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  "body": "\n## Proof\n\nWe build a bijection from the [[power set|0BC7KX51Z8]] to a set of functions.\n\n1. Let $F = \\set{0, 1}^X$ be the set of all functions from $X$ to $\\set{0, 1}$.\n2. Define $\\Phi : \\mathcal{P}(X) \\to F$ by $\\Phi(S) = 1_S$, the [[indicator function|0VSB0XF4G7]] of $S$. So $1_S(x) = 1$ for $x \\in S$ and $1_S(x) = 0$ for $x \\in X \\setminus S$.\n3. Define $\\Psi : F \\to \\mathcal{P}(X)$ by $\\Psi(\\varphi) = \\set{x \\in X : \\varphi(x) = 1}$.\n4. For $S \\subseteq X$, the set $\\Psi(\\Phi(S))$ is the set of $x$ with $1_S(x) = 1$. This is $S$.\n5. For $\\varphi \\in F$, let $T = \\Psi(\\varphi)$. Then $1_T(x) = 1 = \\varphi(x)$ for $x \\in T$.\n6. For $x \\notin T$ we have $\\varphi(x) \\ne 1$. Since $\\varphi$ takes values in $\\set{0, 1}$, this gives $\\varphi(x) = 0 = 1_T(x)$. So $\\Phi(\\Psi(\\varphi)) = \\varphi$.\n7. Thus $\\Phi$ has a two-sided inverse, so it is a bijection by the [[inverse criterion|01VG2A0BHW]].\n8. By the [[number of functions between finite sets|08QQH9F7G0]], $F$ is finite and $|F| = 2^n$.\n9. A numbering of $F$ composed with $\\Psi$ numbers $\\mathcal{P}(X)$. So $\\mathcal{P}(X)$ is finite, and the [[bijection principle|0GV2573FP4]] gives $|\\mathcal{P}(X)| = |F| = 2^n$.\n\n## Idea\n\nA subset is determined by answering \"in or out?\" for each element.\nThe answers form a function from $X$ to $\\set{0, 1}$, and every such function arises from exactly one subset.\nSo subsets are counted by functions, and there are $2^n$ of those.\n",
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