Power Set

So far, we looked into sets that contain various objects. Now, we'll also talk about sets whose elements are other sets!

The power set of a set $S$ is a set containing all the possible subsets of $S$. It is a sort of a multiverse of subsets. We denote the power set as $\mathcal{P}(S)$ (

$\mathcal{P}(S)$

). Examples:

• For $S = \{😀, 😁\}$, we have $\mathcal{P}(S) = \;$$\{\emptyset, \{😀\}, \{😁\}, \,$$\{😀, 😁\}\}$.
• For $N = \{1, 2, 3\}$, we have $\mathcal{P}(N) = \;$$\{\emptyset, \{1\}, \{2\}, \{3\}, \,$$\{1, 2\}, \{1, 3\}, \{2, 3\}, \,$$\{1, 2, 3\}\}$.
• For $A = \{\text{Alice}\}$, we have $\mathcal{P}(A) = \{\emptyset, \{\text{Alice}\}\}$.

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If a set $S$ contains $n$ elements, how many elements does $\mathcal{P}(S)$ have?

From the above examples, we see that (1) when $|S| = 1$, $|\mathcal{P}(S)| = 2$, (2) when $|S| = 2$, $|\mathcal{P}(S)| = 4$, and (3) when $|S| = 3$, $|\mathcal{P}(S)| = 8$. Can you see a pattern? What is the formula for $|\mathcal{P}(S)|$ in the general case of $|S| = n$?