The theory of functions of one Boolean variable
Karl Schmidt · Transactions of the American Mathematical Society · 1922
In the algebra of Boolean f entities as developed by Ernst Schröder % and others, several important theorems regarding functions of Boolean variables are used.They have recently been stated by Eugen Müller.§ The following three I shall use in this paper.P. 1. Boole's theorem.Every function f(x) of one variable can be "developed," i. e., brought into the "normal form" f(x)=ax + bx where a=f(U) and b=f(Z), Uand Z designating the universal and the null class, respectively. P. 2. Proposition of mean value.The values which a function f(x) =ax + bx can take lie "between"** ab and a + b, i. e., ab < f(x) < a + b, for all values of x.+t