By Frank Markham Brown
This ebook is ready the common sense of Boolean equations. Such equations have been important within the "algebra of common sense" created in 1847 through Boole [12, thirteen] and devel oped by means of others, significantly Schroder , within the rest of the 19th century. Boolean equations also are the language wherein electronic circuits are defined at the present time. Logicians within the 20th century have deserted Boole's equation established good judgment in desire of the extra robust predicate calculus. therefore, electronic engineers-and others who use Boole's language routinely-remain principally ignorant of its software as a medium for reasoning. the purpose of this booklet, for this reason, is to is to provide a scientific define of the good judgment of Boolean equations, within the desire that Boole's tools could end up precious 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 strategy was once to symbolize sessions (e. g. , chuffed creatures, issues efficient 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 couple of nineteenth-century logicians, together with Jevons , Poretsky , Schroder , Venn , and Whitehead [212, 213], sought a more robust formula in accordance with ex tensions or transformations of Boole's algebra. those efforts met with simply restricted success.
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Extra resources for Boolean Reasoning: The Logic of Boolean Equations
Xn have concrete interpretations. The symbols may in such cases be used to construct a 22n -element Boolean algebra as follows: each element of the algebra is the disjunction of a subset of the 2n minterms built from Xl, X2, ... , Xn. The null disjunction is the O-element of the Boolean algebra; the disjunction of all of the minterms is the I-element of the algebra. 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 ).
An important task in many applications of Boolean algebra is to select a good formula-given some definition of "good"-to represent a Boolean function. The range of a Boolean function. The range (cf. 7) of a Boolean function f: Bn __ B is the set of images of elements of Bn under f. It was shown by Schroder [178, Vol. 1, Sect. 39) f(A) , Ef(A)]. 4]. 8 Boole's Expansion Theorem The basis for computation with Boolean functions is the expansion theorem given below. Called "the fundamental theorem of Boolean algebra" by Rosenbloom , it was discussed in Chapter V of Boole's Laws of Thought  and was widely applied by Boole and other nineteenth-century logicians.
6). 49) is valid in all Boolean algebras, and to do so without going back to the postulates. We cannot substitute all possible values for the variables x and y, because no limit has been specified for the size of the carrier, B. We have seen, however, that a Boolean function is completely defined by the 0,1 assignments of its arguments. Thus 0,1-substitutions are adequate to verify a Boolean identity. las is valid in an arbitrary Boolean algebra if and only if it is valid in the two-element Boolean algebra.
Boolean Reasoning: The Logic of Boolean Equations by Frank Markham Brown