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Boolean Algebra SCHAUM’S OUTLlNE OF THEORY AND PROBLEMS OF BOOLEAN ALGEBRA and SWITCHING CIRCUITS BY ELLIOTT MENDELSON, Ph.D. Professor of Mathematics Queens College City University of New York SCHAUM’S OUTLINE SERIES MCGRAW-HILL BOOK COMPANY New York, St. Louis, San Francisco, London, Sydney, Toronto, Mexico, and Panama Copyright @ 1970 by McGraw-Hill, Inc. All Rights Reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. 07-041460-2 1234567890 SHSH 7543210 To Julia, Hilary, and Peter Preface This book is devoted to two separate, but related, topics: (1) the synthesis and simplification of switching and logic circuits, and (2) the theory of Boolean algebras. Those people whose primary interest is in switching and logic circuits can read Chapter 4 immediately after a quick perusal of Chapter 1. We have confined our treatment of switching and logic circuits to combinational circuits, i.e. circuits in which the outputs at a given time depend only on the present value of the inputs and not upon the previous values of the inputs. The extensive theory of sequential circuits, in which the outputs depend also upon the history of the inputs, may be pursued by the reader in Introduction to Switching Theory and Logical Design by F. J. Hill and G. R. Peterson (reference 34, page 202), Zntroduction to Switching and Automata Theory by M. A. Harrison (ref. 33, page 202), and other textbooks on switching theory. The treatment of Boolean algebras is somewhat deeper than in most elementary texts. It can serve as an introduction to graduate-level books such as Boolean Algebras by R. Sikorski (ref. 148, page 207) and Lectures on Boolean Algebras by P. R. Halmos (ref. 116, page 207). There is no prerequisite for the reading of this book. Each chapter begins with clear statements of pertinent definitions, principles and theorems together with illus- trative and other descriptive material. This is followed by graded sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory, bring into sharp focus those fine points without which the student continually feels himself on unsafe ground, and provide the repetition of basic principles so vital to effective learning. A few problems which involve modern algebra or point-set topology are clearly labeled. The supplementary problems serve as a complete review of the material of each chapter. Difficult problems are identified by a superscript D following the problem number. The extensive bibliography at the end of the book is divided into two parts, the first on Switching Circuits, Logic Circuits and Minimization, and the second on Boolean Algebras and Related Topics. It was designed for browsing. We have listed many articles and books not explicitly referred to in the body of the text in order to give the reader the opportunity to delve further into the literature on his own. Queens College ELLIOTT MENDELSON July 1970 CONTENTS Page Chapter 1 THE ALGEBRA OF LOGIC 1.1 Truth-Functional Operations ............................................ 1 1.2 Connectives ............................................................ 4 1.3 Statement Forms ...................................................... 4 1.4 Parentheses ............................................................ 5 1.5 Truth Tables .......................................................... 6 1.6 Tautologies and Contradictions .......................................... 7 1.7 Logical Implication and Equivalence .................................... 9 1.8 Disjunctive Normal Form .............................................. 12 1.9 Adequate Systems of Connectives ........................................ 15 Chapter 2 THE ALGEBRA OF SETS 2.1 Sets .................................................................. 30 2.2 Equality and Inclusion of Sets. Subsets ................................ 31 2.3 Null Set. Number of Subsets .......................................... 31 2.4 Union ................................................................ 32 2.5 Intersection ............................................................ 33 2.6 Difference and Symmetric Difference .................................... 34 2.7 Universal Set. Complement ............................................ 35 2.8 Derivations of Relations among Sets .................................... 36 2.9 Propositional Logic and the Algebra of Sets .............................. 37 2.10 Ordered Pairs. Functions .............................................. 38 2.11 Finite, Infinite, Denumerable, and Countable Sets ........................ 39 2.12 Fields of Sets .......................................................... 39 2.13 Number of Elements in a Finite Set .................................... 40 Chapter 3 BOOLEAN ALGEBRAS 3.1 Operations ............................................................ 52 3.2 Axioms for a Boolean Algebra .......................................... 52 3.3 Subalgebras .......................................................... 56 3.4 Partial Orders ........................................................ 57 3.5 Boolean Expressions and Functions. Normal Forms ...................... 60 3.6 Isomorphisms .......................................................... 62 3.7 Boolean Algebras and Propositional Logic ................................ 63 CONTENTS Page Chapter 4 SWITCHING CIRCUITS AND LOGIC CIRCUITS 4.1 Switching Circuits ...................................................... 71 4.2 Simplification of Circuits .............................................. 72 4.3 Bridge Circuits ........................................................ 74 4.4 Logic Circuits ........................................................ 75 4.5 The Binary Number System ........................................... 77 4.6 Multiple Output Logic Circuits .......................................... 78 4.7 Minimization .......................................................... 80 4.8 Don’t Care Conditions .................................................. 82 4.9 Minimal Disjunctive Normal Forms ...................................... 84 4.10 Prime Implicants ...................................................... 84 4.11 The Quine-McCluskey Method for Finding All Prime Implicants .......... 86 4.12 Prime Implicant Tables ................................................ 90 4.13 Minimizing with Don’t Care Conditions .................................. 93 4.14 The Consensus Method for Finding Prime Implicants ...................... 94 4.15 Finding Minimal Dnf’s by the Consensus Method .......................... 96 4.16 Karnaugh Maps ........................................................ 98 4.17 Karnaugh Maps with Don’t Care Conditions .............................. 106 4.18 Minimal Dnf’s or Cnf’s ................................................ 107 Chapter 5 TOPICS IN THE THEORY OF BOOLEAN ALGEBRAS 5.1 Lattices .............................................................. 133 5.2 Atoms ................................................................ 135 5.3 Symmetric Difference. Boolean Rings .................................... 137 5.4 Alternative Axiomatizations ............................................ 140 5.5 Ideals ................................................................ 143 @5.6 Quotient Algebras ...................................................... 146 5.7 The Boolean Representation Theorem .................................... 148 5.8 Infinite Meets and Joins ................................................ 149 5.9 Duality .............................................................. 151 5.10 Infinite Distributivity .................................................. 151 5.11 m-Completeness ........................................................ 153 Appendix A ELIMINATION OF PARENTHESES ....................................... 190 Appendix B PARENTHESIS-FREE NOTATION ........................................ 195 Appendix C THE AXIOM OF CHOICE IMPLIES ZORN’S LEMMA .................... 1% Appendix D A LATTICE-THEORETIC PROOF OF THE SCHRijDER-BERNSTEIN THEOREM 200 BlBLlOGRAPHY ........................................................ 201 INDEX.. ............................................................... 209 INDEX OF SYMBOLS AND ABBREVIATIONS ........................... 213 Chapter 1 The Algebra of logic 1.1 TRUTH-FUNCTIONAL OPERATIONS There are many ways of operating on propositions to form new propositions. We shall limit ourselves to those operations on propositions which are most relevant to mathematics and science, namely, to truth-functional operations. An operation is said to be truth- functional if the truth value (truth or falsity) of the resulting proposition is determined by the truth values of the propositions from which it is constructed. The investigation of truth-functional operations is called the propositional calculus, or, in old-fashioned terminology, the algebra of logic, although its subject matter forms only a small and atypically simple branch of modern mathematical logic. Negation Negation is the simplest common example of a truth-functional operation. If A is a proposition, then its denial, not-A, is true when A is false and false when A is true. We shall use a special sign 1 to stand for negation. Thus, 1 A is the proposition which asserts the denial of A. The relation between the truth values
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