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In boolean logicBoolean logic

Boolean logic is a complete system for logical operations....
, logical nor or joint denial is a truth-functional operator which produces a result that is the inverse of logical orLogical disjunction

In logic and mathematics, logical disjunction is a logical operator that results in true just whenever some of its op...
. 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. In grammarGrammar

Grammar is the study of rules governing the use of language....
, nor is a coordinating conjunctionGrammatical conjunction

In grammar, a conjunction is a part of speech that connects two words, phrases, or clauses together....
.

The NOR operator is also known as Webb-operation or Peirce arrow, named after Charles PeirceCharles Peirce

Charles Sanders Peirce, was an American polymath, born in Cambridge, Massachusetts....
 who demonstrated that any logical operation can be expressed in terms of logical NOR. Thus, as with NAND operator, NOR can be used by itself, without any other logical operator, to constitute a logical formal systemFacts About Formal system

In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or tr...
 (making NOR functionally completeFunctional completeness

In logic, a set of logical connectives is functionally complete if all other possible connectives can be defined in terms of...
). It is also known as QuineWillard Van Orman Quine

Willard Van Orman Quine , usually cited as W.V....
's dagger.

Definition


The NOR operation is a logical operation on two logical valueLogical value

In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a propositi...
s, typically the values of two propositionProposition

In common philosophical language, a proposition is the content of an assertion, that is, it is true-or-false and defined by ...
s, 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 tableTruth table

A truth table is a mathematical table used in logic specifically in connection with boolean algebra, boolean functions, and...
 of p NOR q (also written as p ⊥ q or p ↓ q) is as follows:

Venn diagram

The Venn DiagramVenn diagram

Venn diagrams are illustrations used in the branch of mathematics known as set theory....
 of "Neither A nor B" (in the red area the statement is true)



One way of expressing p NOR q is , where the symbol signifies OR and the bar over the expression signifies the negation of the expression under the bar. In essence, simply . Another way of expressing p NOR q is .

Properties


Nor does not possess any of the five qualities, each of which is required to be absent from at least one member of a set of functionally completeFunctional completeness

In logic, a set of logical connectives is functionally complete if all other possible connectives can be defined in terms of...
 operators. (truth-preserving, false-preserving, linearLinear

The word linear comes from the Latin word linearis, which means created by lines....
, monotonic, self-dual). This is why it suffices alone for a complete set.

Joint denial

NOR has the interesting feature that all other logical operators can be expressed by various functions of NOR.
"not p" is equivalent to "p NOR p" 
"p and q" is equivalent to "(p NOR p) NOR (q NOR q)" 
"p or q" is equivalent to "(p NOR q) NOR (p NOR q)" 
"p implies q" is equivalent to "((p NOR q) NOR q) NOR ((p NOR q) NOR q)" 


The logical NANDSheffer stroke

The Sheffer stroke, written "|" or "↑", denotes a logical operation that is equivalent to the negation of the conjuncti...
 operator also has this ability to express all logical operations.

The computerComputer

A computer is a machine for manipulating data according to a list of instructions known as a program....
 used in the spacecraft that first carried humans to the moonMoon

The Moon is Earth's only natural satellite....
, the Apollo Guidance ComputerApollo Guidance Computer

The Apollo Guidance Computer was the first recognizably modern embedded system, used in real-time by astronaut pilots to col...
, was constructed entirely using NOR gates with three inputs.

See also


  • Bitwise NORBitwise operation

    In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their in...
  • Boolean algebra
  • Boolean domainBoolean domain

    A boolean domain B is a generic 2-element set, say, B = , whose elements are interpreted as logical values, typi...
  • Boolean functionBoolean function

    In mathematics, a finitary boolean function is a function of the form f : Bk → B, where B =&...
  • Functional completenessFunctional completeness

    In logic, a set of logical connectives is functionally complete if all other possible connectives can be defined in terms of...
  • NOR gateNOR gate

    The NOR gate is a digital logic gate that implements logical NOR - it behaves according to the truth table to the right....
  • Propositional logic
  • Sole sufficient operatorSole sufficient operator

    A sole sufficient operator or a sole sufficient connective is an operator that is sufficient by itself to generate all...