Boolean Algebra




Boolean Algebra

In mid 1800, George Boole (1815-1864), an English mathematician, developed algebra for simplifying the representation and manipulation of propositional logic. It is known as Boolean algebra after its developer's name. Later, in the year 1938, Claude E. Shannon proposed the use of Boolean algebra in the design of relay switching circuits. The basic techniques described by shannon were adopted almost universally for the design and analysis of switching circuits. Owing to analogous relationships between the action of relays and modern electronic circuits, designers of modern computers still use the same techniques.

Boolean algebra provides an economical and straightforward approach to the design of relay and other types of switching circuits. Just as basic theorems of algebra help in simplifying an ordinary algebraic expression, Boolean algebra helps in simplifying an expression describing a given switching circuit. Today, designers, use Boolean algebra extensively in designing electronic circuitry of computers.

A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join operators instead of the usual addition and multiplication operators. Explicitly, a Boolean algebra is the partial order on subsets defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra b(A) of a set A is the set of subsets of A that can be obtained by means of a finite number of the set operations union (OR), intersection (AND), and complementation (NOT) (Comtet 1974, p. 185). A Boolean algebra also forms a lattice (Skiena 1990, p. 170), and each of the elements of b(A) is called a Boolean function. There are 2^(2^n) Boolean functions in a Boolean algebra of order n (Comtet 1974, p. 186).

In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. In modern times, Boolean algebra and Boolean functions are therefore indispensable in the design of computer chips and integrated circuits.