Logical NOR

Summary

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 and , where the symbol signifies logical negation, signifies OR, and signifies AND. In grammar, nor is a coordinating conjunction.

Logical NOR
NOR
Venn diagram of Logical NOR
Definition
Truth table
Logic gateNOR ANSI.svg
Normal forms
Disjunctive
Conjunctive
Zhegalkin polynomial
Post's lattices
0-preservingno
1-preservingno
Monotoneno
Affineno

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 Peirce, in unpublished manuscripts, first considered it as a logical operator, and showed that it can express logical NOT, AND, and OR.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,[1] Sheffer,[2] and Nicod[3] were the first to discuss it in print. Quine introduced the symbol for it.[4] As with its dual, the NAND operator (a.k.a. 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). Other terms for the NOR operator include Quine's dagger, the ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways") used by Peirce,[5] and neither-nor. Other ways of notating include, P NOR Q, and "Xpq" (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.[6]

DefinitionEdit

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 tableEdit

The truth table of   (also written as P NOR Q) is as follows:

     
True True False
True False False
False True False
False False True

Logical EquivalencesEdit

The logical NOR   is the negation of the disjunction:

             
               

PropertiesEdit

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 NOREdit

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 alsoEdit

ReferencesEdit

  1. ^ Stamm, Edward (1911). "Beitrag zur Algebra der Logik". Monatshefte für Mathematik und Physik. 22 (1): 137–149. doi:10.1007/BF01742795.
  2. ^ Sheffer, Henry Maurice (1913). "A set of five independent postulates for Boolean algebras, with application to logical constants". Transactions of the American Mathematical Society. 14 (4): 481–488. doi:10.1090/S0002-9947-1913-1500960-1.
  3. ^ Nicod, Jean G. P. (1917). "A reduction in the number of the primitive propositions of logic". Proceedings of the Cambridge Philosophical Society, Mathematical and Physical Sciences. 19: 32–41.
  4. ^ Quine, Willard Van Orman (1940). Mathematical logic (1 ed.). New York: W. W. Norton & Co.
  5. ^ C.S. Peirce, CP 4.264
  6. ^ Hall, Eldon C. (1996), Journey to the Moon: The History of the Apollo Guidance Computer, Reston, Virginia, USA: AIAA, p. 196, ISBN 1-56347-185-X

External linksEdit

  •   Media related to Logical NOR at Wikimedia Commons