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Karnaugh Maps

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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)

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Min Terms • Canonical representation of a Boolean

expression is in the form of ^ v ~ (AND, OR, NOT). – Example: A^B v ~A^~B v A^~B (AB + AB + AB)

• Candidates for canonical representation are taken from the truth table (input-output).

• Candidates are identified where the output is “1”. (Max Term canonical representation candidates are identified by “0”)

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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)

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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)

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K-Map Tables

• K-map tables are organized based on the number of variables. – Example: showing min terms in italic bold.

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K-Map Examples

~B B

A\B 0 1
~A 0 ** 1
**A 1

**f(A,B) = Σ (0, 3)**

*1*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

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K-Map Examples

~B B

A\B 0 1
~A 0 ** 1
**A 1

*1*f(A,B) = Σ (0, 2) f(A,B) = ~B

~B B

A\B 0 1 ~A 0

A 1 ** 1 1
** f(A,B) = Σ (2, 3) f(A,B) = A

~B B

A\B 0 1
~A 0 ** 1
**A 1

*1 1*f(A,B) = Σ (0, 4) f(A,B) = B v A

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K-Map Examples

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
**

~B ~B / C B / C B

A\BC 00 01 11 10
~A 0 ** 1 1
**A 1

*1 1*f(A,B,C) = ~A^~B v A^C

~B ~B / C B / C B

A\BC 00 01 11 10
~A 0 ** 1 1
**A 1

**f(A,B,C) = Σ (0, 2, 4)**

*1*f(A,B,C) = ~A^~C v ~B^~C

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K-Map Examples

f(A,B,C,D) = Σ (5, 7, 13, 15)

~C ~C / D C / D C

AB\CD 00 01 11 10 ~A 00

~A / B 01 ** 1 1
**A / B 11

**A 10**

*1 1*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 10
~A 00 ** 1 1 1 1
**~A / B 01

A / B 11

A 10 *1 1 1 1
*

f(A,B,C) = ~B

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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 10
~A 00 ** 1 1 1
**~A / B 01

A / B 11 ** 1
**A 10

*1 1 1*f(A,B,C) = ~B^~C v ~B^~D v A^B^C^D

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HMWK Due 2/4

1. Build the input-output table from the following min term list of 4-variables: Σ (5, 7, 10, 11, 14, 15)

2. Using a K-map reduce the expression from 1 such that you minimize the number of connectives (AND, OR, NOT). Remember the answer should be in the sum of product form.

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