{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nThe symmetric difference of $A$ and $B$ is the set\n$$A \\mathbin{\\triangle} B = (A \\setminus B) \\cup (B \\setminus A).$$\nAn object $x$ is an element of $A \\mathbin{\\triangle} B$ when $x$ is an element of exactly one of $A$ and $B$.\n\n## Remarks\n\nAnother notation is $A \\ominus B$.\nExample: $\\{1, 2, 3\\} \\mathbin{\\triangle} \\{2, 3, 4\\} = \\{1, 4\\}$.\nAlso $A \\mathbin{\\triangle} B = (A \\cup B) \\setminus (A \\cap B)$.\nThe [[power set|0BC7KX51Z8]] of a set is a [[group|G5S1M9]] under symmetric difference, with identity $\\varnothing$.\n\n## Explanations\n\n### L1 human\n\nThe symmetric difference is the set version of \"exclusive or\".\nIt keeps what is in one set or the other, but not what is in both.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0ME21QN125",
    "kind": "definition",
    "msc": [
      "03E20"
    ],
    "name": "Symmetric difference",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Symmetric_difference"
      }
    ],
    "tag": "0Z49NTR1DZ",
    "uses": [
      "0DT0TCV75N",
      "07KVB4JSC7"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0Z49NTR1DZ/",
  "links": [
    {
      "field": "defines",
      "target": "0ME21QN125"
    },
    {
      "field": "uses",
      "target": "0DT0TCV75N"
    },
    {
      "field": "uses",
      "target": "07KVB4JSC7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0Z49NTR1DZ.md",
  "msc": [
    "03E20"
  ],
  "name": "Symmetric difference",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0Z4/0Z49NTR1DZ.md",
  "status": null,
  "tag": "0Z49NTR1DZ",
  "uri": "m2://0Z49NTR1DZ",
  "url": "https://math2.tamnd.com/wiki/symmetric-difference-definition/",
  "writing": {
    "tag": "0Z49NTR1DZ",
    "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
      }
    ]
  }
}