{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "05GK32H72H"
    },
    {
      "field": "uses",
      "target": "0839XNQXDK"
    }
  ],
  "body": "\n## Definition\n\nLet $I$ be a set.\nA family of sets indexed by $I$ assigns to each $i \\in I$ a [[set|0SJ3RX9TXA]] $A_i$. It is written $(A_i)_{i \\in I}$.\nThe set $I$ is the index set.\n\n## Remarks\n\nFormally, a family indexed by $I$ is a [[function|0XHDTE5YMZ]] with [[domain|093CM69AGS]] $I$.\nThe same set can appear for two different indices: $A_i = A_j$ with $i \\neq j$ is allowed.\nExample: $A_n = \\{0, 1, \\dots, n\\}$ for $n \\in \\mathbb{N}$ is a family indexed by $\\mathbb{N}$.\nSome authors use \"family of sets\" for any set of sets, without an index.\n\n## Explanations\n\n### L1 human\n\nA family is a labelled list of sets, where the labels can be any set.\nWith labels $1, 2, 3$, you get three sets $A_1, A_2, A_3$.\nWith labels in $\\mathbb{N}$, you get an infinite sequence of sets.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0C21Q694AG",
    "kind": "definition",
    "msc": [
      "03E20"
    ],
    "name": "Family of sets",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Family_of_sets"
      }
    ],
    "tag": "05Q6G4EH2G",
    "uses": [
      "0SJ3RX9TXA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/05Q6G4EH2G/",
  "links": [
    {
      "field": "defines",
      "target": "0C21Q694AG"
    },
    {
      "field": "uses",
      "target": "0SJ3RX9TXA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05Q6G4EH2G.md",
  "msc": [
    "03E20"
  ],
  "name": "Family of sets",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05Q/05Q6G4EH2G.md",
  "status": null,
  "tag": "05Q6G4EH2G",
  "uri": "m2://05Q6G4EH2G",
  "url": "https://math2.tamnd.com/wiki/family-of-sets-definition/",
  "writing": {
    "tag": "05Q6G4EH2G",
    "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
      }
    ]
  }
}