{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]].\nThe equivalence relation induced by $f$ is the relation $\\sim_f$ on $A$ with\n$$a \\sim_f a' \\iff f(a) = f(a')$$\nfor all $a, a' \\in A$.\n\n## Remarks\n\nAnother name is the kernel of $f$.\nThe relation $\\sim_f$ is an [[equivalence relation|0SGTMBEWRF]], because equality is one.\nIts classes are the nonempty preimages $f^{-1}(\\{b\\})$ of the points $b \\in B$.\nThe function $f$ is [[injective|0PVG1CRAJN]] exactly when $\\sim_f$ is equality.\n\n## Explanations\n\n### L1 human\n\nTake the function that sends each person to their birthday.\nTwo people are equivalent under it when they have the same birthday.\nThe function forgets everything except its value, and the relation records what it forgets.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0KT8TGXD8K",
    "kind": "definition",
    "msc": [
      "03E20"
    ],
    "name": "Equivalence relation induced by a function",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Kernel_(set_theory)"
      }
    ],
    "tag": "0QE5PFJY7V",
    "uses": [
      "0XHDTE5YMZ",
      "0SGTMBEWRF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0QE5PFJY7V/",
  "links": [
    {
      "field": "defines",
      "target": "0KT8TGXD8K"
    },
    {
      "field": "uses",
      "target": "0XHDTE5YMZ"
    },
    {
      "field": "uses",
      "target": "0SGTMBEWRF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0QE5PFJY7V.md",
  "msc": [
    "03E20"
  ],
  "name": "Equivalence relation induced by a function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0QE/0QE5PFJY7V.md",
  "status": null,
  "tag": "0QE5PFJY7V",
  "uri": "m2://0QE5PFJY7V",
  "url": "https://math2.tamnd.com/wiki/equivalence-relation-induced-by-a-function-definition/",
  "writing": {
    "tag": "0QE5PFJY7V",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}