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Karnaugh Map Simplification Guide

The document discusses Karnaugh maps, which provide a pictorial method for simplifying Boolean expressions. Some key points: - Karnaugh maps arrange truth table rows into a two-dimensional grid, allowing common factors to be grouped visually. - They can be used to simplify expressions with 2, 3, 4, or 5 variables. Larger maps have more cells corresponding to more variables/terms. - Groups in a Karnaugh map must follow rules like containing only 1s, being a power of 2 cells, and not including diagonals or zeros. - The document provides examples of 3 and 4 variable Karnaugh maps and reviews the rules for grouping cells to minimize expressions.

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16 views16 pages

Karnaugh Map Simplification Guide

The document discusses Karnaugh maps, which provide a pictorial method for simplifying Boolean expressions. Some key points: - Karnaugh maps arrange truth table rows into a two-dimensional grid, allowing common factors to be grouped visually. - They can be used to simplify expressions with 2, 3, 4, or 5 variables. Larger maps have more cells corresponding to more variables/terms. - Groups in a Karnaugh map must follow rules like containing only 1s, being a power of 2 cells, and not including diagonals or zeros. - The document provides examples of 3 and 4 variable Karnaugh maps and reviews the rules for grouping cells to minimize expressions.

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© © All Rights Reserved
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Digital Logic Design

Lecture#4
Instructor: Syed Muhammad Usman

Slides Courtesy: Ms. Aunsia Khan Lecturer, SZABIST


Karnaugh Map

 A Karnaugh map provides a pictorial method of grouping


together expressions with common factors and therefore
eliminating unwanted variables
Karnaugh Map

 Simplification of Boolean Expressions


 Doesn’t guarantee simplest form of expression
 Terms are not obvious
 Skills of applying rules and laws

 K-map provides a systematic method


 An array of cells
 Used for simplifying 2, 3, 4 and 5 variable
expressions
3-Variable K-map

 Used for simplifying 3-variable expressions


 K-map has 8 cells representing the 8 minterms and 8
maxterms
 K-map can be represented in row format or column
format
4-Variable K-map

 Used for simplifying 4-variable expressions


 K-map has 16 cells representing the 16 minterms and
16 maxterms
 A 4-variable K-map has a square format
Karnaugh Map

 The Karnaugh map can also be described as a special


arrangement of a truth table.

 The diagram below illustrates the correspondence


between the Karnaugh map and the truth table for the
general case of a two variable problem.
Karnaugh Map

The values inside the squares are copied from the output
column of the truth table, therefore there is one square in
the map for every row in the truth table
Karnaugh Map Rules

 The Karnaugh map uses the following rules for the


simplification of expressions by grouping together
adjacent cells containing ones
 Groups may not include any cell containing a zero

 Groups may be horizontal or vertical, but not diagonal.


Karnaugh Map Rules
 Groups must contain 1, 2, 4, 8, or in general 2n cells.
That is if n = 1, a group will contain two 1's since 21 = 2.
If n = 2, a group will contain four 1's since 22 = 4.
Karnaugh Map Rules

 Each group should be as large as possible.

 Each cell containing a one must be in at least one group.

9
Karnaugh Map Rules

 Groups may overlap.


Karnaugh Map Rules

 Groups may wrap around the table. The leftmost cell in a row
may be grouped with the rightmost cell and the top cell in a
column may be grouped with the bottom cell.
Karnaugh Map Rules

 There should be as few groups as possible, as long as this


does not contradict any of the previous rules.
Summary

i. No zeros allowed.
ii. No diagonals.
iii. Only power of 2 number of cells in each group.
iv. Groups should be as large as possible.
v. Every one must be in at least one group.
vi. Overlapping allowed.
vii. Wrap around allowed.
viii. Fewest number of groups possible.
Practice Problems

 Minimize the following problems using the Karnaugh maps


method.
15 Practice Problems

Minimize the following problems using the Karnaugh maps


method.

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