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  "body": "\n## Summary\n\nA set of operations on a finite set $A$ is functionally complete if every finitary operation on $A$ can be built from it by composition.\nIn other words, the [[clone|PTKR703G6A]] it generates is the clone of all operations.\nFor $A = \\set{0, 1}$ this is the familiar notion for sets of [[logical connectives|QS0XVAN9BG]], such as $\\set{\\wedge, \\neg}$.\nCriteria for functional completeness are given by [[Post's lattice|MA20X7JCET]] in the two-valued case and by maximal clones in general.\n\n## Explanations\n\n### L1 human\n\nA set of logic gates is functionally complete if every truth table can be realised by a circuit built from those gates.\nFor example, AND and NOT suffice: OR can be written as $x \\vee y = \\neg(\\neg x \\wedge \\neg y)$, and every truth table has a disjunctive normal form.\nAND and OR together do not suffice, because they can never turn an input of all ones into a zero.\n",
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