Boolean algebra
Boolean algebra
Boolean algebra
(ˈbuːlɪən)Bool′e•an al′gebra
(ˈbu li ən)n.
Boolean algebra
(bo͞o′lē-ən)| Noun | 1. | Boolean algebra - a system of symbolic logic devised by George Boole; used in computers |
| 单词 | boolean algebra | |||
| 释义 | Boolean algebraBoolean algebraBoolean algebra(ˈbuːlɪən)Bool′e•an al′gebra(ˈbu li ən)n. Boolean algebra(bo͞o′lē-ən)
Boolean algebraBoolean 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. ..... Click the link for more information. (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. ..... Click the link for more information. 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). ..... Click the link for more information. —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]Boolean algebraBoolean algebra(mathematics, logic)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 = 1a ^ 1 = a a V 0 = aa ^ 0 = 0 a V 1 = 1-1 = 0 -0 = 1 There 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. Boolean algebra
Synonyms for Boolean algebra
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