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

By Frank Markham Brown

This e-book is ready the common sense of Boolean equations. Such equations have been primary within the "algebra of common sense" created in 1847 via Boole [12, thirteen] and devel­ oped through others, particularly Schroder [178], within the rest of the 19th century. Boolean equations also are the language through which electronic circuits are defined this present day. Logicians within the 20th century have deserted Boole's equation­ dependent good judgment in prefer of the extra robust predicate calculus. consequently, 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 publication, as a result, is to is to offer a scientific define of the good judgment of Boolean equations, within the desire that Boole's tools could end up beneficial in fixing present-day difficulties. Logical Languages common sense seeks to minimize reasoning to calculation. major languages were constructed to accomplish that item: Boole's "algebra of common sense" and the predicate calculus. Boole's technique used to be to symbolize periods (e. g. , satisfied creatures, issues effective of enjoyment) by way of symbols and to symbolize logical statements as equations to be solved. His formula proved insufficient, even though, to symbolize traditional discourse. a few nineteenth-century logicians, together with Jevons [94], Poretsky [159], Schroder [178], Venn [210], and Whitehead [212, 213], sought a stronger formula in line with ex­ tensions or adjustments of Boole's algebra. those efforts met with purely constrained success.

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V]. 1): f(x,y,z) x'f(O,y,z)+xf(l,y,z) x'[y' f(O, 0, z) + yf(O, 1, z)] + x[y' f(l, 0, z) + yf(l, 1, z)] = x'y'z' f(O, 0, 0) + x'y'zf(O, 0,1) +x'yz' f(O, 1, 0) + x'yzf(O, 1, 1) +xy' z' f(l, 0, 0) + xy'zf(l, 0,1) +xyz' f(l, 1, 0) + xyzf(l, 1, 1) . •. , X n - l , x n ) = + f(O, ... ,0, 0) f(O, ... ,0, 1) x~ + f(l, ... , 1, 1) Xl ••• Xn-lX n • ... , f(l, ... ,l,l) x~ ... x~_l x~ .. 42) The values f(O, ... , 0, 0), f(O, ... , 0,1), are elements of B called the discriminants of the function f; the elementary products are called the minterms of X = (Xl, ••• , x n ).

The resulting structure is called the free Boolean algebra on the n generators Xl, X2, ... , Xn and is denoted by F B(xt, X2, .. , x n ). It is shown by Nelson [148, p. 39] that F B( Xl, X2, ... ,xn ) is isomorphic to the Boolean algebra of switching functions of n variables. 2 The carrier of the free Boolean algebra F B(Xl, X2) is the 16-element set of disjunctions of subsets of the set {x~x~, X~X2' XlX~, XlX2}. Each of these formulas is the representative of an equivalence-class of formulas; thus the disjunction x~ x~ + x~ X2 of minterms is equivalent, by the rules of Boolean algebra, to the formula x~.

3. If 9 and h are Boolean formulas, then so are (a) (g) + (h) (b) (g)(h) (c) (g)' . 4. A string is a Boolean formula if and only if its being so follows from finitely many applications of rules 1, 2, and 3. las. The number of such formulas is clearly infinite. Given the Boolean algebra B = {a, l,a',a}, the strings a (a) (a) (a) + + + (a) «a) «a) + (a» + «a) + (a») ,... for example, are all distinct n-variable Boolean formulas for any value of n. Our definition rejects as Boolean formulas such reasonable-looking strings as b + X2 and aXI because they lack the parentheses demanded by our rules.

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