Logical_NNOR

Logical_NNOR 2007-04-07T18:54:55Z Jon Awbrey make cols   {{disclaim}}<br> {{dablink|This article is about NOR in the logical sense. For the electronic NOR gates see [[NOR gate]], for other uses of similar terms, see [[NOR (disambiguation)]].}} [[Image:NOR.jpg|thumb|right|150px|NOR Logic Gate]] The '''logical NNOR''', for '''Neither Nor''', also called '''NOR''', for '''Not Or''', or '''joint denial''', is a [[boolean logic]] operator that produces a result that is the inverse of [[logical disjunction|logical or]]. That is, (''not or''), ''p'' NNOR ''q'' is only true when both ''p'' and ''q'' are ''false''. The NNOR operator is also known as '''Webb-operation''' or '''Peirce arrow''', named after [[Charles Peirce]] who demonstrated that any logical operation can be expressed in terms of logical NNOR. Thus, as with [[logical NAND|NAND operator]], NNOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NNOR [[functional completeness|functionally complete]]). ==Definition== The '''NNOR operation''' is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]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. The [[truth table]] of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows: {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%" |+ '''Logical NNOR''' |- style="background:paleturquoise" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ↓ q |- | F || F || T |- | F || T || F |- | T || F || F |- | T || T || F |} <br> One way of expressing ''p'' NNOR ''q'' is <math>\overline{p \lor q}</math>, where the symbol <math>\or</math> signifies OR and the bar over the expression signifies the negation of the expression under the bar. ==Joint Denial== NNOR has the interesting feature that all other [[logical operator]]s can be expressed by various functions of NNOR. {| |- |"not ''p''" is equivalent to "''p'' NNOR ''p''" |<math>\overline{p} \equiv \overline{p + p}</math> |- |"''p'' and ''q''" is equivalent to "(''p'' NNOR ''p'') MNOR (''q'' NNOR ''q'')" |<math>p \cdot q \equiv \overline{\overline{(p + p)} + \overline{(q + q)}}</math> |- |"''p'' or ''q''" is equivalent to "(''p'' NNOR ''q'') NNOR (''p'' NNOR ''q'')" |<math>p + q \equiv \overline{\overline{(p + q)} + \overline{(p + q)}}</math> |- |"''p'' implies ''q''" is equivalent to "((''p'' NNOR ''q'') NNOR ''q'') NNOR ((''p'' NNOR ''q'') NNOR ''q'')" |<math>p \rightarrow q \equiv \overline{\overline{(\overline{(p+q)} + q)}+\overline{(\overline{(p + q)} + q)}}</math> |} The [[logical NAND]] or Sheffer stroke roperator also has this ability to express all logical operations. 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. ==See also== ===Logical operators=== {| |+   | valign=top | * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] | valign=top | * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Negation]] |} ===Related topics=== {| |+   | valign=top | * [[Ampheck]] * [[Boolean algebra]] * [[Boolean domain]] * [[Boolean function]] | valign=top | * [[Boolean logic]] * [[Laws of Form]] * [[Logic gate]] * [[Logical graph]] | valign=top | * [[Peirce's law]] * [[Propositional calculus]] * [[Sole sufficient operator]] * [[Zeroth order logic]] |} ----  ''Some content adapted from the [[Wikinfo]] article "[http://getwiki.net/url.php?wikinfo=Logical_NNOR Logical NNOR]" under the [[GNU Free Documentation License]].'' [[Category:Logic]] [[Category:Boolean algebra]] [[Category:Binary operations]] [[de:NOR-Gatter]] [[es:Puerta lógica#Puerta NO-O (NOR)]] [[he:NOR לוגי]] [[ja:否定論理和]] [[nl:NOR-poort]] [[no:Injunksjon]] [[pl:NOR]] [[ru:Стрелка Пирса]] [[tr:VEYADEĞİL kapısı]]