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# Karnaugh map - PowerPoint PPT Presentation

Karnaugh map . Introduction Strategy for Minimization Minimization of Product-of-Sums Forms Minimization of More Complex Expressions Don\'t care Terms . Introduction. Why karnaugh map Example (With Boolean algebra). W = A + . B = A . ( B + ) + . B

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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

Introduction
• Why karnaugh map
• Example (With Boolean algebra)

W = A + . B

= A . ( B + ) + . B

= A . B + A . + . B

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

= A + B

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

As it can be seen, each box of the Karnaugh map corresponds to a row of the truth table and has been numbered accordingly

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

Truth Table

Karnaugh Map

A

0

1

B

0

1

W

This form of representing w in the following example is called a Sum of Product (SOP)

Which will be define in next slides

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

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

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

Strategy for Minimization
• Terminology
• Minimization Procedure
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
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 POScontains all of the variables of the function
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

Karnaugh Map

AB

0 01 11 10

C

0

2

0

1

3

1

7

5

6

4

W =

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

Karnaugh Map

AB

00 01 11 10

C

0

2

1

0

3

1

5

7

6

4

W =

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

Strategy for Minimization
• Terminology
• Minimization Procedure