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?
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Digital Electronics Activity 2.1.4 Circuit Simplification: Boolean Algebra – Page 1
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.
(
F1 = A A + A B
1.
)
F 1 = A( A + AB)
F 1 = A( A + B)
F 1 = AA + AB
F 1 = 0 + AB
F 1 = AB
F2 = X Y Z + X Y Z + X Y Z
2.
F 2 = X Y Z + X Y Z + X YZ
F 2 = Y Z ( X + X ) + X YZ
F 2 = Y Z (1) + X YZ
F 2 = Y Z + X YZ
F2
=
Z (Y + Y X )
F2
=
Z (Y + X )
F 2 = YZ + X Z
3.
F3 = J K + J K
F3 = JK + J K
F3 = J ( K + K )
F3 = J (1)
F3 = J
4.
(
)(
F4 = B + B A B + A B C
)
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Digital Electronics Activity 2.1.4 Circuit Simplification: Boolean Algebra – Page 2
F4 = ( B + B)( AB + A BC )
F4 = (1)( AB + ABC )
F4 = AB(1 + C )
F4 = AB(1)
F4 = AB
5.
(
)
F5 = X + Y (X + Y )
F5 = ( X + Y )( X + Y )
F5 = XX + XY + X Y + YY
F5 = X + XY + X Y + 0
F5 = X + X (Y + Y )
F5 = X + X (1)
F5 = X + X
F5 = X
6.
(
)
F6 = J K + J + K L + J K
F6 = JK + ( J + K ) L + JK
F6 = JK + J L + K L + JK
F6 = JK + J L + K L
F6 = JK + J L( K + K ) + K L
F6 = JK + J KL + J K L + K L
F6 = JK + J KL + K L( J + 1)
F6 = JK + J KL + K L(1)
F6 = JK + J KL + K L
F6 = K ( J + J L) + K L
F6 = K ( J + L) + K L
F6 = JK + KL + K L
F6 = JK + L( K + K )
F6 = JK + L(1)
F6 = JK + L
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Digital Electronics Activity 2.1.4 Circuit Simplification: Boolean Algebra – Page 3
7.
F7 = R S + R (S + T ) + S (S + U)
F 7 = RS + R( S + T ) + S ( S + U )
F 7 = RS + RS + RT + S S + SU
F 7 = RS + RT + 0 + SU
F 7 = RS + RT + SU
8.
(
)(
)
F8 = N + N M N + N M (N + M)
F 8 = ( N + N M )( N + NM )( N + M )
F 8 = ( N + M )( N + M )( N + M )
F 8 = ( N N + NM + N M + MM )( N + M )
F 8 = (0 + MN + M N + M )( N + M )
F 8 = ( M + MN + M N )( N + M )
F 8 = ( M + M N )( N + M )
F 8 = ( M )( N + M )
F 8 = MN + MM
F 8 = MN + M
F8 = M
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.
AB+ AC+BC= A C+BC+ AB
commutative
10.
(D E)(F G) = D (E F) G
associative
11.
((R + S) + T) + U = (R + S) + (T + U)
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Digital Electronics Activity 2.1.4 Circuit Simplification: Boolean Algebra – Page 4
associative
12.
(J + K )(L + M) = J L + J M + K L + K M
distributive
13.
(
)
R ST +S V =RST +RS V
distributive
Conclusion
1. Describe the process that you would use to simplify a logic expression using Boolean
algebra.
Use it to organize then use it to simplify
2. How do you know when you are finished simplifying and have arrived at the simplest
equation?
When there are no mor theorems left you are done
3. Other than using Boolean algebra, how could you prove that two circuits are
equivalent?
Compare truth tables
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?
You would save $300,000 a year
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Digital Electronics Activity 2.1.4 Circuit Simplification: Boolean Algebra – Page 5