Boolean algebra

/booh"lee euhn/1. Logic. a deductive logical system, usually applied to classes, in which, under the operations of intersection and symmetric difference, classes are treated as algebraic quantities.2. Math. a ring with a multiplicative identity in which every element is an idempotent.[188590; named after G. BOOLE; see AN]
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Symbolic system used for designing logic circuits and networks for digital computers.Its chief utility is in representing the truth value of statements, rather than the numeric quantities handled by ordinary algebra. It lends itself to use in the binary system employed by digital computers, since the only possible truth values, true and false, can be represented by the binary digits 1 and 0. A circuit in computer memory can be open or closed, depending on the value assigned to it, and it is the integrated work of such circuits that give computers their computing ability. The fundamental operations of Boolean logic, often called Boolean operators, are "and," "or," and "not"; combinations of these make up 13 other Boolean operators.* * *
symbolic system of mathematical logic that represents relationships between entities—either ideas or objects. The basic rules of this system were formulated in 1847 by George Boole of England and were subsequently refined by other mathematicians and applied to set theory. Today, Boolean algebra is of significance to the theory of probability, geometry of sets, and information theory. Furthermore, it constitutes the basis for the design of circuits used in electronic digital computers (digital computer).In a Boolean algebra a set of elements is closed under two commutative binary operations that can be described by any of various systems of postulates, all of which can be deduced from the basic postulates that an identity element exists for each operation, that each operation is distributive over the other, and that for every element in the set there is another element that combines with the first under either of the operations to yield the identity element of the other.The ordinary algebra (in which the elements are the real numbers and the commutative binary operations are addition and multiplication) does not satisfy all the requirements of a Boolean algebra. The set of real numbers is closed under the two operations (that is, the sum or the product of two real numbers also is a real number); identity elements exist—0 for addition and 1 for multiplication (that is, a + 0 = a and a × 1 = a for any real number a); and multiplication is distributive over addition (that is, a × [b + c] = [a × b] + [a × c]); but addition is not distributive over multiplication (that is, a + [b × c] does not, in general, equal [a + b] × [a + c]).The advantage of Boolean algebra is that it is valid when truthvalues (truthvalue)—i.e., the truth or falsity of a given proposition or logical statement—are used as variables instead of the numeric quantities employed by ordinary algebra. It lends itself to manipulating propositions that are either true (with truthvalue 1) or false (with truthvalue 0). Two such propositions can be combined to form a compound proposition by use of the logical connectives, or operators, AND or OR. (The standard symbols for these connectives are ∧ and ∨, respectively.) The truthvalue of the resulting proposition is dependent on the truthvalues of the components and the connective employed. For example, the propositions a and b may be true or false, independently of one another. The connective AND produces a proposition, a ∧ b, that is true when both a and b are true, and false otherwise.* * *
Universalium. 2010.
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Boolean algebra — statusas T sritis automatika atitikmenys: angl. Boolean algebra vok. Boolesche Algebra, f rus. булева алгебра, f pranc. algèbre de Boole, f ryšiai: sinonimas – Bulio algebra … Automatikos terminų žodynas
Boolean algebra — This article discusses the subject referred to as Boolean algebra. For the mathematical objects, see Boolean algebra (structure). Boolean algebra, as developed in 1854 by George Boole in his book An Investigation of the Laws of Thought,[1] is a… … Wikipedia
Boolean Algebra — A division of mathematics which deals with operations on logical values. Boolean algebra traces its origins to an 1854 book by mathematician George Boole. The distinguishing factor of Boolean algebra is that it deals only with the study of binary … Investment dictionary
Boolean algebra — A Boolean algebra is a system consisting of a set S and two operations, n and ? (cap and cup), subject to the following axioms. For all sets a,b,c, that are members of S: 1 a n (b n c) = (a n b) n c. Also a ? (b ? c) = (a ? b) ? c (associativity) … Philosophy dictionary
Boolean algebra — noun a system of symbolic logic devised by George Boole; used in computers • Syn: ↑Boolean logic • Hypernyms: ↑symbolic logic, ↑mathematical logic, ↑formal logic * * * noun Usage: usually capitalized B … Useful english dictionary
Boolean algebra — noun a) An algebra in which all elements can take only one of two values (typically 0 and 1, or true and false ) and are subject to operations based on AND, OR and NOT The set of divisors of 30, with binary operators: g.c.d. and l.c.m., unary… … Wiktionary
Boolean algebra — noun Date: circa 1889 an algebraic system that consists of a set closed under two binary operations and that can be described by any of various systems of postulates all of which can be deduced from the postulates that each operation is… … New Collegiate Dictionary
Boolean algebra — See comparative sociology ; qualitative comparative analysis … Dictionary of sociology
boolean algebra — mathematical set with operations whose rules are any of various equivalent systems of postulates … English contemporary dictionary
Boolean algebra — Bool′e•an al′gebra [[t]ˈbu li ən[/t]] n. pho a system of symbolic logic dealing with the relationship of sets: the basis of logic gates in computers • Etymology: 1885–90; after G. Boole (1815–64), English mathematician; see an I … From formal English to slang