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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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.