A monomial is defined to be either a single letter representing a set, with or without a prime, or an indicated product of two or more such symbols representing the intersection of these sets; X, Y', and XY'Z are examples of monomials. A polynomial is an indicated sum of monomials. each of which is called a term of the polynomial. The polynomial represents the union of the sets corresponding to the separate terms; X - Y' - X Y'Z is an example of apolynomial. In any expression representing an intersection of sets, each such set will be termed a factor of the set of intersection. The factors of the set X'(Y + Z) are X' and Y + Z. In particular, a factor is said to be linear if it is a single letter, with or without a prime, or a sum of such symbols: X + Y' is linear, while Z -, XY and (X - Y)' are not linear.

In general, all useful terminology from the algebra of numbers will be carried over to the algebra of sets.

X (Y + Z) = X (Y + XZ )                                             (*)

Many algebraic expressions arising in the algebra of sets lend themselves to remarkable simplifications. One may perform the usual operations of factoring, or expanding products, familiar from working with numbers, in any expression. These processes are based on applications of the first distributive law, number and are illustrated in the following examples.

EXAMPLE 1. Expand (X + Y) (Z' + TV) into a polynomial.

Solution. The steps in the expansion are as follows:

                                                     (X + Y) (Z' +

مواضيع ذات صلة

Zero Polynomial

Kronecker,s Polynomial Theorem

THE ALGEBRA OF SETS-The number of elements in a set

THE ALGEBRA OF SETS-Solution of equations

THE ALGEBRA OF SETS-Conditional equations

THE ALGEBRA OF SETS-Properties of set inclusion

THE ALGEBRA OF SETS-Fundamental laws

THE ALGEBRA OF SETS-Venn diagrams

THE ALGEBRA OF SETS-The combination of sets

THE ALGEBRA OF SETS-Element and set

THE ALGEBRA OF SETS-Introduction

SYMBOLIC LOGIC AND THE ALGEBRA OF PROPOSITIONS-Special problems