aljabar boolean, penyederhanaan logika dan peta...

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)


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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


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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


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ACDABCDBA

CBA

CBCA

CBA

CBA

Try This:

The Standard SOP FormDCBADBACBA

DD CC

CCDBA

DDCBA


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CCDBA

DDCBA

DCBADCBACDBADCBADCBA

The Products-of-Sum (POS)FormWhen two or more sum terms are multiplied.

CACBABA

CBABA


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CACBABA

CBABA

The Standard POS Form DCBADCBCBA

DD AA

AADCB

DDCBA

Rule 12!


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AADCB

DDCBA

DCBADCBADCBADCBADCBA

Boolean Expressionand Truth Table


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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


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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


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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


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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


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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


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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


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The 4-Variable K-Map


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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


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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


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Solution:

Example:Map the following standard SOP expression on a K-Map:

DCBADCBADCABABCDDCABDCBACDBA


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Solution:

Exercise:

CBACBABCA

Map the following standard SOP expression on a K-Map:


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ABCDDCABDABCDBCA

Answer:


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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


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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


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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.


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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


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DCDCACAB

1 DC


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Example: Use a K-Map to minimize thefollowing standard SOP expression

CBACBACBABCACBA


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CAB

Example: Use a K-Map to minimize thefollowing standard SOP expression

DCBADABCDBCADCBACDBACBDBADCABDCBADCB


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CBD

Mapping Directly from a TruthTable


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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


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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


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Exercise: Use K-Map to find theminimum SOP from

CBBCACBBCA

CBACBACBA

1

2


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Thank You“Gagal Setelah Mencuba Seribu Kali

Lebih Baik Daripada Tidak PernahMencuba. Keperitan dan Kepayahan

Adalah Jalan Menuju Kebenaran”


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“Gagal Setelah Mencuba Seribu KaliLebih Baik Daripada Tidak Pernah

Mencuba. Keperitan dan KepayahanAdalah Jalan Menuju Kebenaran”

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