In logic, a functionally complete set of logical connectives or Boolean operators is one that can be used to express all possible truth tables by combining members of the set into a Boolean expression.[1][2] A well-known complete set of connectives is { AND, NOT }. Each of the singleton sets { NAND } and { NOR } is functionally complete. However, the set { AND, OR } is incomplete, due to its inability to express NOT.
A gate (or set of gates) that is functionally complete can also be called a universal gate (or a universal set of gates).
In a context of propositional logic, functionally complete sets of connectives are also called (expressively) adequate.[3]
From the point of view of digital electronics, functional completeness means that every possible logic gate can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates can be assembled from either only binary NAND gates, or only binary NOR gates.
^Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3. ("Complete set of logical connectives").
^Nolt, John; Rohatyn, Dennis; Varzi, Achille (1998), Schaum's outline of theory and problems of logic (2nd ed.), New York: McGraw–Hill, ISBN 978-0-07-046649-4. ("[F]unctional completeness of [a] set of logical operators").
^Smith, Peter (2003), An introduction to formal logic, Cambridge University Press, ISBN 978-0-521-00804-4. (Defines "expressively adequate", shortened to "adequate set of connectives" in a section heading.)
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