Logical NOR

Logical NOR
In Boolean logic, logical NOR or 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 of p and q are false. It is logically equivalent to and , where the symbol signifies logical negation, signifies OR, and signifies AND.
Non-disjunction is usually denoted as or or (prefix) or .
As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either , or ), 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.
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 is as follows:
Logical equivalences
The logical NOR is the negation of the disjunction:
Alternative notations and names
Peirce is the first to show the functional completeness of non-disjunction while he doesn't publish his result. Peirce used for non-conjunction and for non-disjunction (in fact, what Peirce himself used is and he didn't introduce while Peirce's editors made such disambiguated use). Peirce called as (from Ancient Greek "cutting both ways").
In 1911, was the first to publish a description of both non-conjunction (using , the Stamm hook), and non-disjunction (using , the Stamm star), and showed their functional completeness.</ref> Note that most uses in logical notation of use this for negation.
In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used for non-conjunction, and for non-disjunction.
In 1935, Webb described non-disjunction for -valued logic, and use for the operator. So some people call it Webb operator, Webb operation or Webb function.
In 1940, Quine also described non-disjunction and use for the operator. So some people call the operator Peirce arrow or Quine dagger.
In 1944, Church also described non-disjunction and use for the operator.
In 1954, Bocheński used in for non-disjunction in Polish notation.
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 , the usual operators of propositional logic are:
See also
Bitwise NOR
Boolean algebra
Boolean domain
Boolean function
Functional completeness
NOR gate
Propositional logic
Sole sufficient operator
Sheffer stroke as symbol for the logical NAND
References
External links
NOR
Charles Sanders Peirce
de:Peirce-Funktion
it:Algebra di Boole#NOR