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KARNAUGH MAP

• Introduction• Strategy for Minimization• Minimization of Product-of-Sums Forms• Minimization of More Complex Expressions• Don't care Terms

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Introduction

• Why karnaugh map

• Example (With Boolean algebra)

W = A + . B = A . ( B + ) + . B = A . B + A . + . B = A . ( B + ) + B ( A+ ) = A + B

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Introduction ( cont. )• Using Boolean algebra for minimization causes it’s own problem because of it mainly being a trial and error process, and we can almost never be sure that we have reached a minimal representation.

• If we can form a graphical notation for our Boolean algebra the insight need for the minimization will be less vital in solving the problems.

We can come close to our aim by using a graphical notation named Karnaugh Map that

will be defined in next slides

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Introduction ( cont. )• Comparing Karnaugh Map and Boolean Algebra

A B W

0 0 0

0 1 1

1 0 1

1 1 1

Truth Table

0 1

1 1

BA

0

0

1

1

W

W = . B + A . + A . B =

W = . B + A . B + A . + A .B=

W= B ( + A ) + A ( + B ) = A + B

Karnaugh Map

As it can be seen, each box of the Karnaugh map

corresponds to a row of the truth table and has

been numbered accordingly

This form of representing w in the following

example is called a Sum of Product

(SOP)Which will be define in next

slides

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Strategy for Minimization

• Terminology

• Minimization Procedure

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Terminology• Implicant : Product term that implies function

• Prime Implicant : An Implicant that is not completely covered by any other Implicant but itself

• Essential prime Implicant : A prime Implicant that has a minter not covered by any other prime Implicant

• Product term : An and expression

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Terminology • Minterm : We define a Minterm to be a product that

contains all variables of that particular switching function in either complemented or non-complemented form

• Maxterm : We define a Maxterm to be a sum that contains all variables of that particular switching function in either complemented or non-complemented form

• Standard SOP(Sum Of Products) : In standard SOP, the products are obtained directly from the Karnaugh map or truth table, so the SOP contains all of the variables of the function

• Standard POS(Product Of Sums) : In standard POS, the products are obtained directly from the Karnaugh map or truth table, so the POS contains all of the variables of the function

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Terminology ( cont. )• A simpler shorthand form of representing a SOP is to use the number of the Minterms that appear in that representation. In the following example for instance we could have written

0 0 0 1

1 1 0 1

0 01 11 10

0

1

Karnaugh Map

1 3 2

4 5 7 6

0

ABC

W =

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Terminology ( cont. )• Sometimes writing an expression in a POS form is easier as seen in the following example:

0 1 1 0

1 1 1 1

00 01 11 10

0

1

Karnaugh Map

W =

1 3 2

5 7 64

0

w = (a + b + c) . (+ b + c)

ABC

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Strategy for Minimization• Terminology

• Minimization Procedure