{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]] and let $S$ and $T$ be [[subsets|0FE8Q5S42Y]] of $A$.\nThen the [[image|0Q3F20XB0A]] of $S \\cap T$ is a subset of the intersection of the images:\n$$f(S \\cap T) \\subseteq f(S) \\cap f(T).$$\n\n## Remarks\n\nThe inclusion can be strict. For the constant function $f \\colon \\{1, 2\\} \\to \\{0\\}$, $S = \\{1\\}$ and $T = \\{2\\}$, the left side is $\\varnothing$ and the right side is $\\{0\\}$.\nEquality holds for all $S$ and $T$ if and only if $f$ is [[injective|0PVG1CRAJN]].\nFor unions, equality always holds: $f(S \\cup T) = f(S) \\cup f(T)$.\n\n## Explanations\n\n### L1 human\n\nIf $b = f(x)$ with $x$ in both $S$ and $T$, then $b$ is in both images.\nBut $b$ can be in both images for another reason: $b = f(s) = f(t)$ with $s \\in S$, $t \\in T$ and $s \\neq t$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "03E20"
    ],
    "name": "Image of an intersection",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Image_(mathematics)"
      }
    ],
    "tag": "0RX7DDBYCK",
    "uses": [
      "0Q3F20XB0A",
      "0HBRW4E6G1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0RX7DDBYCK/",
  "links": [
    {
      "field": "uses",
      "target": "0Q3F20XB0A"
    },
    {
      "field": "uses",
      "target": "0HBRW4E6G1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0RX7DDBYCK.md",
  "msc": [
    "03E20"
  ],
  "name": "Image of an intersection",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0RX/0RX7DDBYCK.md",
  "status": null,
  "tag": "0RX7DDBYCK",
  "uri": "m2://0RX7DDBYCK",
  "url": "https://math2.tamnd.com/wiki/image-of-an-intersection/",
  "writing": {
    "tag": "0RX7DDBYCK",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}