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Aljabar Boolean,Penyederhanaan Logika
dan Peta Karnaugh
Program Studi T. ElektroFT - UHAMKA
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ENDY SAENDY SAProgram Studi Teknik Elektro
Fakultas TeknikUniversitas Muhammadiyah Prof. Dr. HAMKA
Standard Forms ofBoolean Expressions
Sum of Product (SOP)Product of Sum (POS)
Program Studi T. ElektroFT - UHAMKA
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Sum of Product (SOP)Product of Sum (POS)
The Sum-of-Products (SOP)Form
DCBCDEABC
ABCAB
When two or more product terms are summed byBoolean addition
Program Studi T. ElektroFT - UHAMKA
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DCBCDEABC
ABCAB
Conversion of a GeneralExpression to SOP Form
Any logic expression can be change into SOP form byapplying Boolean Algebra techniques
Program Studi T. ElektroFT - UHAMKA
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ACDABCDBA
CBA
CBCA
CBA
CBA
Try This:
The Standard SOP FormDCBADBACBA
DD CC
CCDBA
DDCBA
Program Studi T. ElektroFT - UHAMKA
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CCDBA
DDCBA
DCBADCBACDBADCBADCBA
The Products-of-Sum (POS)FormWhen two or more sum terms are multiplied.
CACBABA
CBABA
Program Studi T. ElektroFT - UHAMKA
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CACBABA
CBABA
The Standard POS Form DCBADCBCBA
DD AA
AADCB
DDCBA
Rule 12!
Program Studi T. ElektroFT - UHAMKA
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AADCB
DDCBA
DCBADCBADCBADCBADCBA
Boolean Expressionand Truth Table
Program Studi T. ElektroFT - UHAMKA
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Examine each of the products to determine wherethe product is equal to a 1. Set the remaining row outputs to 0.
Converting SOP to Truth Table
Program Studi T. ElektroFT - UHAMKA
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Opposite process from the SOP expressions. Each sum term results in a 0. Set the remaining row outputs to 1.
Converting POS to Truth Table
Program Studi T. ElektroFT - UHAMKA
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Converting from Truth Table toSOP and POS
Inputs Output
A B C X
0 0 0 0
0 0 1 0ABCCABCBABCAX
Program Studi T. ElektroFT - UHAMKA
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0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
CBACBACBACBAX
The Karnaugh Map
Program Studi T. ElektroFT - UHAMKA
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The Karnaugh Map
Provides a systematic method for simplifyingBoolean expressions
Produces the simplest SOP or POSexpression
Similar to a truth table because it presents allof the possible values of input variables
Program Studi T. ElektroFT - UHAMKA
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Provides a systematic method for simplifyingBoolean expressions
Produces the simplest SOP or POSexpression
Similar to a truth table because it presents allof the possible values of input variables
The 3-Variable K-Map
Program Studi T. ElektroFT - UHAMKA
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The 4-Variable K-Map
Program Studi T. ElektroFT - UHAMKA
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K-Map SOP Minimization A 1 is placed on the K-
Map for each productterm in the expression.
Each 1 is placed in acell corresponding tothe value of a productterm
Program Studi T. ElektroFT - UHAMKA
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A 1 is placed on the K-Map for each productterm in the expression.
Each 1 is placed in acell corresponding tothe value of a productterm
Example:Map the following standard SOP expression on a K-Map:
ABCCABCBACBA
Program Studi T. ElektroFT - UHAMKA
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Solution:
Example:Map the following standard SOP expression on a K-Map:
DCBADCBADCABABCDDCABDCBACDBA
Program Studi T. ElektroFT - UHAMKA
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Solution:
Exercise:
CBACBABCA
Map the following standard SOP expression on a K-Map:
Program Studi T. ElektroFT - UHAMKA
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ABCDDCABDABCDBCA
Answer:
Program Studi T. ElektroFT - UHAMKA
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K-Map Simplification of SOPExpressions
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 donot have to be adjacent to each other
Always include the largest possible number 1s in agroup in accordance with rule 1
Each 1 on the map must be included in at least onegroup. The 1s already in a group can be included inanother group as long as the overlapping groupsinclude noncommon 1s
Program Studi T. ElektroFT - UHAMKA
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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 donot have to be adjacent to each other
Always include the largest possible number 1s in agroup in accordance with rule 1
Each 1 on the map must be included in at least onegroup. The 1s already in a group can be included inanother group as long as the overlapping groupsinclude noncommon 1s
To maximize the size of the groups and to minimize the number of groups
Example: Group the 1s in each K-Maps
Program Studi T. ElektroFT - UHAMKA
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Determining the minimum SOPExpression from the Map
Groups the cells that have 1s. Each group ofcells containing 1s create one product termcomposed of all variables that occur in onlyone form (either uncomplemented orcomplemented) within the group. Variablethat occurs both uncomplemented andcomplemented within the group areeliminated. These are called contradictoryvariables.
Program Studi T. ElektroFT - UHAMKA
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Groups the cells that have 1s. Each group ofcells containing 1s create one product termcomposed of all variables that occur in onlyone form (either uncomplemented orcomplemented) within the group. Variablethat occurs both uncomplemented andcomplemented within the group areeliminated. These are called contradictoryvariables.
Example: Determine the product term for the K-Map below and write the resulting minimumSOP expression
DCACAB
Program Studi T. ElektroFT - UHAMKA
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DCDCACAB
1 DC
Program Studi T. ElektroFT - UHAMKA
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Example: Use a K-Map to minimize thefollowing standard SOP expression
CBACBACBABCACBA
Program Studi T. ElektroFT - UHAMKA
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CAB
Example: Use a K-Map to minimize thefollowing standard SOP expression
DCBADABCDBCADCBACDBACBDBADCABDCBADCB
Program Studi T. ElektroFT - UHAMKA
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CBD
Mapping Directly from a TruthTable
Program Studi T. ElektroFT - UHAMKA
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Don’t Care (X) Conditions
A situation arises in which input variablecombinations are not allowed
Don’t care terms either a 1 or a 0 may beassigned to the output
Program Studi T. ElektroFT - UHAMKA
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A situation arises in which input variablecombinations are not allowed
Don’t care terms either a 1 or a 0 may beassigned to the output
Don’t Care (X) ConditionsExample of the use of “don’tcare” conditions to simplify anexpression
Program Studi T. ElektroFT - UHAMKA
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Exercise: Use K-Map to find theminimum SOP from
CBBCACBBCACBACBACBA
1
2
Program Studi T. ElektroFT - UHAMKA
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Thank You“Gagal Setelah Mencuba Seribu Kali
Lebih Baik Daripada Tidak PernahMencuba. Keperitan dan Kepayahan
Adalah Jalan Menuju Kebenaran”
Program Studi T. ElektroFT - UHAMKA
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“Gagal Setelah Mencuba Seribu KaliLebih Baik Daripada Tidak Pernah
Mencuba. Keperitan dan KepayahanAdalah Jalan Menuju Kebenaran”