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The stroke is named after Henry Maurice Sheffer, who in 1913 published a paper in the Transactions of the American Mathematical Society [10] providing an axiomatization of Boolean algebras using the stroke, and proved its equivalence to a standard formulation thereof by Huntington employing the familiar operators of propositional logic (AND, OR, NOT).
It is one of 25 candidate axioms for this property identified by Stephen Wolfram, by enumerating the Sheffer identities of length less or equal to 15 elements (excluding mirror images) that have no noncommutative models with four or fewer variables, and was first proven equivalent by William McCune, Branden Fitelson, and Larry Wos.
The vertical bar is a punctuation mark used in computing and mathematics to denote absolute value, logical OR, and pipe commands.
Because Sheffer's stroke (also known as NAND operator) is functionally complete, it can be used to create an entire formulation of propositional calculus. NAND formulations use a rule of inference called Nicod's modus ponens:
Sheffer stroke: Is the contemporary logical NAND (NOT-AND), i.e., "incompatibility", meaning: "Given two propositions p and q, then ' p | q ' means "proposition p is incompatible with proposition q", i.e., if both propositions p and q evaluate as true, then and only then p | q evaluates as false." After section 8 the Sheffer stroke sees no usage.
The composition of two relations. Also used for the Sheffer stroke in *8 appendix A of the second edition. *34.01 R 2, R 3: R n is the composition of R with itself n times. *34.02, *34.03 is the relation R with its domain restricted to α *35.01
Sheffer stroke; Sole sufficient operator; Symmetric Boolean function; Symmetric difference; Zhegalkin polynomial; Examples of Boolean algebras. Boolean domain;
NAND or Sheffer stroke - true when it is not the case that all inputs are true ("not both") NOR or logical nor - true when none of the inputs are true ("neither") XNOR or logical equality - true when both inputs are the same ("equal") An example of a more complicated function is the majority function (of an odd number of inputs).