In Section (Venn diagrams) some of the basic identities which are valid in the algebra of sets (and in any Boolean algebra) were mentioned in connection with Venn diagrams. These laws and others which will be used throughout this book are listed below. The laws are numbered, for convenience of reference. The names given are those commonly used, although some of the names reflect one particular application, rather than Boolean algebra in general. For example, "complementation" suggests the application to algebra of sets, whereas "tautology" suggests the application to symbolic logic. No proofs for these laws are given, but each may be justified intuitively by the use of an appropriate Venn diagram.

If 1 denotes the universal set and 0 denotes the null set, the following

identities are valid in the algebra of sets for arbitrary sets X, Y, and Z:

                                                   COMMUTATIVE LAWS

(la) XY = YX                                                                                   (lb) X + Y = Y + X

                                                       ASSOCIATIVE LAWS

(2a) X(YZ) = (XY)Z                                                               (2b) X + (Y + Z) = (X + Y) + Z

                                                             DISTRIBUTIVE LAWS

(3a) X (Y + Z) = X Y + XZ                                                           (3b) X + YZ = (X + Y) (X +Z)

                                                        LAWS OF TAUTOLOGY

(4a) XX = X                                                                                            (4b) X + X = X

                                                            LAWS OF ABSORPTION

(5a) X(X + Y) = X                                                                                      (5b) X + XY = X

                                                         LAWS OF COMPLEMENTATION

(6a) XX' = 0                                                                                               (6b) X + X' = 1

                                            LAW OF DOUBLE COMPLEMENTATION

                                                                            (7) (X')' = X

                                                           LAWS OF DE MORGAN

(8a) (XY)' = X' + Y'                                                                          (8b) (X + Y)' = X'Y'

                                                          OPERATIONS WITH 0 AND 1

(9a) OX = 0                                                                                          (9b) 1 + X = 1

(10a) 1X = X                                                                                         (10b) 0 + X = X

 (11a) 0' = 1                                                                                                 (11b) 1' = 0

Note that many of these laws are already familiar as laws which hold in the algebra of real numbers. However, (3b), (4a), (4b), (5a), and (5b)  are not valid for numbers, and laws involving complementation obviously do not apply to numbers. It is perhaps surprising that any similarity at all is evident. Since some similarity does exist, it is especially important to study the ways in which this algebra differs from ordinary algebra.

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- Expanding, factoring, and simplifying

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