Logical NOR

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 A \downarrow B is as follows: Logical equivalences The logical NOR \downarrow 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 \overline{\curlywedge} for non-conjunction and \curlywedge for non-disjunction (in fact, what Peirce himself used is \curlywedge and he didn't introduce \overline{\curlywedge} while Peirce's editors made such disambiguated use). Note that most uses in logical notation of \sim use this for negation. In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used \mid for non-conjunction, and \wedge for non-disjunction. In 1935, Donald L. Webb described non-disjunction for n-valued logic, and use \mid for the operator. So some people call it Webb operator, So some people call the operator Peirce arrow or Quine dagger. In 1944, Church also described non-disjunction and use \overline{\vee} for the operator. In 1954, Bocheński used X in Xpq for non-disjunction in Polish notation. APL uses a glyph that combines a with a . ==Properties==

NOR is commutative but not associative, which means that P \downarrow Q \leftrightarrow Q \downarrow P but (P \downarrow Q) \downarrow R \not\leftrightarrow P \downarrow (Q \downarrow R). Functional completeness The logical NOR, taken by itself, is a functionally complete set of connectives. This can be proved by first showing, with a truth table, that \neg A is truth-functionally equivalent to A \downarrow A. Then, since A \downarrow B is truth-functionally equivalent to \neg (A \lor B), and A \lor B is equivalent to \neg(\neg A \land \neg B), the logical NOR suffices to define the set of connectives \{\land, \lor, \neg\}, which is shown to be truth-functionally complete by the Disjunctive Normal Form Theorem. This may also be seen from the fact that 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. ==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 \downarrow, the usual operators of propositional logic are: ==See also==