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Karnaugh Maps

BCS Certificate Level

Computer & Network Technology

Week 9

Nundloll Arvin

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Karnaugh Maps

K-Maps are a convenient way to simplify Boolean

Expressions.

They can be used for up to 4 or 5 variables.

They are a visual representation of a truth table. Expression are most commonly expressed in

sum of products form.

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Truth table to K-Map

A B P

0 0 1

0 1 1

1 0 0

1 1 1

BA 0 1

0 1 1

1 1

minterms are represented by a 1

in the corresponding location inthe K map.

The expression is:

A.B + A.B + A.B

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K-Maps

Adjacent 1s can be paired off

Any variable which is both a 1 and a zero in this

pairing can be eliminated

Pairs may be adjacent horizontally or vertically

BA 0 1

0 1 1

1 1

a pair

another pair

B is eliminated,

leaving A as the term

A is eliminated,

leaving B as the termThe expressionbecomes A + B

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K-Maps (Cont.)

Two Variable K-Map

A B C P

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

A.B.C + A.B.C + A.B.C

BC

A 00 01 11 100 1

1 1 1

One square filled in for each

minterm.Notice the code sequence:

00 01 11 10 a Gray code.

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Grouping the Pairs

BC

A 00 01 11 10

0 1

1 1 1

equates to B.C as Ais eliminated.

Here, we can wrap

around and this pairequates to A.C as Bis eliminated.

Our truth table simplifies toA.C + B.C as before.

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Groups of 4

BC

A 00 01 11 10

0 1 11 1 1

Groups of 4 in a block can be used to eliminate two

variables:

The solution is B because it is a 1 over the whole block

(vertical pairs) = BC + BC = B(C + C) = B.

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Karnaugh Maps

Three Variable K-Map

Extreme ends of same row considered adjacent

ABC 00 01 11 10

0

1

A.B.C A.B.C A.B.C A.B.C

A.B.C A.B.C A.B.C A.B.C

0010

A.B.C

A.

B.

C

A.B.C

A.

B.

C

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Karnaugh Maps

Three Variable K-Map example

X A.B.CA.B.CA.B.CA.B.C

A BC 00 01 11 10

0

1

X =

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The Block of 4, again

ABC 00 01 11 10

0 1 1

11 1

X = C

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Karnaugh Maps

Two Variable K-Map

A B C P

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 0

1 0 0 1

1 0 1 0

1 1 0 1

1 1 1 0

A.B.C + A.B.C + A.B.C

AB

C 00 01 11 10

0 1 1 1

1

There is more than one way to label the axes of the

K-Map, some views lead to groupings which are

easier to see.

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Dont Care States Sometimes in a truth table it does

not matter if the output is a zero or a

one

Traditionally marked with an x.

We can use these as 1s if it helps.

AB

C00 01 11 10

0 1 11 x x 1

A B C P

0 0 0 0

0 0 1 x0 1 0 1

0 1 1 x

1 0 0 1

1 0 1 01 1 0 1

1 1 1 0

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Karnaugh Maps

Four Variable K-Map

Four corners adjacent

ABCD 00 01 11 10

00

01

11

10

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D A.B.C.D A.B.C.D A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

A.B.C.D

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Karnaugh Maps

Four Variable K-Map exampleFA.B.C.DA.B.C.D+A.B.C.DA.B.C.DA.B.C.DA.B.C.DA.B.C.D

ABCD 00 01 11 10

0001

11

10

F =

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ABCD 00 01 11 10

00 1 101 1 1

11

10 1 1

Karnaugh Maps

Four Variable K-Map solutionFA.B.C.DA.B.C.D+A.B.C.DA.B.C.DA.B.C.DA.B.C.DA.B.C.D

F = B.D + A.C

1

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Product-of-SumsWe have populated the maps with 1s using

sum-of-products extracted from the truth table.

We can equally well work with the 0s

AB

C 00 01 11 10

0 1 1 1

1 1

A B C P

0 0 0 00 0 1 0

0 1 0 1

0 1 1 1

1 0 0 11 0 1 0

1 1 0 1

1 1 1 0

AB

C 00 01 11 10

0 0

1 0 0 0

P = (A + B).(A + C)

P = A.B + A.C equivalent

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Inverted K Maps

In some cases a better simplification can beobtained if the inverse of the output is considered

i.e. group the zeros instead of the ones

particularly when the number and patterns of zeros is

simpler than the ones