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  "body": "\n## Statement\n\nLet $A$ be a [[set|0SJ3RX9TXA]].\nThen the [[power set|0BC7KX51Z8]] of $A$ is equinumerous with the set $\\{0, 1\\}^A$ of functions from $A$ to $\\{0, 1\\}$, so\n$$|\\mathcal{P}(A)| = 2^{|A|}.$$\n\n## Remarks\n\nThe bijection sends a subset $S$ to its [[indicator function|0VSB0XF4G7]] $\\mathbf{1}_S$.\nFor a finite set with $n$ elements, the power set has $2^n$ elements.\nFor $A = \\mathbb{N}$, $|\\mathcal{P}(\\mathbb{N})| = 2^{\\aleph_0}$, the [[cardinality of the continuum|0WC28W9DNM]].\n\n## Explanations\n\n### L1 human\n\nA subset is a choice of \"in\" or \"out\" for each element.\nWith $n$ elements, there are $2 \\cdot 2 \\cdots 2 = 2^n$ such choices.\n",
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  "name": "Cardinality of the power set",
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