Power Set

The Power Set \mathcal{P}(S) of a set S is the set of all subsets of S, including the empty set \varnothing and S itself. It is the building block for defining the collection of Events on a Sample Space in Set Theory.

\mathcal{P}(S) = \{\, A : A \subseteq S \,\}

|\mathcal{P}(S)| = 2^{|S|}

• The cardinality |\mathcal{P}(S)| = 2^{n} for a finite set with n = |S| elements, because each element is independently either in or out of a given subset.
• It always contains both the empty set \varnothing and the full set S.
• Grouping subsets by their size links the count to Combinatorics: \sum_{k=0}^{n} \binom{n}{k} = 2^{n}, so the Combination counts of every size sum to the power-set total.
• On a finite Sample Space, \mathcal{P}(S) is the natural event space — every event is one of its members.

Counting Policy Rider Combinations An auto policy offers 5 distinct optional riders (roadside, rental, glass, gap, accident-forgiveness). How many different coverage packages can a customer build, including the bare policy with no riders?

Answer Each package is a subset of the 5 riders, so the number of packages is the size of the power set: |\mathcal{P}(S)| = 2^{5} = 32 This counts the empty set (no riders) and the full set (all five riders) among the 32 possible packages.

Decomposing the Power Set by Subset Size List the power set of S = \{A, B, C\} and verify the count by summing the number of subsets of each size.

Answer The 2^{3} = 8 subsets, grouped by size, are:

\begin{align*}
\text{size } 0 &: \ \varnothing \\
\text{size } 1 &: \ \{A\},\ \{B\},\ \{C\} \\
\text{size } 2 &: \ \{A,B\},\ \{A,C\},\ \{B,C\} \\
\text{size } 3 &: \ \{A,B,C\}
\end{align*}

Summing the Combination counts of each size reproduces the total: \binom{3}{0} + \binom{3}{1} + \binom{3}{2} + \binom{3}{3} = 1 + 3 + 3 + 1 = 8