Lec-4/logic/Boolean algebra and logic gates

Boolean Algebra And Logic Gates
1. Introduction
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. Boolean Algebra deals with the rules by which logical operations are carried out.
2. 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.