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.

[1885-90; named after G. BOOLE; see -AN]

* * *

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 truth-values (truth-value)—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 truth-value 1) or false (with truth-value 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 truth-value of the resulting proposition is dependent on the truth-values 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.

* * *