Logical NOR

In Boolean logic, logical NOR, non-disjunction, or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both p and q are false. It is logically equivalent to ${\displaystyle \neg (p\lor q)}$ and ${\displaystyle \neg p\land \neg q}$, where the symbol ${\displaystyle \neg }$ signifies logical negation, ${\displaystyle \lor }$ signifies OR, and ${\displaystyle \land }$ signifies AND.

Logical NOR

NOR
Definition	${\displaystyle {\overline {x+y}}}$
Truth table	${\displaystyle (0001)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle {\overline {x}}\cdot {\overline {y}}}$
Conjunctive	${\displaystyle {\overline {x}}\cdot {\overline {y}}}$
Zhegalkin polynomial	${\displaystyle 1\oplus x\oplus y\oplus xy}$
Post's lattices
0-preserving	no
1-preserving	no
Monotone	no
Affine	no

Non-disjunction is usually denoted as ${\displaystyle \downarrow }$ or ${\displaystyle {\overline {\vee }}}$ or ${\displaystyle X}$ (prefix) or ${\displaystyle \operatorname {NOR} }$.

As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either ${\displaystyle \uparrow }$, ${\displaystyle \mid }$ or ${\displaystyle /}$), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete).

The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.