Functional completeness

Logical connectives
NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ └ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

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.