Karnaugh Maps - Discrete Mathematical Structures - Lecture Slides, Slides of Discrete Mathematics

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Karnaugh Maps

Karnaugh Maps

• K-maps provide a simple approach to reducing

Boolean expressions from a input-output

table.

• The output from the table is used to fill-in the

K-map.

– 1’s are used to create a Sum of Product (SOP)

solution. (min terms)

– 0’s are used to create a Product of Sum (POS)

solution. (max terms)

Min Terms

Min terms are taken directly from the truth tables. Where ever there is a “1”

for an output, F(), we note the min term value and place a “1” in the K-map

corresponding to the min term value of the table.

Min term short hand is often used to replace a full input-output table. The

short hand indicate the variables and the min terms that are “1”.

Example: f(A,B,C) = Σ (1, 5, 7)

Examples

f(A,B,C) = Σ (0, 1, 5, 7)

Input Output min term A B C F(A,B,C) 0 0 0 0 1 1 0 0 1 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 1 6 1 1 0 7 1 1 1 1 Input Output min term A B F(A,B) 0 0 0 1 0 1 1 2 1 0 1 3 1 1

f(A,B) = Σ (1, 2)

K-Map Examples

~B B

A\B 0

~A 0^1

A 1^1

f(A,B) = Σ (0, 3)

Reducing a Boolean expression using K-map

1. Identify min terms (from table or function form)

2. Fill-in appropriate K-map.

3. Group min terms in largest grouping using 4-neighbor rule.

1. a min term is a number if it is either to the right, left, top, or bottom.

2. K-map edges are connected as neighbors.

4. Write out the groupings as the reduced expression (circuit).

f(A,B) = ~A^~B v A^B

K-Map Examples

~B B

A\B 0

~A 0^1

A 1^1

f(A,B) = Σ (0, 2) f(A,B) = ~B

~B B

A\B 0

~A 0

A 1^1

f(A,B) = Σ (2, 3) f(A,B) = A

~B B

A\B 0

~A 0^1

A 1^1

f(A,B) = Σ (0, 4) f(A,B) = B v A

K-Map Examples

f(A,B,C,D) = Σ (5, 7, 13, 15)

~C ~C / D C / D C

AB\CD 00 01 11

~A 00

~A / B 01^1

A / B 11^1

A 10

f(A,B,C) = B^D

f(A,B,C,D) = Σ (0,1,2,3,8,9,10,11)

~C ~C / D C / D C

AB\CD 00 01 11

~A 00^1 1 1

~A / B 01

A / B 11

A 10^1 1 1

f(A,B,C) = ~B

K-Map Examples

f(A,B,C,D) = Σ (0,1,2,8,9,10,15)

~C ~C / D C / D C

AB\CD 00 01 11

~A 00^1 1

~A / B 01

A / B 11^1

A 10^1 1

f(A,B,C) = ~B^~C v ~B^~D v A^B^C^D