Aljabar Boolean, Penyederhanaan Logika dan Peta

Aljabar Boolean,
Penyederhanaan Logika
dan Peta Karnaugh
ENDY SA
Program Studi Teknik Elektro
Fakultas Teknik
Universitas Muhammadiyah Prof. Dr. HAMKA
Program Studi T. Elektro
FT - UHAMKA
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Standard Forms of
Boolean Expressions
Sum of Product (SOP)
Product of Sum (POS)
Program Studi T. Elektro
FT - UHAMKA
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The Sum-of-Products (SOP)
Form
When two or more product terms are summed by
Boolean addition
AB  ABC
ABC  CDE  BC D
Program Studi T. Elektro
FT - UHAMKA
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Conversion of a General
Expression to SOP Form
Any logic expression can be change into SOP form by
applying Boolean Algebra techniques
AB  CD   AB  ACD
Try This:
A  B   C


 A B C
  A  B C
 AC  B C
Program Studi T. Elektro
FT - UHAMKA
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The Standard SOP Form
ABC  ABD  ABC D
D
D
C C
 
A BD  C  C 
ABC  D  D
ABC D  ABC D  ABCD  ABC D  ABC D
Program Studi T. Elektro
FT - UHAMKA
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The Products-of-Sum (POS)
Form
When two or more sum terms are multiplied.
 A  B  A  B  C 
 A  B  A  B  C  A  C 
Program Studi T. Elektro
FT - UHAMKA
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The Standard POS Form
A  B  C B  C  D A  B  C  D 
D
D
A
Rule 12!
A
A
B
 D D
D  A A
 B  C
 C 
A  B  C  D A  B  C  D A  B  C  D A  B  C  D A  B  C  D 
Program Studi T. Elektro
FT - UHAMKA
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Boolean Expression
and Truth Table
Program Studi T. Elektro
FT - UHAMKA
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Converting SOP to Truth Table
 Examine each of the products to determine where
the product is equal to a 1.
 Set the remaining row outputs to 0.
Program Studi T. Elektro
FT - UHAMKA
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Converting POS to Truth Table
 Opposite process from the SOP expressions.
 Each sum term results in a 0.
 Set the remaining row outputs to 1.
Program Studi T. Elektro
FT - UHAMKA
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Converting from Truth Table to
SOP and POS
Inputs
Output
A
B
C
X
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
1
Program Studi T. Elektro
FT - UHAMKA
X  ABC  ABC  ABC  ABC



X  A  B  C  A  B  C A  B  C A  B  C
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
The Karnaugh Map
Program Studi T. Elektro
FT - UHAMKA
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The Karnaugh Map



Provides a systematic method for simplifying
Boolean expressions
Produces the simplest SOP or POS
expression
Similar to a truth table because it presents all
of the possible values of input variables
Program Studi T. Elektro
FT - UHAMKA
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The 3-Variable K-Map
Program Studi T. Elektro
FT - UHAMKA
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The 4-Variable K-Map
Program Studi T. Elektro
FT - UHAMKA
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K-Map SOP Minimization


Program Studi T. Elektro
FT - UHAMKA
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A 1 is placed on the KMap for each product
term in the expression.
Each 1 is placed in a
cell corresponding to
the value of a product
term
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Example:
Map the following standard SOP expression on a K-Map:
ABC  ABC  ABC  ABC
Solution:
Program Studi T. Elektro
FT - UHAMKA
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Example:
Map the following standard SOP expression on a K-Map:
ABCD  ABC D  ABC D  ABCD  ABC D  ABC D  ABC D
Solution:
Program Studi T. Elektro
FT - UHAMKA
Slide - 6
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Exercise:
Map the following standard SOP expression on a K-Map:
ABC  A BC  A BC
ABC D  ABC D  AB C D  ABCD
Program Studi T. Elektro
FT - UHAMKA
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Answer:
Program Studi T. Elektro
FT - UHAMKA
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K-Map Simplification of SOP
Expressions




A group must contain either 1, 2, 4, 8 or 16 cells.
Each cell in group must be adjacent to one or more
cells in that same group but all cells in the group do
not have to be adjacent to each other
Always include the largest possible number 1s in a
group in accordance with rule 1
Each 1 on the map must be included in at least one
group. The 1s already in a group can be included in
another group as long as the overlapping groups
include noncommon 1s
Program Studi T. Elektro
FT - UHAMKA
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To maximize the size of the groups and to minimize the number of groups
Example: Group the 1s in each KMaps
Program Studi T. Elektro
FT - UHAMKA
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Determining the minimum SOP
Expression from the Map

Groups the cells that have 1s. Each group of
cells containing 1s create one product term
composed of all variables that occur in only
one form (either uncomplemented or
complemented) within the group. Variable
that occurs both uncomplemented and
complemented within the group are
eliminated. These are called contradictory
variables.
Program Studi T. Elektro
FT - UHAMKA
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Example: Determine the product term for the KMap below and write the resulting minimum
SOP expression
B  AC  AC D
CD
1
B  AC  AC D  C D
Program Studi T. Elektro
FT - UHAMKA
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Program Studi T. Elektro
FT - UHAMKA
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Example: Use a K-Map to minimize the
following standard SOP expression
A BC  ABC  A BC  A BC  A BC
B  AC
Program Studi T. Elektro
FT - UHAMKA
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Example: Use a K-Map to minimize the
following standard SOP expression
BC D  ABC D  ABC D  ABCBD  ABCD  ABC D  ABC D  ABC D  ABC D
D  BC
Program Studi T. Elektro
FT - UHAMKA
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Mapping Directly from a Truth
Table
Program Studi T. Elektro
FT - UHAMKA
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Don’t Care (X) Conditions


A situation arises in which input variable
combinations are not allowed
Don’t care terms either a 1 or a 0 may be
assigned to the output
Program Studi T. Elektro
FT - UHAMKA
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Don’t Care (X) Conditions
Example of the use of “don’t
care” conditions to simplify an
expression
Program Studi T. Elektro
FT - UHAMKA
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Exercise: Use K-Map to find the
minimum SOP from
1
A B C  A BC  A BC
ABC  B C   ABC  B C 
Program Studi T. Elektro
FT - UHAMKA
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Thank You
“Gagal Setelah Mencuba Seribu Kali
Lebih Baik Daripada Tidak Pernah
Mencuba. Keperitan dan Kepayahan
Adalah Jalan Menuju Kebenaran”
Program Studi T. Elektro
FT - UHAMKA
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