BOOLEAN ALGEBRA AND LOGIC GATES

Binary logic deals with variables that have two discrete values: 1 for TRUE and 0 for
FALSE. A simple switching circuit containing active elements such as a diode and
transistor can demonstrate the binary logic, which can either be ON (switch closed) or
OFF (switch open). Electrical signals such as voltage and current exist in the digital
system in either one of the two recognized values, except during transition.
The switching functions can be expressed with Boolean equations. Complex Boolean
equations can be simplified by a new kind of algebra, which is popularly called
Switching Algebra or Boolean Algebra, invented by the mathematician George Boole
in 1854. Boolean Algebra deals with the rules by which logical operations are carried
out.
BASIC DEFINITIONS
Boolean algebra, like any other deductive mathematical system, may be defined with
a set of elements, a set of operators, and a number of assumptions and postulates. A
set of elements means any collection of objects having common properties. If S
denotes a set, and X and Y are certain objects, then X ?¸ S denotes X is an object of
set S, whereas Y . S denotes Y is not the object of set S. A binary operator defined on
a set S of elements is a rule that assigns to each pair of elements from S a unique
element from S. As an example, consider this relation X*Y = Z. This implies that * is
a binary operator if it specifies a rule for finding Z from the objects ( X, Y ) and also
if all X, Y, and Z are of the same set S. On the other hand, * can not be binary
operator if X and Y are of set S and Z is not from the same set S.
The postulates of a mathematical system are based on the basic assumptions, which
make possible to deduce the rules, theorems, and properties of the system. Various
algebraic structures are formulated on the basis of the most common postulates, which
are described as follows:
1. Closer: A set is closed with respect to a binary operator if, for every pair of
elements of S, the binary operator specifies a rule for obtaining a unique element of S.
For example, the set of natural numbers N = {1, 2, 3, 4, ...} is said to be closed with
respect to the binary operator plus ( + ) by the rules of arithmetic addition, since for
any X,Y ?¸ N we obtain a unique element Z ?¸ N by the operation X + Y = Z.
However, note that the set of natural numbers is not closed with respect to the binary
operator minus (.) by the rules of arithmetic subtraction because for 1 . 2 = .1, where
.1 is not of the set of naturals numbers.