K map

Shri S’ad Vidya Mandal
Institute Of Technology
BHARUCH - 392001, GUJARAT
[KarnaughMap]
Presented By:
Gauresh Mehta ,
Shivang Mehta.
Guided By:-
Nishant N. Parmar

What is K-map ?
 K-maps provide an alternative way of simplifying
logic circuits.
 Instead of using Boolean algebra simplification
techniques, you can transfer logic values from a
Boolean statement or a truth table into a k-map.
 The arrangement of 0's and 1's within the map
helps you to visualize the logic relationships
between the variables and leads directly to a
simplified Boolean statement.

 K-map simplification of Boolean equation can
be done easily and systematically.
 This method is used to given simplification of
two, three or four-variable switching function.
 This method can be used for more than four
variables also but it becomes complicated.
 It gives the most simplified form of the
expression.

Method Of Constructing K-map
 K-maps consists of cell or squares.
 If there are n variable in function, the number of
cell become 2n, so for two variable, no of cell will
be 22 =4.
 For three variable it will 8 and for variables it will
be 16, with the help of K-MAP simplification of
switching function can be given by very easily
and systematically.
 Each cell is represented by binary number or its
decimal number.

 Simplification of logic expression using
Boolean algebra is awkward because:.
 It lacks specific rule to predict the most suitable next
step in the simplification process.
 It is difficult to determine whether the simplest form
has been achieved.
 K-map is a graphical method used to obtained
the most simplified form of an expression in a
standard form (Sum-of-Products or Product-of-
Sums).
 The simplest form of an expression is the one
that has the minimum number of terms with the
least number of literals (variables) in each term.

 By simplifying an expression to the one that uses
the minimum number of terms, we ensure that the
function will be implemented with the minimum
number of gates.
 By simplifying an expression to the one that uses
the least number of literals for each terms, we
ensure that the function will be implemented with
gates that have the minimum number of inputs.

Grouping of K-map variables
There are some rules to follow while we are grouping
the variables in K-maps.
1. The square that contains ‘1’ should be taken in simplifying,
at least once.
2. The square that contains ‘1’ can be considered as many
times as the grouping is possible with it.
3. Group shouldn’t include any zeros (0).
4. A group should be the as large as possible.
5. Groups can be horizontal or vertical. Grouping of variables
in diagonal manner is not allowed.

Types of k-map
I. Two - variable k-map
II. Three - variable k-map
III. Four - variable k-map

Two Variable K-MAP
 K-map for two variables is show in figure. It has two
rows and two columns, so there are four cells.
 The possible min terms with 2 variables (A and B)
are A.B, A.B’, A’.B and A’.B’.
 The following table shows the positions of all the
possible outputs of 2-variable Boolean function on a
K-map.

 we group the adjacent cells with possible sizes as
2 or 4. In case of larger k-maps, we can group
the variables in larger sizes like 8 or 16.
 The groups of variables should be in rectangular
shape, that means the groups formed by either
vertically or horizontally.
 Diagonal shaped or “L” shaped groups are not
allowed.

Example
First, let’s construct the truth
table for the given equation
We put 1 at the output terms
given in equation.
In this K-map, we can create 2 groups by following the rules for
grouping,
→ one is by combining (X’, Y) and (X’, Y’) terms and the other
is by combining (X, Y’) and (X’, Y’) terms.
the next step is determining the minimized expression such as by
taking out the common terms from two groups
i.e. X’ and Y’. So the reduced equation will be X’ +Y’.

Three Variable K-MAP
 For a 3-variable Boolean function, there is a possibility
of 8 output min terms. The general representation of all
the min terms using 3-variables is shown below.

 A typical plot of a 3-variable K-map is shown below. It
can be observed that the positions of columns 10 and
11 are interchanged so that there is only change in one
variable across adjacent cells.
 Up to 8 cells can be grouped in case of a 3-variable K-
map.

Example
First, let’s construct the truth
table for the given equation
We put 1 at the output terms
given in equation.
→ The largest group size will be
8 but we can also form the
groups of size 4 and size 2, by
possibility.
→ So the size 4 group is formed
as shown below.

And in both the terms, we have ‘Y’ in common. So the group
of size 4 is reduced as the conjunction Y.
The 2 size group has no common variables, so they are written
with their variables and its conjugates. So the reduced
equation will be Y’ + ZX + Z’X’.

Four Variable K-MAP
There are 16 possible min terms in case of a 4-variable
Boolean function. The general representation of minterms
using 4 variables is shown below.

Example
Simplify the given 4-variable Boolean equation by using k-map.
F (W, X, Y, Z) = (1, 5, 12, 13)
By preparing k-map, we can minimize the given Boolean
equation as
F = W Y’ Z + W ‘Y’ Z

K map

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