Fundamentals of Boolean mathematics. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / Beginner radio amateur

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The analysis of combinational devices and digital logic circuits is most easily carried out using Boolean mathematics, which operates with only two concepts: true (logical 1) and false (logical 0). As a result, functions that display information take only the values ​​0 or 1 at any given time. Such functions are called logical functions. Logical functions Y of several variables (X0, X1, ..., Xn-1) determine the nature of logical operations, as a result of which the variable Y is assigned to a set of input variables

Y=f(X0, X1, ..., Xn-1).

The transformation function is most clearly characterized by a table, in the rows of which each combination of input variables X corresponds to the value of the variable Y. It is called a truth table.

The main logical functions are logical multiplication (conjunction), logical addition (disjunction) and logical negation (inversion). In logical multiplication, input variables (two or more) are connected by an AND union. This operation is denoted by /\ or the multiplication sign (*). The function Y1=X1*X2 takes the value of logical 1 only if all input variables are equal to 1. If at least one variable is equal to 0, then the output function is equal to 0 (table 1).

In logical addition, two or more statements are connected by the union OR (OR). This operation is denoted by the \/ symbol or the addition sign (+) The truth table for the disjunction looks like this.

The statement (X1 + X2) is true if at least one of the statements included in it is true.

With logical negation, the function is NOT (NOT), the value of the output function is opposite to the input variable (Table 3). This operation is denoted by X (read "NOT X").

Conjunction, disjunction and inversion can express any other more complex operations on statements. Therefore, the system of functions Y1=X1*X2, Y2=X1+X2 and Y3=-X has functional completeness. As an example, consider several functions implemented using elements of computer technology. Equivalence (or equivalent) is the function Y of two arguments X1 and X2, which takes the value=1 when X1=X2=1 or when X1=X2=0. For different values ​​of the arguments X1≠X2, the value of the function Y=0. It can be shown that the function Y has the form Y=X1*X2+(-X1)*(-X2), which is confirmed by substituting the corresponding values ​​of the arguments into the expression. Inequivalence is a function Y of two arguments X1 and X2, which takes the value 1 for X1≠X2, and the value 0 for X1=X2=0 or,. at X1=X2=1. In this case, we will have Y=X1*X2+X1*X2. The operation of unequalness is often called summation modulo 2 and denoted by Y=X1(+)X2. There are also functionally complete systems consisting of only one function. These include, in particular, the AND-NOT functions (Y= -(X1*X2) and OR-NOT (Y=-(X1+X2)), which are widely used in modeling digital devices. OR-NOT of two variables X1 and X2.

Boolean mathematics allows you to transform formulas that describe complex statements in order to simplify them. This helps in the end to determine the optimal structure of a digital device that implements any complex function. Under the optimal structure, it is customary to understand the construction of a device in which the number of elements included in its composition is minimal.

Author: -=GiG=-, gig@sibmail; Publication: cxem.net

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