Related Articles

# Introduction of K-Map (Karnaugh Map)

In many digital circuits and practical problems, we need to find expressions with minimum variables. We can minimize Boolean expressions of 3, 4 variables very easily using K-map without using any Boolean algebra theorems.

K-map can take two forms:

1. Sum of product (SOP)
2. Product of Sum (POS)

According to the need of problem. K-map is a table-like representation, but it gives more information than the TABLE. We fill a grid of the K-map with 0â€™s and 1â€™s then solve it by making groups.

## Steps to Solve Expression using K-map

1. Select the K-map according to the number of variables.
2. Identify minterms or maxterms as given in the problem.
3. For SOP put 1â€™s in blocks of K-map respective to the minterms (0â€™s elsewhere).
4. For POS put 0â€™s in blocks of K-map respective to the max terms (1â€™s elsewhere).
5. Make rectangular groups containing total terms in power of two like 2,4,8 ..(except 1) and try to cover as many elements as you can in one group.
6. From the groups made in step 5 find the product terms and sum them up for SOP form.

## SOP FORM

### 1. K-map of 3 variables

K-map SOP form for 3 variables

`Z= ?A,B,C(1,3,6,7)  `

From red group we get product termâ€”

`Aâ€™C `

From green group we get product termâ€”

`AB `

Summing these product terms  we get- Final expression (Aâ€™C+AB)

### 2. K-map for 4 variables

K-map 4 variable SOP form

`F(P,Q,R,S)=?(0,2,5,7,8,10,13,15)  `

From red group we get product termâ€”

`QS `

From green group we get product termâ€”

`Qâ€™Sâ€™ `

Summing  these product terms  we get- Final expression (QS+Qâ€™Sâ€™).

## POS FORM

### 1. K-map of 3 variables

K-map 3 variable POS form

`F(A,B,C)=?(0,3,6,7)`

From red group we find terms

`A    B   `

Taking complement of these two

`A'     B'    `

Now sum up them

`(A' + B') `

From brown group we find terms

`B   C `

Taking complement of these two terms

`Bâ€™  Câ€™ `

Now sum up them

`(Bâ€™+Câ€™) `

From yellow group we find terms

`A' B' Câ€™ `

Taking complement of these two

`A B C `

Now sum up them

`(A + B + C) `

We will take product of these three terms : Final expression –

`(A' + Bâ€™) (Bâ€™ + Câ€™) (A + B + C) `

### 2. K-map of  4 variables

K-map 4 variable POS form

`F(A,B,C,D)=?(3,5,7,8,10,11,12,13) `

From green group we find terms

`Câ€™  D  B `

Taking their complement and summing them

`(C+Dâ€™+Bâ€™) `

From red group we find terms

`C  D  Aâ€™ `

Taking their complement and summing them

`(Câ€™+Dâ€™+A) `

From blue  group we find terms

`A  Câ€™  Dâ€™ `

Taking their complement and summing them

`(Aâ€™+C+D) `

From brown  group we find terms

`A  Bâ€™  C `

Taking their complement and summing them

`(Aâ€™+B+Câ€™) `

Finally we express these as product –

`(C+Dâ€™+Bâ€™).(Câ€™+Dâ€™+A).(Aâ€™+C+D).(Aâ€™+B+Câ€™) `

PITFALL  *Always remember POS ? (SOP)’

*The correct form is (POS of F)=(SOP of F’)’

Quiz on K-MAP

Previous
Next