By Frank Markham Brown

This booklet is set the common sense of Boolean equations. Such equations have been primary within the "algebra of good judgment" created in 1847 through Boole [12, thirteen] and devel­ oped via others, significantly Schroder [178], within the rest of the 19th century. Boolean equations also are the language in which electronic circuits are defined this day. Logicians within the 20th century have deserted Boole's equation­ established common sense in want of the extra robust predicate calculus. therefore, electronic engineers-and others who use Boole's language routinely-remain mostly ignorant of its software as a medium for reasoning. the purpose of this booklet, for that reason, is to is to give a scientific define of the common sense of Boolean equations, within the desire that Boole's equipment may possibly turn out worthy in fixing present-day difficulties. Logical Languages common sense seeks to lessen reasoning to calculation. major languages were constructed to accomplish that item: Boole's "algebra of common sense" and the predicate calculus. Boole's process used to be to symbolize periods (e. g. , satisfied creatures, issues effective of enjoyment) through symbols and to symbolize logical statements as equations to be solved. His formula proved insufficient, even though, 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 keeping with ex­ tensions or changes of Boole's algebra. those efforts met with in basic terms constrained success.

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Additional resources for Boolean Reasoning: The Logic of Boolean Equations

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Suppose a = b. Then a'b + ab' = a' a + aa' = 0 + 0 = o. Suppose on the other hand that a'b+ab' = o. If a is added to both sides, the result, after simplification, is a + b = aj if instead b is added to both sides, the simplified result is a + b b. Thus a b. 2 enables an arbitrary Boolean equation to be recast equivalently in the standard form f = o. We will make frequent use of this form. 2 also provides a direct means of verifying Boolean identities of the form a = bj it is more convenient in many cases to evaluate a'b + ab' than it is to manipulate one side of an identity until it becomes the same as the other.

F is the set of functions, that is, that map D into R. Express terms of # D and # R. # F in EXERCISES 21 11. Let S be a set comprising k elements, and let n be a positive integer. (a) How many elements are there in sn? (b) How many n-variable functions are there from S into S? 12. Decide, for each of the following sets of ordered pairs, whether the set is a function. (a) {(x, y) (b) {(x, y) (c) {(x, y) (d) {(x, y) I x and y are people and x is the mother of y} I x and yare people and y is the mother of x} I x and yare real numbers and x 2 + y2 = I} I [x = 1 and y = 2] or [x = -1 and y = 2]} Chapter 2 Boolean Algebras We outline in this chapter the ideas concerning Boolean algebras that we shall need in the remaining chapters.

Then we call S a universal set and we call its subsets the classes of S. If S = {a, b}, for example, then the classes of S are 0, {a}, {b}, and {a,b}. The algebra of classes consists of the set 25 (the set of subsets of S), together with two operations on 25 , namely, U (set-union) and n (setintersection). 2. , the system (28,U,n,0,S) is a Boolean algebra. The "algebra of logic" of Boole [13], Carroll [34], Venn [210], and other nineteenth-century logicians was formulated in terms of classes. Carroll's problems, involving classes such as "my poultry," "things able to manage a crocodile," and "persons who are despised," remain popular today as logical puzzles.

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