CHAPTER 4
Applications of Boolean Algebra/
Minterm and Maxterm Expansions
This chapter in the book includes:
Objectives
Study Guide
4.1
Conversion of English Sentences to Boolean Equations
4.2
Combinational Logic Design Using a Truth
Table
4.3
Minterm and Maxterm Expansions
4.4
General Minterm and Maxterm Expansions
4.5
Incompletely Specified Functions
4.6
Examples of Truth Table Construction
4.7
Design of Binary Adders and Subtracters
1
Objective
•
Conversion of English Sentences to Boolean Equations
•
Combinational Logic Design Using a Truth Table
•
Minterm and Maxterm Expansions
•
General Minterm and Maxterm Expansions
•
Incompletely Specified Functions (Don’t care term)
•
Examples of Truth Table Construction
•
Design of Binary Adders(Full adder) and Subtracters
2
4.1 Conversion of English Sentences to Boolean Equations
- Steps in designing a single-output combinational switching circuit
1. Find switching function which specifies the desired behavior of the circuit
2. Find a simplified algebraic expression for the function
3. Realize the simplified function using available logic elements
1. F is ‘true’ if A and B are both ‘true’  F=AB
3
4.1 Conversion of English Sentences to Boolean Equations
1. The alarm will ring(Z) iff the alarm switch is turned on(A) and the
door is not closed(B’), or it is after 6PM(C) and window is not
closed(D’)
2. Boolean Equation
Z  AB'CD'
3. Circuit realization
4
4.2 Combinational Logic Design Using a Truth Table
- Combinational Circuit with Truth Table
When expression for f=1 
f  A' BC  AB' C ' AB' C  ABC ' ABC
5
4.2 Combinational Logic Design Using a Truth Table
Original equation 
f  A' BC  AB' C ' AB' C  ABC ' ABC
Simplified equation 
f  A' BC  AB' AB  A' BC  A  A  BC
Circuit realization 
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4.2 Combinational Logic Design Using a Truth Table
- Combinational Circuit with Truth Table
When expression for f=0 
f  ( A  B  C )( A  B  C ' )( A  B'C )
f  ( A  B)( A  B'C )  A  B( B'C )  A  BC
When expression for f ’=1 
and take the complement of f ‘
f '  A' B ' C ' A' B ' C  A' BC '
f  ( A  B  C )( A  B  C ' )( A  B'7C )
4.3 Minterm and Maxterm Expansions
- literal is a variable or its complement (e.g. A, A’)
- Minterm, Maxterm for three variables
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4.3 Minterm and Maxterm Expansions
- Minterm of n variables is a product of n literals in which each variable appears exactly
once in either true (A) or complemented form(A’), but not both. ( m0)
-Minterm expansion,
-Standard Sum of Product 
f  A' BC  AB' C ' AB' C  ABC ' ABC
f ( A, B, C )  m3  m4  m5  m6  m7
f ( A, B, C)   m(3,4,5,6,7)
9
4.3 Minterm and Maxterm Expansions
- Maxterm of n variables is a sum of n literals in which each variable appears exactly
once in either true (A) or complemented form(A’) , but not both.( M0)
- Maxterm expansion,
- Standard Product of Sum 
f  ( A  B  C )( A  B  C ' )( A  B'C )
f ( A, B, C )  M 0 M1M 2
f ( A, B, C)   M (0,1,2)
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4.3 Minterm and Maxterm Expansions
f ( A, B, C )  m3  m4  m5  m6  m7
f '  m0  m1  m2   m(0,1,2)
f ( A, B, C )  M 0 M1M 2
f '   M (3,4,5,6,7)  M 3M 4 M 5 M 6 M 7
- Minterm and Maxterm expansions are complement each other
f '  (m3  m4  m5  m6  m7 )'  m'3 m'4 m'5 m'6 m'7  M 3 M 4 M 5 M 6 M 7
f '  ( M 0 M1M 2 )'  M '0  M '1  M '2  m0  m1  m2
11
4.4 General Minterm and Maxterm Expansions
