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Publisher:

Dover Publications

Publication Date:

2011

Number of Pages:

144

Format:

Paperback

Price:

12.95

ISBN:

9780486483856

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

10/11/2012

This book is primarily about Boolean algebras and Boolean rings. It starts out with a simple exposition of symbolic logic and proof methods, but this is not developed further and is primarily to provide a foundation for what follows. The book is an unaltered reprint of the 1962 Prentice-Hall work.

This is a brief introduction to the subject and is aimed at upper-division undergraduates. It has few prerequisites, but does assume familiarity with set theory and proofs. The approach is generally abstract, with abstract structures and operations being introduced, but then immediately illustrated with examples, especially from set theory. The book is well-supplied with a large number of problems, most of them fairly easy, but with some harder proof problems.

The terminology was a little unsettling. Lattices are defined in terms of supremum and infimum rather than meet and join (and the set theory union and intersection symbols are used rather than the up and down carets). Partially-ordered sets are named “ordered”, and there's no discussion of total ordering (which is not used in the book, but it would still be useful as a contrast to partial ordering).

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

- 1 Some Concepts of Intuitive Logic
- 1-1 Introduction
- 1-2 Sentences and Connectives
- 1-3 Variables and Functions
- 1-4 Related Sentential Functions; Truth Value Functions
- 1-5 Some Methods of Proof

- 2 Boolean Functions
- 2-1 Introduction
- 2-2 Basic Definitions; Use of Tables
- 2-3 Equivalence Relations
- 2-4 Correspondence between Truth Value Functions and Boolean Functions

- 3 Ordered Sets
- 3-1 Introduction
- 3-2 Basic Definition; Graphical Representation
- 3-3 Isomorphism
- 3-4 The Ordering in
*B*_{n} - 3-5 Connection with Logic

- 4 Lattices
- 4-1 Introduction
- 4-2 Basic Definitions
- 4-3 Further Properties of Lattices
- 4-4 The Lattice
*B*_{n}and Applications to Logic

- 5 Boolean Algebras
- 5-1 Introduction
- 5-2 Basic Definition and Properties
- 5-3 Representation in Terms of Subsets of a Set
- 5-4
*B*_{n}as a Boolean Algebra; Applications to Logic

- 6 Boolean Rings
- 6-1 Introduction
- 6-2 Definitions and Basic Properties
- 6-3 Boolean Rings with a Unit Element are Equivalent to Boolean Algebras
- 6-4 Representation of a Finite Boolean Ring in Terms of
*n*-tuples of Boolean Constants

- 7 Normal Forms, Duality
- 7-1 Introduction
- 7-2 Disjunctive and Conjunctive Normal Forms
- 7-3 The Ring Normal Form
- 7-4 Normal Forms in
*B*_{n} - 7-5 Duality

- 8 Some Applications of Boolean Algebra
- 8-1 Introduction
- 8-2 Applications to Electrical Networks
- 8-3 Applications to Computer Design
- 8-4 Applications to Logic

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