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  "body": "\n## Definition\n\nLet $A$ be a [[set|0SJ3RX9TXA]].\nThe power set of $A$ is the set\n$$\\mathcal{P}(A) = \\{S \\mid S \\subseteq A\\}$$\nof all [[subsets|0FE8Q5S42Y]] of $A$.\n\n## Remarks\n\nOther notations are $2^A$ and $\\wp(A)$.\nExample: $\\mathcal{P}(\\{1, 2\\}) = \\{\\varnothing, \\{1\\}, \\{2\\}, \\{1, 2\\}\\}$.\nA finite set with $n$ elements has a power set with $2^n$ elements. See [[cardinality of the power set|0GPFKS96T7]].\nBy [[Cantor's theorem|0WTEMGYV2C]], no set is in bijection with its power set.\n\n## Explanations\n\n### L1 human\n\nTo list the power set, choose for each element whether it is in or out of the subset.\nEach choice gives one subset, so a set with $3$ elements has $2^3 = 8$ subsets.\nThe empty set and the set itself are always among them.\n",
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