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  "body": "\n## Definition\n\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nA function from $A$ to $B$ is a [[binary relation|08EJC6Q25H]] $f \\subseteq A \\times B$ such that, for each $a \\in A$, there is exactly one $b \\in B$ with $(a, b) \\in f$.\nThis $b$ is the value of $f$ at $a$, written $f(a)$.\nThe notation $f \\colon A \\to B$ says that $f$ is a function from $A$ to $B$.\n\n## Remarks\n\nThe set $A$ is the [[domain|093CM69AGS]] and the set $B$ is the [[codomain|0F9JHP8AJN]] of $f$.\nMany authors take the triple $(A, B, f)$ as the function. Then two functions with the same pairs but different codomains are different.\nTwo functions $f, g \\colon A \\to B$ are equal when $f(a) = g(a)$ for each $a \\in A$.\nOther names are map and mapping.\n\n## Explanations\n\n### L1 human\n\nA function is a machine: you put in an element $a$ of $A$, and it gives back one element $f(a)$ of $B$.\nEach input gives exactly one output, but two inputs can give the same output.\nFor example, $f(x) = x^2$ on $\\mathbb{R}$ sends both $2$ and $-2$ to $4$.\n",
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