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a system of symbolic logic devised by George Boole; used in computers

a system of symbolic logic devised by George Boole; used in computers

Studies binary variables that may only have two values which are true (1) and false (0), and also studies the mathematical operations performed on them. A set of rules known as Boolean logic are provided and are indispensable in digital switching circuit and computer circuit design. Operations are done via algebraic operators known as operators. Appropriately named after George Boole. (181565).

(After the logician George Boole)1. Commonly, and especially in computer science and digitalelectronics, this term is used to mean two-valued logic.2. This is in stark contrast with the definition used by puremathematicians who in the 1960s introduced "Boolean-valuedmodels" into logic precisely because a "Boolean-valuedmodel" is an interpretation of a theory that allows morethan two possible truth values!Strangely, a Boolean algebra (in the mathematical sense) isnot strictly an algebra, but is in fact a lattice. ABoolean algebra is sometimes defined as a "complementeddistributive lattice".Boole's work which inspired the mathematical definitionconcerned algebras of sets, involving the operations ofintersection, union and complement on sets. Such algebrasobey the following identities where the operators ^, V, - andconstants 1 and 0 can be thought of either as setintersection, union, complement, universal, empty; or astwo-valued logic AND, OR, NOT, TRUE, FALSE; or any otherconforming 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 = 1There are several common alternative notations for the "-" orlogical complement operator.If a and b are elements of a Boolean algebra, we define a <= bto mean that a ^ b = a, or equivalently a V b = b. Thus, forexample, if ^, V and - denote set intersection, union andcomplement then <= is the inclusive subset relation. Therelation <= is a partial ordering, though it is notnecessarily a linear ordering since some Boolean algebrascontain incomparable values.Note that these laws only refer explicitly to the twodistinguished constants 1 and 0 (sometimes written as LaTeX\top and \bot), and in two-valued logic there are no others,but according to the more general mathematical definition, insome systems variables a, b and c may take on other values aswell.