Introduction to Industrial Automation

Figure 4.2  Waveform of a digital signal.

b

‘1’

‘1’

Z

b

Z

(a)

(b)

Figure 4.3  The pushbutton as a binary variable: (a) NO button and (b) NC button.

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  Introduction to Industrial Automation

4.2.1   Postulates and Theorems of Boolean Algebra

Boolean algebra, as well as classical algebra, were founded on some basic postulates like those 

of commutative and distributive property of the two logical operations of AND and OR, the 

existence of the neutral elements 0 and 1, and many more. From these postulates, it is possible 

to extract a series of theorems that can be utilized in the simplification of the logical functions. 

In Figure 4.4, some of the most fundamental postulates and theorems are presented in graphical 

(contact symbol) form as well as in Boolean form. The profound logic behind these theorems is 

that they are not only useful in their mathematical form in the simplification of the logical func-

tions (i.e., during the logical design that is going to be described subsequently), but they are also 

very useful in the empirical design of automation circuits.

Α

Short circuit

A+1=1

Short circuit

A+0=A

Α

Α

Open circuit

Α

Α

Α

A•A=A

A+A=A

Α

Α

Α

Short circuit

A+A=1

Α

Α

Α

Α

Open circuit

A•A=0

A+A•B=A

Α

Α

B

Α

A•(A+B)=A

Α

Α

A

B

A•1=A

Α

Short circuit

Α

A•0=0

Α

Open circuit

Open circuit