Logical NOR
In boolean logic, logical nor 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 of "p" and "q" are "false". It is logically equivalent to formula_1 and formula_2, where the symbol formula_3 signifies logical negation, formula_4 signifies OR, and formula_5 signifies AND. In grammar, nor is a coordinating conjunction. The NOR operator is also known as Peirce's arrow. Peirce, in unpublished manuscripts, first considered  it as a logical operator, and showed that it can express logical NOT, AND, and OR. Stamm, Sheffer, and Nicod were the first to discuss it in print. Quine introduced the symbol formula_6 for it. As with its dual, the NAND operator (a.k.a. the Sheffer stroke—symbolized as either formula_7, formula_8 or formula_9), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete). Other terms for the NOR operator include Quine's dagger, the (from Ancient Greek , , "cutting both ways") used by Peirce, and neither-nor. Other ways of  notating formula_10 include, P NOR Q, and "X"pq"" (in Bocheński notation). 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. Definition. The NOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of "true" if and only if both operands are false. In other words, it produces a value of "false" if and only if at least one operand is true. Truth table. The truth table of formula_10 (also written as P NOR  Q) is as follows: Logical Equivalences. The logical NOR formula_6 is the negation of the disjunction: Properties. Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, linear, monotonic, self-dual) required to be absent from at least one member of a set of functionally complete operators. Thus, the set containing only NOR suffices as a complete set. Other Boolean Operations in terms of the Logical NOR. NOR has the interesting feature that all other logical operators can be expressed by interlaced NOR operations. The logical NAND operator also has this ability. Expressed in terms of  NOR formula_6, the usual operators of propositional logic are: