{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nThe complement operation is defined by the following two laws.\n\n$$\\begin{aligned} \u0026\\text{Complementation 1} \u0026 x \\wedge \\neg x \u0026 = 0 \\\\ \u0026\\text{Complementation 2} \u0026 x \\vee \\neg x \u0026 = 1 \\end{aligned}$$\n\nAll properties of negation including the laws below follow from the above two laws alone.\n\nIn both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law (also called involution law)\n\n$$\\begin{aligned} \u0026\\text{Double negation} \u0026 \\neg{(\\neg{x})} \u0026 = x \\end{aligned}$$\n\nBut whereas *ordinary algebra* satisfies the two laws\n\n$$\\begin{aligned} (-x)(-y) \u0026 = xy \\\\ (-x) + (-y) \u0026 = -(x + y) \\end{aligned}$$\n\nBoolean algebra satisfies [[De Morgan's laws|09PKJKA6J0]]:\n\n$$\\begin{aligned} \u0026\\text{De Morgan 1} \u0026 \\neg x \\wedge \\neg y \u0026 = \\neg{(x \\vee y)} \\\\ \u0026\\text{De Morgan 2} \u0026 \\neg x \\vee \\neg y \u0026 = \\neg{(x \\wedge y)} \\end{aligned}$$\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "inventory": [
      "V056JPD4BA"
    ],
    "kind": "statement",
    "name": "Nonmonotone laws",
    "role": "proposition",
    "sources": [
      {
        "locator": "Boolean algebra, section Laws \u003e Nonmonotone laws",
        "mention": "wikipedia://en/54476844#SYNI",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:54476844@r1368725040"
      }
    ],
    "tag": "SX87GXZBGG"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/SX87GXZBGG/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/SX87GXZBGG.md",
  "msc": null,
  "name": "Nonmonotone laws",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SX8/SX87GXZBGG.md",
  "status": null,
  "tag": "SX87GXZBGG",
  "uri": "m2://SX87GXZBGG",
  "url": "https://math2.tamnd.com/wiki/nonmonotone-laws/",
  "writing": {
    "tag": "SX87GXZBGG",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}