Sheffer stroke

Sheffer stroke

NAND
Definition	${\displaystyle {\overline {x\cdot y}}}$
Truth table	${\displaystyle (0111)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle {\overline {x}}+{\overline {y}}}$
Conjunctive	${\displaystyle {\overline {x}}+{\overline {y}}}$
Zhegalkin polynomial	${\displaystyle 1\oplus xy}$
Post's lattices
0-preserving	no
1-preserving	no
Monotone	no
Affine	no
v t e

Logical connectives
AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ 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
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
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In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, or alternative denial^[1] (since it says in effect that at least one of its operands is false), or NAND ("not and").^[1] In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as ${\displaystyle \mid }$ or as ${\displaystyle \uparrow }$ or as ${\displaystyle {\overline {\wedge }}}$ or as ${\displaystyle Dpq}$ in Polish notation by Łukasiewicz (but not as ||, often used to represent disjunction).

Its dual is the NOR operator (also known as the Peirce arrow, Quine dagger or Webb operator). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical formal system (making NAND functionally complete). This property makes the NAND gate crucial to modern digital electronics, including its use in computer processor design.