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  "body": "\n## Definition\n\nLet $A$ be a [[finite set|070R982G4B]] with at least two elements.\nA set $F$ of finitary [[operations|EJ8YJVM9BW]] on $A$ is *functionally complete* if the [[clone|PTKR703G6A]] generated by $F$ contains every operation $A^n \\to A$ for every $n \\ge 1$.\n\n## Remarks\n\nExample: $F = \\set{\\wedge, \\neg}$ on $\\set{0, 1}$ is functionally complete.\nCheck: every Boolean function of $n \\ge 1$ variables has a disjunctive normal form, with the constant $0$ written as $x_1 \\wedge \\neg x_1$.\nThe disjunctions are removed with $x \\vee y = \\neg(\\neg x \\wedge \\neg y)$.\nNon-example: $F = \\set{\\wedge, \\vee}$ on $\\set{0, 1}$ is not functionally complete.\nEvery operation built from $\\wedge$ and $\\vee$ maps $(1, \\dots, 1)$ to $1$, so $\\neg$ is not in the generated clone.\nSome texts, especially in logic, allow constants to be used for free; this changes which sets count as complete.\n",
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