EE2504 Elec. Sys. : Karnaugh maps
Course EE2504 – Lecturer: Dr F. Verdicchio
1
K-maps
Truth-Table
Example: FULL ADDER
A
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
0
1
1
01
10
1
0
0
10
01
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Sum = ( A’ * B’ * Cin ) + ( A’ * B * Cin’ ) + ( A * B’ * Cin’ ) + ( A * B * Cin )
Cout = ( A’ * B * Cin ) + ( A * B’ * Cin ) + ( A * B * Cin’ ) + ( A * B * Cin )
2
K-maps
K-map: example table structure for 3 boolean variables A, B, C
3
K-maps
Aim: Map each output column (example: Cout ) to a matrix
A
AB
A
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
.
.
.
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
B
4
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
5
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
So, let’s pick a 2x4 table
AB
C
.
.
.
.
.
.
.
.
6
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
7
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
Gray code indexing scheme
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
8
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
B
9
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
C
B
10
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A’
A
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
C
B
11
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A’
A
AB
C
00
01
11
10
C’
0
.
.
.
.
C
1
.
.
.
.
B
12
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A’
A
AB
C
00
01
11
10
C’
0
.
.
.
.
C
1
.
.
.
.
B’
B
B’
13
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
A
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
C
B
14
K-maps
K-map: example table structure for 3 boolean variables A, B, C
First: The map has 8 entries, one for each possible input combination.
At most two variables can be associated with each row (or col).
Second: arrange variables along rows/cols following gray-code indexing
Third: fill the table (with 0 and 1 from the truth table)
A
AB
00
01
11
10
0
.
.
.
.
1
.
.
.
.
C
C
B
15
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
.
.
.
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
16
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
.
.
.
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
17
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
.
.
.
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
18
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
1
.
.
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
19
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
.
.
1
0
0
10
01
1
.
1
.
1
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
20
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
1
.
1
0
0
10
01
1
.
1
.
1
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
21
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
.
.
1
.
1
0
0
10
01
1
.
1
1
1
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
22
K-maps
K-map example: Cout for Full adder.
A
AB
B
Cin
Cout
Sum
Sum
Cout
0
0
0
00
00
0
0
1
10
01
0
1
0
10
01
00
01
11
10
0
1
1
01
10
0
0
0
1
0
1
0
0
10
01
1
0
1
1
1
1
0
1
01
10
1
1
0
01
10
1
1
1
11
11
Cin
Cin
A
B
23
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
Cout = ( A*B*Cin’ ) + (A’*B*Cin) + (A*B*Cin) + (A*B’*Cin)
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
24
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
25
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
26
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
Cout = ( B*Cin ) + ...
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
27
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
Cout = ( B*Cin ) + ( A*B ) +...
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
28
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
Cout = ( B*Cin ) + ( A*B ) + ( A*Cin )
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
29
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
(Any leftover square needs to be included separately.)
Cout = ( B*Cin ) + ( A*B ) + ( A*Cin )
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
30
K-maps
Minimization using K-map example: Cout for Full adder.
Sum of products:
Each square in the K-map corresponds to a product of 3 literals.
The objective is to group squares (of all 1s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the product of 2 literals.
(Any leftover square needs to be included separately.)
Cout = ( B*Cin ) + ( A*B ) + ( A*Cin )
Cout = ( A’ * B * Cin ) + ( A * B’ * Cin ) + ( A * B * Cin’ ) + ( A * B * Cin )
31
K-maps
Minimization using K-map example: Cout for Full adder.
Product of sums:
Each square (with a 0) in the K-map corresponds to a sum of 3 literals.
The objective is to group squares (of all 0s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the sum of 2 literals.
Mind the index (reversed compared to sum of products)!!
Cout = ( A+Cin ) * ...
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
32
K-maps
Minimization using K-map example: Cout for Full adder.
Product of sums:
Each square (with a 0) in the K-map corresponds to a sum of 3 literals.
The objective is to group squares (of all 0s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the sum of 2 literals.
Mind the index (reversed compared to sum of products)!!
Cout = ( A+Cin ) * (A+B) *...
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
33
K-maps
Minimization using K-map example: Cout for Full adder.
Product of sums:
Each square (with a 0) in the K-map corresponds to a sum of 3 literals.
The objective is to group squares (of all 0s ) that are horizontally or
vertically adjacent.
A group of two adjacent blocks corresponds to the sum of 2 literals.
Mind the index (reversed compared to sum of products)!!
