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Lexikon

Boolean algebra

(After the logician ref="module.php?name=Lexikon&file=search&eid=1&query=George Boole">George Boole) 1. Commonly, and especially in computer science and digital electronics, this term is used to mean ref="module.php?name=Lexikon&file=search&eid=1&query=two-valued logic">two-valued logic. 2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced "Boolean-valued ref="module.php?name=Lexikon&file=search&eid=1&query=models">models" into logic precisely because a "Boolean-valued model" is an interpretation of a ref="module.php?name=Lexikon&file=search&eid=1&query=theory">theory that allows more than two possible truth values! Strangely, a Boolean algebra (in the mathematical sense) is not strictly an ref="module.php?name=Lexikon&file=search&eid=1&query=algebra">algebra, but is in fact a ref="module.php?name=Lexikon&file=search&eid=1&query=lattice">lattice. A Boolean algebra is sometimes defined as a "complemented ref="module.php?name=Lexikon&file=search&eid=1&query=distributive lattice">distributive lattice". Boole' s work which inspired the mathematical definition concerned ref="module.php?name=Lexikon&file=search&eid=1&query=algebras">algebras of ref="module.php?name=Lexikon&file=search&eid=1&query=set">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 ref="module.php?name=Lexikon&file=search&eid=1&query=logical complement">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 ref="module.php?name=Lexikon&file=search&eid=1&query=partial ordering">partial ordering, though it is not necessarily a ref="module.php?name=Lexikon&file=search&eid=1&query=linear ordering">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 ref="module.php?name=Lexikon&file=search&eid=1&query=LaTeX">LaTeX op and ot), and in ref="module.php?name=Lexikon&file=search&eid=1&query=two-valued logic">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)

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