Boolean algebra


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Boolean algebra

(bo͞o`lēən), an abstract mathematical system primarily used in computer science and in expressing the relationships between setsset,
in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
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 (groups of objects or concepts). The notational system was developed by the English mathematician George BooleBoole, George,
1815–64, English mathematician and logician. He became professor at Queen's College, Cork, in 1849. Boole wrote An Investigation of the Laws of Thought (1854) and works on calculus and differential equations.
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 c.1850 to permit an algebraic manipulation of logical statements. Such manipulation can demonstrate whether or not a statement is true and show how a complicated statement can be rephrased in a simpler, more convenient form without changing its meaning. In his 1881 treatise, Symbolic Logic, the English logician and mathematician John Venn interpreted Boole's work and introduced a new method of diagramming Boole's notation; this was later refined by the English mathematician Charles Dodgson (better known as Lewis CarrollCarroll, Lewis,
pseud. of Charles Lutwidge Dodgson,
1832–98, English writer, mathematician, and amateur photographer, b. near Daresbury, Cheshire (now in Halton).
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—this method is now know as the Venn diagram. When used in set theory, Boolean notation can demonstrate the relationship between groups, indicating what is in each set alone, what is jointly contained in both, and what is contained in neither. Boolean algebra is of significance in the study of information theory, the theory of probability, and the geometry of sets. The expression of electrical networks in Boolean notation has aided the development of switching theory and the design of computers.

Boolean algebra

[′bü·lē·ən ′al·jə·brə]
(mathematics)
An algebraic system with two binary operations and one unary operation important in representing a two-valued logic.

Boolean algebra

a system of symbolic logic devised by George Boole to codify logical operations. It is used in computers

Boolean algebra

(mathematics, 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 \top and \bot), 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.
References in periodicals archive ?
Definition 1: O family of parts P of a set M is a Boolean algebra if and only if:
Boolean algebra is the type of mathematics on which computers and digital equipment are based.
One of his biographers explained Boolean algebra with a simple example: If the symbol x represents the class of all "white objects" and the symbol y the class of all "round objects," the symbol xy represents the class of objects that are simultaneously white and round.
The "propositions" about the classical system are then classified by the Boolean algebra of subsets of the phase space (taking, in the case of a particle, the form of propositions concerning the ranges of values of coordinates and momenta).
Thus F is a complete Boolean algebra with 1 = S and 0 = [Phi].
Properties of Boolean algebra are obeyed, and the exact-match semantics of Boolean retrieval are preserved.
In semantics and logics of programs, Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) [Fischer and Ladher 1979], in which it is mixed with Boolean algebra and modal logic to give a theoretically appealing and practical system for reasoning about computation at the propositional level.
As a benchmark test of his system, McCune turned to the Robbins conjecture, which states that a set of three equations in logic is equivalent to a Boolean algebra (see box).
Gregory's use of Venn diagrams and references to Boolean algebra do not reveal any logical scandals either.
s], but that it be a partial Boolean algebra of projections (Kochen and Specker [1967]).
Boolean algebra can map language for computers to work on, but cannot map the rich connotations we associate with language.
Using Boolean algebra, we limited our search to pages that contained the phrase "electronic warfare" and "US military" and were dated between January 1, 1996, and May 6, 1996.