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Simplifying statements in Boolean algebra using De Morgan's laws

Introduction
We have defined De Morgan's laws in a previous section. The key to understanding the different ways you can use De Morgan's laws and Boolean algebra is to do as many examples as you can. We will now look at some examples that use De Morgan's laws. As you get more experienced, you will see that De Morgan's rules don't just apply to single variables. They apply to where there are more than just two variables in the expression and by working through as many examples as you can find, you will begin to understand this and look out for it. Also, don't forget that the rules can be applied in reverse as well e.g. XY ≡  X + Y as well as X + Y ≡ XY so look out for that being used. Don't forget:

An easy way to remember De Morgan's rules is that each term is complemented, and 
then the ORs become ANDs, and the ANDs become ORs.

Example 1
Use De Morgan's law on the expression NOT(A AND B AND C).

We can represent this as ¬(A Λ B Λ C) or our preferred notation

ABC

Applying the De Morgan's rule that states XY   X + Y we get 

ABC  ≡  A + B + C                     


Example 2
Use De Morgan's law on the expression NOT(A OR B OR C).

We can represent this as ¬(A V B V C) or our preferred notation

A + B + C

Applying the De Morgan's rule that states X + Y   X Y we get 

A + B + C  ≡  A B C 

 

Example 3
Use De Morgan's law on the expression NOT(E AND F AND G AND H).

We can represent this as ¬(E Λ F Λ G Λ H ) or our preferred notation

EFGH

Applying the De Morgan's rule that states XY   X + Y we get 

EFGH  ≡  E + F + G H                      


Example 4
Use De Morgan's law on the expression NOT(E OR F OR G OR H).

We can represent this as ¬(E V F V G V H) or our preferred notation

E + F + G + H

Applying the De Morgan's rule that states X + Y   X Y we get 

E + F + G + H ≡  E F G H

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