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BOOLEAN ALGEBRA
Boolean algebra , or the algebra of logic, was devised by the English mathematician
George Boole (1815-64), and embodies the first successful application of algebraic methods to
logic. Boole seems initially to have conceived of each of the basic symbols of his algebraic
system as standing for the mental operation of selecting just the objects possessing some given
attribute or included in some given class. Later he conceived of these symbols as standing for the
attributes or classes themselves. He also recognized that the algebraic laws he proposed are
essentially those of binary arithmetic, i.e., if the basic symbols are interpreted as taking just the
number values 0 and 1. In each of these interpretations the basic symbols are conceived as being
capable of combination under certain operations: multiplication, corresponding to conjunction of
attributes or intersection of classes, addition, corresponding to (exclusive) disjunction or
(disjoint) union, and subtraction, corresponding to "excepting" or diifference. Boole's ideas as
outlined here have since undergone extensive development, and the resulting mathematical
concept of Boolean algebra now plays a central role in mathematical logic, probability theory
and computer design.
A Boolean algebra is a structure (B,+B, iB ,–B, 0B, 1B), where B is a nonempty set, +B and
i B are binary operations on B, –B is a unary operation on B, and 0B, 1B are distinct elements of B
satisfying the following laws: for all x, y, z in B,
associativity x + (y + z) = (x + y) + z x i (y i z) = (x i y) i z
commutativity x + y = y + x x i y = y i x
absorption x + (x i y) = x x i (x + y) = x
distributivity x + (y i z) = (x + y) i (x + z) x i (y + z) = (x i y) + (x i z)
complementation x + (–x) = 1 x i (–x) = 0. | 