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  "body": "\n## Summary\n\nThe distributive property links two [[binary operations|0M418FE7BB]] on one set: one of them can be applied term by term over the other.\nThe basic case is $a \\cdot (b + c) = a \\cdot b + a \\cdot c$ for numbers, the [[distributive law|0KBEMAJAYA]].\nDistributivity of multiplication over addition is an axiom of rings and fields, and distributivity of meet over join defines [[distributive lattices|0HM0HQCK6N]].\n\n## Explanations\n\n### L1 human\n\nTo compute $3 \\cdot 12$, one can split $12$ as $10 + 2$ and compute $3 \\cdot 10 + 3 \\cdot 2 = 36$.\nThat this always works is the distributive property of multiplication over addition.\nFor sets, intersection distributes over union: $A \\cap (B \\cup C) = (A \\cap B) \\cup (A \\cap C)$.\nAddition does not distribute over multiplication: $1 + 2 \\cdot 3 = 7$ but $(1 + 2) \\cdot (1 + 3) = 12$.\n",
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