Cout = ( A+Cin ) * (A+B) *(Cin+B)
A
AB
00
01
11
10
0
0
0
1
0
1
0
1
1
1
Cin
Cin
B
34
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
35
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y=B
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
0
C
Cin
B
36
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y=B
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
0
C
Cin
B
37
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y=
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
1
C
Cin
B
38
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y = B + ...
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
1
C
Cin
B
39
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y = B + A B’ C
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
1
C
Cin
B
40
K-maps
Minimization using K-map : Where to stop.
Once all the adjacent blocks have been grouped the form is minimal.
Note: when four adjacent square exists, for example in the following table,
then a one-literal term product appears
A
Y = B + AC
AB
00
01
11
10
0
0
1
1
0
1
0
1
1
1
C
Cin
B
41
Observation:
• Up to 4 (input) variables can be arranged in a two-dimensional Kmap. If you try to place more than two variables along the rows (or the
columns) of the matrix below, adjacent blocks of 1s or 0s loose their
meaning − which makes a K-map pointless.
A
AB
CD
00
11
01
10
00
01
C
D
11
10
B
42
Observation:
• Up to 4 (input) variables can be arranged in a two-dimensional Kmap. If you try to place more than two variables along the rows (or the
columns) of the matrix below, adjacent blocks of 1s or 0s loose their
meaning − which makes a K-map pointless.
X
A
ABX
CD
000
110
010
100
101
011
111
001
00
01
C
D
11
10
B
B
43
Observation:
• Up to 4 (input) variables can be arranged in a two-dimensional Kmap. If you try to place more than two variables along the rows (or the
columns) of the matrix below, adjacent blocks of 1s or 0s loose their
meaning − which makes a K-map pointless.
• For more than 4 variables you need tri-dimensional K-map (not
discussed in this course): instead apply SoP (or PoS) to the truth table.
A
AB
CD
00
11
01
10
00
01
C
D
11
10
B
44
Questions?!
(otherwise let’s move on)
45
Application ─ 1-bit multiplexer (MUX)
Boolean function: 1 output, 3 inputs In0, In1 and Sel.
When Sel=0 then out=In0; when Sel=1 then out=In1.
In1
In0
Sel=1
out
Sel=0
Sel
Block scheme
46
Application ─ 1-bit multiplexer (MUX)
Boolean function: 1 output, 3 inputs In0, In1 and Sel.
When Sel=0 then out=In0; when Sel=1 then out=In1.
In1
In0
Sel=1
out
Sel=0
Sel
Block scheme
47
Application ─ 1-bit multiplexer (MUX)
Boolean function: 1 output, 3 inputs In0, In1 and Sel.
When Sel=0 then out=In0; when Sel=1 then out=In1.
In1
In0
Sel=1
Sel=0
Sel
Block scheme
out
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
Truth Table
48
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
49
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
Using Sum of Products (SoP):
out = (Sel’ * In1’ * In0) + (Sel’ * In1 * In0) + (Sel * In1 * In0’) + (Sel * In1 * In0)
50
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
Using Sum of Products (SoP):
out = [(Sel’ * In1’ * In0) + (Sel’ * In1 * In0)] + [(Sel * In1 * In0’) + (Sel * In1 * In0)]
51
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
Using Sum of Products (SoP) and a little Boolean Algebra:
out = [ (Sel’ * In0)*(In1’ + In1) ] + [(Sel * In1)*(In0’ + In0) ]
52
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
Using Sum of Products (SoP) and a little Boolean Algebra:
out = [ (Sel’ * In0)
] + [ (Sel * In1) ]
53
Application ─ 1-bit multiplexer (MUX)
Sel
In1
In0
out
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
In0
Using the k-map:
In0 In1
00
01
11
10
0
0
0
1
1
1
0
1
1
0
Sel
Sel
In1
54
Application ─ 1-bit multiplexer (MUX)
out = ( Sel’ * In0
) + ( Sel * In1 )
In0
Using the k-map:
In0 In1
00
01
11
10
0
0
0
1
1
1
0
1
1
0
Sel
Sel
In1
55
Application of a MUX
Function F(A) based on a conditional test Ctest(A) performed on A.
The test can be either true or false (e.g A>0).
If the test is true:
F(A) performs FT(A),
Otherwise:
F(A) performs FF(A).
FT(A) if Ctest (A) = true
F(A) =
FF(A) if Ctest (A) = false
56
Application of a MUX
Function F(A) based on a conditional test Ctest(A) performed on A.
The test can be either true or false (e.g A>0).
If the test is true:
F(A) performs FT(A),
Otherwise:
F(A) performs FF(A).
FT(A)
In1
A
FF(A)
In0
Out
out
Sel
Ctest(A)
F(A)
57