BOOLEAN ALGEBRA

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.