Introduction to Boolean Algebra || Boolean Algebra and Logic Gates

Introduction to Boolean Algebra:

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
algebraic system now called Boolean algebra. In 1938 C. E. Shannon introduced a two-valued Boolean
algebra called switching algebra, in which he demonstrated that the properties of bistable electrical
switching circuits can be represented by this algebra. Thus, the mathematical system of binary logic is known as Boolean or switching algebra.

Postulates
Boolean algebra is an algebraic structure defined on a set of elements B (Boolean system) together with
two binary operators + (OR) and • (AND) and unary operator ‘ (NOT), provided the following postulates
are satisfied:
P1 Closure: Boolean algebra is closed under the AND, OR, and NOT operations.
P2 Commutativity: The • and + operators are commutative i.e. x + y = y + x and x • y = y • x, for all
x, y ∈ B.
P3 Distribution: • and + are distributive with respect to one another i.e.
X • (y + z) = (x • y) + (x • z).
x + (y • z) = (x + y) • (x + z), for all x, y, z ∈ B.
P4 Identity: The identity element with respect to • is 1 and + is 0 i.e. x + 0 = 0 + x = x and x • 1=1•
x = x. There is no identity element with respect to logical NOT.
P5 Inverse: For every value x there exists a value x’ such that x • x’ = 0 and x + x’ = 1. This value is
the logical complement (or NOT) of x.
P6 There exists at least two elements x, y ∈ B such that x ≠ y.
One can formulate many Boolean algebras (viz. set theory, n-bit-vectors algebra), depending on the
choice of elements of B and the rules of operation. Here, we deal only with a two-valued Boolean
algebra, i.e., B = {0, 1}. Two-valued Boolean algebra has applications in set theory and in propositional
logic. Our interest here is with the application of Boolean algebra to gate-type circuits.