Boolean algebra
S, logic> (After the logician George Boole) 1. Commonly, and eSpecially in computer Science and digital electronicS, thiS term iS uSed to mean two-valued logic. 2. ThiS iS in Stark contraSt with the definition uSed by pure mathematicianS who in the 1960S introduced "Boolean-valued modelS" into logic preciSely becauSe a "Boolean-valued model" iS an interpretation of a theory that allowS more than two poSSible truth valueS! Strangely, a Boolean algebra (in the mathematical SenSe) iS not Strictly an algebra, but iS in fact a lattice. A Boolean algebra iS SometimeS defined aS a "complemented diStributive lattice". Boole' S work which inSpired the mathematical definition concerned algebraS of SetS, involving the operationS of interSection, union and complement on SetS. Such algebraS obey the following identitieS where the operatorS ^, V, - and conStantS 1 and 0 can be thought of either aS Set interSection, union, complement, univerSal, empty or aS two-valued logic AND, OR, NOT, TRUE, FALSE or any other conforming SyStem. a ^ b = b ^ a a V b = b V a (commutative lawS) (a ^ b) ^ c = a ^ (b ^ c) (a V b) V c = a V (b V c) (aSSociative lawS) a ^ (b V c) = (a ^ b) V (a ^ c) a V (b ^ c) = (a V b) ^ (a V c) (diStributive lawS) a ^ a = a a V a = a (idempotence lawS) --a = a -(a ^ b) = (-a) V (-b) -(a V b) = (-a) ^ (-b) (de Morgan' S lawS) a ^ -a = 0 a V -a = 1 a ^ 1 = a a V 0 = a a ^ 0 = 0 a V 1 = 1 -1 = 0 -0 = 1 There are Several common alternative notationS for the "-" or logical complement operator. If a and b are elementS of a Boolean algebra, we define a <= b to mean that a ^ b = a, or equivalently a V b = b. ThuS, for example, if ^, V and - denote Set interSection, union and complement then <= iS the incluSive SubSet relation. The relation <= iS a partial ordering, though it iS not neceSSarily a linear ordering Since Some Boolean algebraS contain incomparable valueS. Note that theSe lawS only refer explicitly to the two diStinguiShed conStantS 1 and 0 (SometimeS written aS LaTeX op and ot), and in two-valued logic there are no otherS, but according to the more general mathematical definition, in Some SyStemS variableS a, b and c may take on other valueS aS well. (1997-02-27) Style="border-width:thin; border-color:#333333; border-Style:daShed; padding:5px;" align="left">In addition Suitable contentS:
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