- THE THREE VARIABLE KARNAUGH MAP. - THE FOUR VARIABLE KARNAUGH MAP. - Karnaugh Map Simplification of SOP Expressions. - KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION 5-VARIABLE K-MAPS. - DON'T CARE CONDITIONS.

1. CHAPTER 4
THE KARNAUGH-VEITCHMAP

2. Contents






THE THREE VARIABLE KARNAUGH MAP
THE FOUR VARIABLE KARNAUGH MAP
Karnaugh Map Simplification of SOP Expressions
KARNAUGH MAP PRODUCT OF SUM (POS)
SIMPLIFICATION
5-VARIABLE K-MAPS
DON'T CARE CONDITIONS
2

3. THE THREE VARIABLE
KARNAUGH MAP
Karnaugh Maps

4. THE FOUR VARIABLE
KARNAUGH MAP
Karnaugh Maps

5. Karnaugh Map Simplification of SOP
Expressions
 Grouping
the 1s
1. A group must contain either 1, 2, 4, 8,
or 16 cells.
 2. Each cell in a group must be adjacent to
one or more cells in that same group.
 3. Always include the largest possible
number of 1s in a group.
4. Each 1 on the map must be included in at
least one group.


Karnaugh Maps

6. Determining the Minimum SOP
Expression from the Map



1. Group the cells that have 1s.
2. Determine the minimum product terms for each group.
For a 3-variable map:






?
For a 5-variable map


A 1-cell group yields a 3-variable product term
A 2-cell group yields a 2-variable product term
A 4-cell group yields a 1-variable term
An 8-cell group yields a value of 1 for the expression
For a 4-variable map


(1)
(2)
(3)
(4)
?
3. When all the minimum product terms are derived from
the Karnaugh map, they are summed to form the minimum
SOP expression.
Karnaugh Maps

7. EXAMPLE

Simplify the Boolean expression:

F(x,y,z) = Σ (0,1,6,7)
7

8. EXAMPLE:

Simplify the Boolean expression:
F(x,y,z) = Σ (0,2,5,7)
8

9. EXAMPLE:

Group the 1's in each of the following
Karnaugh maps:
9

10. EXAMPLE:
10

11. KARNAUGH MAP PRODUCT OF
SUM (POS) SIMPLIFICATION
If we mark the empty squares by 0's and
combine them into valid adjacent squares,
we obtain a simplified expression of the
complement of the function, i.e., of F'.
 The complement of F' gives us back the
function F.
 Because of Demorgan's theorem, the
function so obtained is automatically in the
product of sums form.

11

12. EXAMPLE


Simplify the following Boolean function in (a) sum
of products and (b) product of sums.
F(w,x,y,z) = Σ (0,1,2,3,10,11,14)
12

13. EXAMPLE




Use a Karnaugh map to
minimize the following
POS expression.
(x+y+z)(w+x+y'+z)
(w'+x+y+z')
(w+x'+y+z)
(w'+x'+y+z)
(w+x+y+z)(w'+x+y+z)
(w+x+y'+z)
(w'+x+y+z')
(w+x'+y+z)
(w'+x'+y+z)
=Π(0,8,2,9,4,12)
13

14. 5-VARIABLE K-MAPS (1/2)

K-maps of more than 4 variables are more
difficult to use because the geometry (hypercube
configurations) for combining adjacent squares
becomes more involved.

For 5 variables, e.g. v, w, x, y, z, we need 25 =
32 squares.

Each square has 5 neighbours.
ELC224C
Karnaugh Maps

15. 5-VARIABLE K-MAPS (2/2)

Organized as two 4-variable K-maps. One for v'
and the other for v.
v'
wx
v
y
yz
00
01
11
m0
m1
m3
0 m4
1
m5
m7
00
m12
w
wx
10
00
01
11
10
m2
00 m16
m17
m19
m18
m6
0 m20
1
m21
m23
m22
m28
m29
m31
m30
m24
m25
m27
m26
x
m13
m15
m14
w
11
m8
m9
m11
y
yz
11
m10
10
x
10
z
z
Corresponding squares of each map are adjacent.
Can visualise this as one 4-variable K-map being on
TOP of the other 4-variable K-map.
ELC224C
Karnaugh Maps

16. DON'T CARE CONDITIONS





Sometimes a situation arises in which some input
variable combinations are not allowed.
Since these unallowed states will never occur in
an application involving the BCD code, they can
be treated as "don't care" terms
for each "don't care" term, an X is placed in the
cell.
When grouping the 1's, Xs can be treated as 1's
to make a larger grouping or as 0s if they cannot
be used to advantage.
Be careful do not make a group entirely of
x's.
16

17. Example
x
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
Inputs
y
y
0
0
0
0
0
1
0
1
1
0
1
0
1
1
1
1
0
0
0
0
0
1
0
1
1
0
1
0
1
1
1
1
w
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Output
Y
0
0
0
0
0
0
0
1
1
1
X
X
X
X
X
X
17

18. Example



Simplify the following
Boolean function F,
where d represents the
set of do not care
conditions
F(w,x,y,z) = Σ
(0,1,2,8,10,11)
d(w,x,y,z) = Σ
(4,6,12,13)
18