K-map and its types. Rules for simplifying K-map

1. KARNAUGH MAP (K-MAP)
NAME : ADEEN ALI
BSIT 3RD SEMESTER

2. K-MAP INVENTOR:
Maurice Karnaugh (October 4, 1924 (age 96 years) ) introduced
Karnaugh map in 1953. The Karnaugh map (KM or K-map) is a
method of simplifying Boolean algebra expressions.
A Karnaugh map (K-map) is a pictorial method used to
minimize Boolean expressions without having to use Boolean
algebra theorems and equation manipulations. A K-map can be
thought of as a special version of a truth table .

3. KARNAUGH MAP:
• Karnaugh map is an array of cells in which each cell represents a
binary value of input variables.
• The cells are managed in a way so that simplification of a given
expression is simply a matter of properly grouping the cells.
• Karnaugh maps can be used for expression with two, three, four and
five variables.

4. STEPS TO SOLVE EXPRESSION USING K-MAP:
1. Select K-map according to the number of variables.
2. Identify minters or midterms as given in problem.
3. For SOP put 1’s in blocks of K-map respective to the minters (0’s
elsewhere).
4. For POS put 0’s in blocks of K-map respective to the midterms(1’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.

5. NO. CELL IN K-MAP:
The number of cells in K-MAP is equal to the total number of possible
input variable combinations as is the number of row in a truth table.
• For two variable, the number of cell are 22 =4.
• For three variable, the number of cell are 23 =8.
• For four variables, the number of cell are 24 = 16.
• For five variables, the number of cells are 25 =32.

6. 2 VARIABLE KARNAUGH MAP:
A two variable has four minters, hence it has 4 squares one for
each term, as shown below,

7. 3 VARIABLE KARNAUGH MAP:
• The 3-variable karnaugh map is an array of eight cells.
• In this case A, B and C are used for the variables although other
letters could be used.
• Binary values of A is along the left side and the value of B and C
is across the top.
• The value of a given cell is the binary values of A at the left in
the same row combined with the value of B and C at the top in
the same column.

8. 3 VARIABLE K-MAP
For example: the cell in the upper left corner has a binary value
of 000 and the cell in the lower right corner has a binary value of
110.

9. 4 VARIABLE KARNAUGH MAP:
The 4-variable karnaugh map is an array of sixteen cells.
Binary values of A and B are along the left side and the values of
C and D are across the top.
The value of a given cell is the binary value of A and B at the left
in the same row combined with the binary values of C and D at
the top of the same column.

10. 4 VARIABLE KARNAUGH MAP:

11. 5 VARIABLE KARNAUGH MAP:
A 5 variable K-Map is having 25 cells = 32 cells arranged in 4
rows and 8 columns or 8 rows and 4 columns .

12. 5 VARIABLE KARNAUGH MAP:
• There are 32 minters (squares) for a Boolean function with 5
variable.
• It consist of 2- four variables.
• Variable distinguishes between the two map, the left hand four
variable map represent 16 square where, A=0.
• And the other four represent the square A=1.

13. GROUPING OF CELLS FOR SIMPLIFICATION:
Adjacent cells which have 1’s or 0’s can be grouped together in
2’s power.
e.g.;
21=2 adjacent cell can be grouped (pair).
22=4 adjacent cell can be grouped (quad).
23 =8 adjacent cell can be grouped (octal).
24 =16 adjacent cell can be grouped (hex).

14. RULES FOR GROUPING IN K-MAP:
The karnaugh map uses the following rules for the simplification of
expressions by grouping together adjacent cells containing ones:
Groups may not include any cell containing a zero.
Groups may be horizontal or vertical, but not diagonal.

15. Groups must contain 1, 2, 4, 8, or in general 2n cells.
That is if n = 1, a group will contain two 1's since 21 = 2.
If n = 2, a group will contain four 1's since 22 = 4.
Each group should be as large as possible.

16. Groups may overlap.
Groups may wrap around the table. The leftmost cell in a row may be
grouped with the rightmost cell and the top cell in a column may be
grouped with the bottom cell.

17. There should be as few groups as possible, as long as this does not
contradict any of the previous rules.