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  "body": "\n## Statement\n\nLet $A$ and $B$ be finite sets.\nIf there is a bijection $f : A \\to B$, then $|A| = |B|$.\nConversely, if $|A| = |B|$, then there is a bijection from $A$ to $B$.\n\n## Remarks\n\nA bijective proof shows that two sets have the same size by a bijection between them. The proof of the [[symmetry of binomial coefficients|0R7JS3DP3W]] is a standard example. It uses the map $S \\mapsto X \\setminus S$ on the subsets of a finite set $X$.\n[[Dyck words|0R9JR0465V]] and [[binary trees|02A8EMC83C]] are both counted by [[Catalan numbers|092DRBFKMG]], and a bijection explains this.\n\n## Explanations\n\n### L1 human\n\nTo show that two collections have the same size, you can pair each element of one with exactly one element of the other.\nFor example, the subsets of $\\{1, 2, 3\\}$ pair with the words of length 3 in the letters 0 and 1. A subset gives the word with 1 in the positions of its elements.\n",
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