^3=8$ elements in $\mathcal{P}(S)$ . In other words, $\vert\mathcal{P}(S)\vert = 2^{\vert S\vert}$ .

If $Q \subset \mathcal{P}(S)$ , then $Q$ is called a ``family of sets over $S$ ''.