Introduction to Industrial Automation

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  Introduction to Industrial Automation

4.2   Boolean Logic Components

The binary system of numbers has only two values, 0 and 1, while it is utilized for the mathemati-

cal description of various physical systems characterized from a binary logic of two states. In elec-

trical systems, the condition “in operation” (ON) is indicated with a “1” and the condition “not in 

operation” (OFF) is indicated with a “0”. Similarly, the values 0 and 1 represent correspondingly 

the open and the closed state of an electrical switching contact (e.g., relay, hand switch, button, 

etc.) as depicted in Figure 4.1a. Every switching component, with two possible values, is repre-

sented with a capital letter and constitutes a binary system variable, while the necessary tool for 

the mathematical foundation of these principles is Boolean algebra.

Boolean algebra, first introduced by the English mathematician Boole, is an algebra of “logic”, har-

monized with a human-based way of thinking. Due to the fact that the variables of Boolean algebra can 

have only two values (0 and 1), this type of algebra is ideal for the binary system, especially in the way 

that the switches are operating. The Boolean values of 0 and 1 are not necessarily the arithmetical values 

of an arithmetic system, but in this case, these can represent symbols of a certain state. As an example, 

the values of a Boolean variable could be “white” or “black”, “low” or “high”, and “true” or “false”.

In its general form, Boolean algebra is defined as the set of the elements a, b, c, …, or B={a, b, 

c, …} in which the equality, as well as the following operations, are valid:

  1. The operation of the logical OR, which is represented by the (+) operator

  2. The operation of the logical AND, which is represented by the (

⋅

) operator

  3.  The operation of the inversion or the complement (NOT), which is represented by the 

( )

 operator

In the previous definition, the elements a, b, c, … are not specifically defined, but in the case of 

digital systems and digital logic, set B is the set of the utilized switches (switching algebra).

The application of Boolean algebra in digital systems initially took place in 1938 by C. 

Shannon, who in “A Symbolic Analysis of Relay and Switching Circuits” introduced a way of 

representing telecommunication circuits through mathematical expressions. With the help of the 

mathematical modeling of these circuits, containing switches and relays, their methodological 

design and calculation has been achieved.

In Figure 4.1b, two electrical switching contacts, one open and one closed, with different sym-

bols are displayed. If “A” is denoted as the open contact (A=0), then the complement 

A

 is a closed 

contact, or 

A

=

1

. There are three other possible assignments of the Boolean variable A to the open 

and closed contacts, shown in Figures 4.1c–e. Subsequently, the symbols and the corresponding 

assignment of Figure 4.1c will be utilized as symbols for contacts, since these are closer in the 

inversion operation from the design point of view, and because they are similar to the switching 

symbols that are utilized in the programming of the PLCs.

Α=0

Α=1

Α=0

Α=1

Α=1

Α=0

Α=0

Α=1

Α=1

Α=0

(a)

(b)

(c)

(d)

(e)