{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0V3CFG1KK9"
    }
  ],
  "body": "\n## Definition\n\nLet $S$ be a set and let $\\ast$ be a [[binary operation|0M418FE7BB]] on $S$.\nThe operation $\\ast$ is associative when $(a \\ast b) \\ast c = a \\ast (b \\ast c)$ for all $a, b, c \\in S$.\n\n## Remarks\n\nAddition and multiplication of real numbers are associative.\nSubtraction of integers is not associative: $(1 - 1) - 1 = -1$, but $1 - (1 - 1) = 1$.\nComposition of maps from a set to itself is associative.\nFor an associative operation, each way to put parentheses in $a_1 \\ast a_2 \\ast \\cdots \\ast a_n$ gives the same result.\n\n## Explanations\n\n### L1 human\n\nAssociativity says that you can do the operations in a chain in any grouping.\nFor example, $(2 + 3) + 4$ and $2 + (3 + 4)$ are both $9$.\nThus we can write $2 + 3 + 4$ without parentheses.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0K8F7PVDMX",
    "kind": "definition",
    "msc": [
      "08A02"
    ],
    "name": "Associativity",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Associative_property"
      }
    ],
    "tag": "0NNQ4KFEX7",
    "uses": [
      "0580ZJSA3Q"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0NNQ4KFEX7/",
  "links": [
    {
      "field": "defines",
      "target": "0K8F7PVDMX"
    },
    {
      "field": "uses",
      "target": "0580ZJSA3Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NNQ4KFEX7.md",
  "msc": [
    "08A02"
  ],
  "name": "Associativity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NN/0NNQ4KFEX7.md",
  "status": null,
  "tag": "0NNQ4KFEX7",
  "uri": "m2://0NNQ4KFEX7",
  "url": "https://math2.tamnd.com/wiki/associativity-definition/",
  "writing": {
    "tag": "0NNQ4KFEX7",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}