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  "body": "\n## Statement\n\nLet $X$ be a finite set with $|X| = n$.\nLet $\\mathcal{P}(X)$ be the power set of $X$, the set of all subsets of $X$.\nThen $|\\mathcal{P}(X)| = 2^n$.\n\n## Remarks\n\nThe 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|08QQH9F7G0]].\nGrouped by size, the count gives the identity in the node [[sum of binomial coefficients|0KAMF19NV8]].\n\n## Explanations\n\n### L1 human\n\nTo build a subset, decide for each element whether it is in or out.\nThat is 2 options per element, so $2 \\cdot 2 \\cdots 2 = 2^n$ subsets. For $\\{a, b\\}$ these are $\\varnothing$, $\\{a\\}$, $\\{b\\}$ and $\\{a, b\\}$.\n",
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