Why does a set with n elements have a power set containing 2^n subsets?

To understand why a set with $n$ elements has a power set containing $2^n$ subsets, consider the process of constructing any possible subset from a set $S$ that contains $n$ distinct elements.

For each individual element in the set $S$, you have exactly two choices when forming a subset: either include that element in the subset or exclude it. Since there are $n$ elements in total and each choice is independent, you multiply the number of choices for each element together:

$$2\times 2\times 2\times \dots \times 2\text{ ((n) times)}=2^n$$

This calculation accounts for all possible combinations, including the empty set (where every element is excluded) and the set $S$ itself (where every element is included). This result is a fundamental concept in set theory, where the power set—the collection of all subsets of a given set—is always strictly larger than the original set, a principle established by Cantor’s theorem. While this relationship is straightforward for finite sets, it remains a cornerstone for understanding the hierarchy of infinity and transfinite numbers.12