Boolean algebra, like any other deductive mathematical system, may be defined with a set of elements, a set of operators and a number of unproved axioms or postulates. In 1854 George Boole introduced a systematic treatment of logic and developed for this purpose an
\nalgebraic system now called Boolean algebra. In 1938 C. E. Shannon introduced a two-valued Boolean
\nalgebra called switching algebra, in which he demonstrated that the properties of bistable electrical
\nswitching circuits can be represented by this algebra. Thus, the mathematical system of binary logic is known as Boolean or switching algebra.<\/p>\n
Postulates<\/strong> <\/p>\n You may also like types of a number system<\/a><\/p>\n
\nBoolean algebra is an algebraic structure defined on a set of elements B (Boolean system) together with
\ntwo binary operators + (OR) and \u2022 (AND) and unary operator ‘ (NOT), provided the following postulates
\nare satisfied:
\nP1\uf0e0 Closure: Boolean algebra is closed under the AND, OR, and NOT operations.
\nP2\uf0e0 Commutativity: The \u2022 and + operators are commutative i.e. x + y = y + x and x \u2022 y = y \u2022 x, for all
\nx, y \u2208 B.
\nP3\uf0e0 Distribution: \u2022 and + are distributive with respect to one another i.e.
\nX \u2022 (y + z) = (x \u2022 y) + (x \u2022 z).
\nx + (y \u2022 z) = (x + y) \u2022 (x + z), for all x, y, z \u2208 B.
\nP4\uf0e0 Identity: The identity element with respect to \u2022 is 1 and + is 0 i.e. x + 0 = 0 + x = x and x \u2022 1=1\u2022
\nx = x. There is no identity element with respect to logical NOT.
\nP5\uf0e0 Inverse: For every value x there exists a value x’ such that x \u2022 x’ = 0 and x + x’ = 1. This value is
\nthe logical complement (or NOT) of x.
\nP6\uf0e0 There exists at least two elements x, y \u2208 B such that x \u2260 y.
\nOne can formulate many Boolean algebras (viz. set theory, n-bit-vectors algebra), depending on the
\nchoice of elements of B and the rules of operation. Here, we deal only with a two-valued Boolean
\nalgebra, i.e., B = {0, 1}. Two-valued Boolean algebra has applications in set theory and in propositional
\nlogic. Our interest here is with the application of Boolean algebra to gate-type circuits.<\/p>\n