{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01MGYYBB9R"
    },
    {
      "field": "uses",
      "target": "01SWBKDAWA"
    },
    {
      "field": "uses",
      "target": "0230ECX1Q5"
    },
    {
      "field": "uses",
      "target": "0CYRECG2YQ"
    },
    {
      "field": "uses",
      "target": "0JCSQNNP2M"
    },
    {
      "field": "uses",
      "target": "0N9SA25Z3V"
    },
    {
      "field": "uses",
      "target": "0PEJH7AC74"
    },
    {
      "field": "uses",
      "target": "0XEGNW851S"
    },
    {
      "field": "uses",
      "target": "0XHDTE5YMZ"
    }
  ],
  "body": "\n## Definition\n\nLet $A$ and $B$ be [[sets|0SJ3RX9TXA]].\nA binary relation from $A$ to $B$ is a [[subset|0FE8Q5S42Y]] $R$ of the [[Cartesian product|0C81YBVTWH]] $A \\times B$.\nFor $a \\in A$ and $b \\in B$, write $a \\mathrel{R} b$ when $(a, b) \\in R$.\nA binary relation on $A$ is a binary relation from $A$ to $A$.\n\n## Remarks\n\nExamples on $\\mathbb{Z}$ are $\\le$, $=$ and divisibility $\\mid$.\nA relation on a finite set can be drawn as a directed graph, with an arrow from $a$ to $b$ when $a \\mathrel{R} b$.\nA [[function|0XHDTE5YMZ]] is a special kind of binary relation.\n\n## Explanations\n\n### L1 human\n\nA relation is a rule that says which pairs belong together.\nFor the relation \"divides\" on $\\{1, 2, 3, 4\\}$, the pair $(2, 4)$ is in the relation and the pair $(3, 4)$ is not.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "05AVARPXA2",
    "kind": "definition",
    "msc": [
      "03E20"
    ],
    "name": "Binary relation",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Binary_relation"
      }
    ],
    "tag": "08EJC6Q25H",
    "uses": [
      "0C81YBVTWH",
      "0FE8Q5S42Y"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/08EJC6Q25H/",
  "links": [
    {
      "field": "defines",
      "target": "05AVARPXA2"
    },
    {
      "field": "uses",
      "target": "0C81YBVTWH"
    },
    {
      "field": "uses",
      "target": "0FE8Q5S42Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/08EJC6Q25H.md",
  "msc": [
    "03E20"
  ],
  "name": "Binary relation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/08E/08EJC6Q25H.md",
  "status": null,
  "tag": "08EJC6Q25H",
  "uri": "m2://08EJC6Q25H",
  "url": "https://math2.tamnd.com/wiki/binary-relation-definition/",
  "writing": {
    "tag": "08EJC6Q25H",
    "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
      }
    ]
  }
}