Number of subsets of a finite set

Statement

Let $X$ be a finite set with $|X| = n$. Let $\mathcal{P}(X)$ be the power set of $X$, the set of all subsets of $X$. Then $|\mathcal{P}(X)| = 2^n$.

Remarks

The map that sends a subset $S$ to its indicator function $1_S : X \to \{0, 1\}$ is a bijection. Thus the count follows from the number of functions between finite sets. Grouped by size, the count gives the identity in the node sum of binomial coefficients.

Explanations

L1 human

To build a subset, decide for each element whether it is in or out. That is 2 options per element, so $2 \cdot 2 \cdots 2 = 2^n$ subsets. For $\{a, b\}$ these are $\varnothing$, $\{a\}$, $\{b\}$ and $\{a, b\}$.