Activity 2.1.4 Circuit Simplification:
Boolean Algebra
Introduction
Have you ever had an idea that you thought was so unique that when you told someone
else about it, you simply could not believe they thought you were wasting your time with
it? If so, you know how the mathematician George Boole felt in the 1800s when he
designed a math system that, at the time, had no practical application. Today, however,
his math system is the most important mathematical tool used in the design of digital
logic circuits. Boole introduced the world to Boolean algebra when he published his
work called ​“An Investigation of the Laws of Thought, on Which Are Founded the
Mathematical Theories of Logic and Probabilities.”​
In the same way that ​normal​ algebra has rules that allow you to simplify algebraic
expressions, Boolean algebra has theorems and laws that allow you to simplify
expressions used to create logic circuits.
By simplifying the logic expression, we can convert a logic circuit into a simpler version
that performs the same function. The advantage of a simpler circuit is that it will contain
fewer gates, will be easier to build, and will cost less to manufacture.
In this activity you will learn how to apply the theorems and laws of Boolean algebra to
simplify logic expressions and digital logic circuits.
The moral of the story is to keep dreaming. Someday your grandchildren may be using
something that you’re thinking about right now. When your grandparents were kids, do
you think that they imagined someday that we would all have 10,000 songs in our
pockets or a telephone in our backpacks?
Procedure
Using the theorems and laws of Boolean algebra, simplify the following logic
expressions. Note the Boolean theorem/law used at each simplification step. Be sure to
put your answer in Sum-Of-Products (SOP) form.
1.
AND Distributive Law => F=(A.A’+A(A.B))
AND Absorption Law => F=(A.A’+A.B)
Complement Law => F=(0+A.B)
2.
F=XZ+X’Z+X’Y’Z
F=X’Z+Y’Z
3.
F​3​=J(K+K’)
F​3​=J
4.
(AB’+AB’C’)(1)
AB’(C+1)
AB’
5.
6.
7.
8.
Almost as important as being able to use the laws of Boolean algebra (i.e.,
associative, commutative, or distributive) to simplify logic expressions, it is also
critical that you are able to identify them. Identify the law of Boolean algebra upon
which the following equalities are based.
9.
10.
11.
12.
13.
Now that you’ve practiced simplifying logic expressions, apply your knowledge to
simplifying an actual circuit.
14. Shown below is a ​VERY​ poorly designed AOI circuit that is part of a coffee vending
machine. Write the ​UN-SIMPLIFIED ​logic expression for the output ​Brew Cut Off​.
“If the temperature is too high or the pressure is not below the safe value with water
present,
the brew sensor cuts off the brew process.”
15. Using the theorems and laws of Boolean algebra, simplify the logic expression
Brew Cut Off​. Be sure to put your answer in Sum-Of-Products (SOP) form.
16. In the space provided, draw an AOI circuit that implements the simplified logic
expression ​Brew Cut Off.​ For your implement, assume that only 2-input AND
gates (74LS08), 2-input OR gates (74LS32), and inverters (74LS04) are available.
Draw this circuit in the space provided.
Brew Cut Off Circuit
Conclusion
1. Describe the process that you would use to simplify a logic expression using
Boolean algebra.
2. How do you know when you are finished simplifying and have arrived at the
simplest equation?
3. Other than using Boolean algebra, how could you prove that two circuits are
equivalent?
4. If you worked for a company that manufactured the coffee vending machine that
used the poorly designed circuit, how much money would your new design save
the company annually if each GATE cost 15¢ and the company made 500,000
vending machines per year?