Boolean Reasoning: The Logic of Boolean Equations by Frank Markham Brown

By Frank Markham Brown

This ebook is set the common sense of Boolean equations. Such equations have been vital within the "algebra of common sense" created in 1847 by way of Boole [12, thirteen] and devel­ oped by way of others, particularly Schroder [178], within the rest of the 19th century. Boolean equations also are the language in which electronic circuits are defined this present day. Logicians within the 20th century have deserted Boole's equation­ established common sense in desire of the extra robust predicate calculus. hence, electronic engineers-and others who use Boole's language routinely-remain principally blind to its software as a medium for reasoning. the purpose of this booklet, consequently, is to is to offer a scientific define of the common sense of Boolean equations, within the desire that Boole's equipment may well end up important in fixing present-day difficulties. Logical Languages good judgment seeks to lessen reasoning to calculation. major languages were built to accomplish that item: Boole's "algebra of good judgment" and the predicate calculus. Boole's method used to be to symbolize sessions (e. g. , satisfied creatures, issues efficient of delight) through symbols and to symbolize logical statements as equations to be solved. His formula proved insufficient, despite the fact that, to symbolize usual discourse. a few nineteenth-century logicians, together with Jevons [94], Poretsky [159], Schroder [178], Venn [210], and Whitehead [212, 213], sought a more robust formula in line with ex­ tensions or transformations of Boole's algebra. those efforts met with merely restricted success.

Show description

Read Online or Download Boolean Reasoning: The Logic of Boolean Equations PDF

Best algebra & trigonometry books

An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra

During this advent to commutative algebra, the writer choses a path that leads the reader in the course of the crucial rules, with no getting embroiled in technicalities. he's taking the reader speedy to the basics of complicated projective geometry, requiring just a simple wisdom of linear and multilinear algebra and a few basic crew concept.

Inequalities : a Mathematical Olympiad approach

This e-book is meant for the Mathematical Olympiad scholars who desire to organize for the examine of inequalities, an issue now of common use at a number of degrees of mathematical competitions. during this quantity we current either vintage inequalities and the extra important inequalities for confronting and fixing optimization difficulties.

Recent Progress in Algebra: An International Conference on Recent Progress in Algebra, August 11-15, 1997, Kaist, Taejon, South Korea

This quantity offers the court cases of the overseas convention on ""Recent growth in Algebra"" that was once held on the Korea complicated Institute of technology and know-how (KAIST) and Korea Institute for complex learn (KIAS). It introduced jointly specialists within the box to debate development in algebra, combinatorics, algebraic geometry and quantity concept.

Additional info for Boolean Reasoning: The Logic of Boolean Equations

Sample text

Thus, the Boolean formula (b) + (X2) yields, by removal of a parenthesis-pair, the Boolean formula (b) + X2; the latter, by another removal, yields the Boolean formula b + X2. We are accustomed to thinking of the formulas (g) + (h) and (g)(h) as representing operations (addition and multiplication) in a Boolean algebra. CHAPTER 2. , rules for the formation of strings of symbols. We shall shortly need to view the same formulas as representing functions. In cases where it is necessary to distinguish Boolean formulas from Boolean functions, we indicate that (g) + (h) is a formula by calling it a disjunction (rather than a sum) and that (g) (h) is a formula by calling it a conjunction (rather than a product).

Xl + (g(O, ... )),)(xi + (g(l, ... ))') = xi[(g')(O, .. )] + xl[(g')(1, .. )] + [(g')(O, ... )][(g')(1, .. )] = xi[(g')(O, .. )]. 32), the rule of consensus. x2, ... ) = [xi + f(1,x2, . )]. [Xl + f(0,X2, .. )]. 41) If f: B n ---+ B is an n-variable Boolean function and if a is an element of g: Bn-l---+B defined by B, then the (n - I)-variable function is also a Boolean function (the proof is left as an exercise). Thus the functions f(O, X2, ... , xn) and f(l, X2, . 1 and its corollary are Boolean.

Rudeanu [172, p. 17] defines a useful class of functions, the simple Boolean functions, that lie between the switching functions and general Boolean functions. A function f: Bn __ B, B being an arbitrary Boolean algebra, is a simple Boolean function if f can be represented by a formula built from variables by superposition of the basic operations. Constants, except 0 and 1, are not allowed. 51) 46 CHAPTER 2. BOOLEAN ALGEBRAS where g: Bn---+ Band h: Bn---+ B are Boolean functions such that g(X) ::; heX) for all X E Bn.

Download PDF sample

Rated 4.79 of 5 – based on 47 votes