Boolean Algebra - Quick Reference

  

Boolean Algebra, also known as the 'algebra of logic', is a branch of mathematics that is similar in form to algebra, but dealing with logical instead of numerical relationships. It was invented by George Boole, after whom this system was named. Thus, instead of variables that represent numerical quantities as in conventional algebra, Boolean algebra handles variables that represent two types of logic propositions: 'true' and 'false'.

  

Boolean algebra has become the main cornerstone of digital electronics, since the latter also operates with two logic states, '1' and '0', represented by two distinct voltage levels. Boolean algebra's formal interpretation of logical operators AND, OR, and NOT has allowed the systematic development of complex digital systems from simple logic gates, that now not only include circuits that perform mathematical operations, but intricate data processing as well.  Tables 1 to 4 summarize the definitions of logical operators and their basic mathematical properties as represented in Boolean algebra.

    

    

Table 1. Elementary Logic Gate Actions

OR

0+0=0

0+1=1

1+0=1

1+1=1

AND

00=0

01=0

10=0

11=1

NOT

0=1

1=0

NOR

0+0=1

0+1=0

1+0=0

1+1=0

NAND

00=1

01=1

10=1

11=0

                      

Table 2. Single-Variable Logic Gate Actions

OR

A+0=A

A+1=1

A+A=A

A+A=1

AND

A0=0

A1=A

AA=A

AA=0

NOT

A=NOT(A)

NOT(A)=A

NOR

A+0=A

A+1=0

A+A=A

A NOR A=0

NAND

A0=1

A1=A

AA=A

A NAND A=1

                      

Table 3. Multi-Variable Boolean Equalities

A+B=B+A

(A+B)+C=A+(B+C)

AB=BA

(AB)C=A(BC)

A(B+C)=AB+AC

                      

Table 4. De Morgan's Theorem

AB=A+B

A+B=AB

                        

See Also:  Logic GatesWhat is a Semiconductor?

 

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