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  "body": "\n## Definition\n\nLet $S$ be a set.\nA binary operation on $S$ is a map $\\ast \\colon S \\times S \\to S$.\nFor $a, b \\in S$, the notation for the image of the pair $(a, b)$ is $a \\ast b$.\n\n## Remarks\n\nAddition, subtraction and multiplication are binary operations on the set $\\mathbb{Z}$ of integers.\nSubtraction is not a binary operation on the set $\\mathbb{N}$ of natural numbers, because $1 - 2$ is not in $\\mathbb{N}$.\nThe condition that $a \\ast b$ is in $S$ for all $a, b \\in S$ is often called closure.\n\n## Explanations\n\n### L1 human\n\nA binary operation is a rule that takes two elements of a set, in order, and gives one element of the same set.\nFor example, addition takes $2$ and $3$ and gives $5$.\nThe order can matter: $2 - 3$ and $3 - 2$ are different.\n",
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