- Minterm expansion for general function
7
F  a0 m0  a1m1  a2 m2  ...  a7 m7   ai mi
i 0
ai =1, minterm mi is present
ai =0, minterm mi is not present
- Maxterm expansion for general function
7
F  (a0  M 0 )( a1  M 1 )( a2  M 2 )...( a7  M 7 )   (ai  M i )
-General truth table
for 3 variables
- ai is either ‘0’ or ‘1’
i 0
ai =1, ai + Mi =1 , Maxterm Mi is not present
ai =0, Maxterm is present
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4.4 General Minterm and Maxterm Expansions
7
7
7
i 0
i 0
i 0
F '  [ ( ai  M i )]'   a 'i M 'i   a 'i mi
 All minterm which are not present in F are present in F ‘
7
7
7
i 0
i 0
i 0
F '  [ ai mi ]'   (a 'i  m'i )   (a 'i  M i )
 All maxterm which are not present in F are present in F ‘
F 
2 n 1
a m
i 0
i
i
2 1
n
F'
 a'
i 0
i
2 n 1
  (ai  M i )
i 0
2 n 1
mi   (a 'i  M i )
i 0
13
4.4 General Minterm and Maxterm Expansions
If i and j are different, mi mj = 0
f1 
2 n 1
f1 f 2  (
2 n 1
a m
i 0
i
2 n 1
i
i 0
i
j
j 0
b m
f2 
i
 a m )(  b m
2 n 1
j
j 0
j
2 n 1 2 n 1
j)

2 n 1
 a b m m  a b m
i
j
i
j
i 0 j 0
i i
i
i 0
Example
f1   m(0,2,3,5,9,11) and f 2  m(0,3,9,11,13,14)
f1 f 2   m(0,3,9,11)
14
Conversion between minterm and maxterm
expansions of F and F’
Example
15
4.5 Incompletely Specified Functions
Truth Table with Don’t Cares
If N1 output does not generate all possible
combination of A,B,C, the output of N2(F) has
‘don’t care’ values.
16
4.5 Incompletely Specified Functions
Finding Function:
Case 1: assign ‘0’ on X’s
F  A' B' C' A' BC  ABC  A' B' C'BC
Case 2: assign ‘1’ to the first X and ‘0’ to the second ‘X’
F  A' B' C' A' B' C  A' BC  ABC  A' B'BC
Case 3: assign ‘1’ on X’s
F  A' B' C ' A' B' C  A' BC  ABC ' ABC  A' B' BC  AB
 The case 2 leads to the simplest function
17
4.5 Incompletely Specified Functions
- Minterm expansion for incompletely specified function
F   m(0,3,7)   d (1,6)
Don’t Cares
- Maxterm expansion for incompletely specified function
F   M (2,4,5) D(1,6)
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4.6 Examples of Truth Table Construction
Example 1 : Binary Adder
A
B
X
Y
0+0=0
0
0
0
0
01
0+1=1
0
1
0
1
0
01
1+0=1
1
10
1+1=2
1
0
0
1
1
1
1
0
a
b
Sum
0
0
00
0
1
1
1
X  AB, Y  A' B  AB '  A  B
19
4.7 Design of Binary Adders and Subtracters
Parallel Adder for 4 bit Binary Numbers
Parallel adder composed of four full adders  Carry Ripple Adder (slow!)
20
4.7 Design of Binary Adders and Subtracters
Truth Table for a Full Adder
21
4.7 Design of Binary Adders and Subtracters
Sum  X ' Y ' Cin  X ' YC 'in  XY ' C 'in  XYCin
 X ' (Y ' Cin  YC 'in )  X (Y ' C 'in YCin )
 X ' (Y  Cin )  X (Y  Cin )'  X  Y  Cin
Cout  X ' YCin  XY ' Cin  XYC 'in  XYCin
 ( X ' YCin  XYCin )  ( XY ' Cin  XYCin )  ( XYC 'in  XYCin )
 YCin  XCin  XY
22
When 1’s complement is used, the end-around carry is accomplished by
connecting C4 to C0 input.
Overflow(V) when adding two signed binary number
V  A'3 B'3 S3  A3 B3 S '3
23
Subtracters
Binary Subtracter using full adder
- Subtraction is done by adding the 2’s complemented number to be subtracted
2’s complemented number
24
Parallel Subtracters using Full Subtracter
b n+1
dn
di
Full
Subtracter
Full
Subtracter
xn
b n b i+1
yn
Truth Table for a Full Subtracter
d2
bi b
d1
Full
Subtracter
b
3
x3
y3
Full
Subtracter
b 1=0
2
x2
y2
x1
xi
yi bi
0
0
0
0
0
0
0
1
1
1
0
1
0
1
1
0
1
1
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
0
0
1
1
1
1
1
y1
bi+1 di
25