Chapter2-4web (1).powerpoint presentation

Boolean Algebra and Logic Gates 1
DIGITAL LOGIC DESIGN
by
Dr. Fenghui Yao
Tennessee State University
Department of Computer Science
Nashville, TN

Definitions
Definitions
 Switching network
Switching network
 One or more inputs
One or more inputs
 One or more outputs
One or more outputs
 Two Types
Two Types
 Combinational
Combinational
 The output depends only on the
The output depends only on the
present values of the inputs
present values of the inputs
 Logic gates are used
Logic gates are used
 Sequential
Sequential
 The output depends on present
The output depends on present
and past input values
and past input values

Boolean Algebra
Boolean Algebra
 Boolean Algebra
Boolean Algebra is used to describe the
is used to describe the
relationship between inputs and outputs
relationship between inputs and outputs
 Boolean Algebra
Boolean Algebra is the logic
is the logic
mathematics used for understanding of
mathematics used for understanding of
digital systems
digital systems
Network
.
.
.
.
.
.
Inputs Outputs

Basic Operations
Basic Operations
 COMPLEMENT (INVERSE)
1
if
0
and
0
if
1
0
1
and
1
0
'
'
'
'






A
A
A
A
0 is low voltage
1 is high voltage

Basic Operations
Basic Operations
 AND
F is 1 if and only if
A and B are both 1

Basic Operations
Basic Operations
 OR
F is 1 if and only if
A or B (or both) are 1

Basic Theorems
Basic Theorems
Let’s prove each one

Simplification Theorems
Simplification Theorems
Y
X
Y
XY
XY
Y
Y
X
X
Y
X
X
X
XY
X
X
Y
X
Y
X
X
XY
XY














'
'
'
'
6.
)
(
5.
)
(
4.
3.
)
)(
(
2.
1.

 Proof 6.
Proof 6.
R.H.S. = X+Y
R.H.S. = X+Y
= X(Y+Y’)+Y(X+X’)
= X(Y+Y’)+Y(X+X’)
= XY+XY’+XY+X’Y
= XY+XY’+XY+X’Y
= XY+XY’+X’Y
= XY+XY’+X’Y
= (X+X’)Y+XY’
= (X+X’)Y+XY’
= Y+XY’
= Y+XY’
= L.H.S.
= L.H.S.

Truth Table
Truth Table
 It can represent a boolean function
It can represent a boolean function
 For possible input combinations it
For possible input combinations it
shows the output value
shows the output value
 There are rows (
There are rows (n
n is the number of
is the number of
input variables)
input variables)
 It ranges from 0 to
It ranges from 0 to
n
2
1
2 
n

Examples
Examples
 Show the truth table for
Show the truth table for
YZ
X
F 
 '
 Show the followings by constructing
Show the followings by constructing
truth tables
truth tables
)
)(
(
)
(
Z
X
Y
X
YZ
X
XZ
XY
Z
Y
X








Example
Example
 Draw the network diagram for
Draw the network diagram for
YZ
X
F 
 '

Example
Example
 Draw the network diagram for
Draw the network diagram for
'
'
'
'
Y
X
Z
XY
XYZ
F 



Operator Precedence
Operator Precedence
 Parenthesis
Parenthesis
 NOT
NOT
 AND
AND
 OR
OR

Inversion
Inversion
'
'
'
'
'
'
)
(
)
(
Y
X
XY
Y
X
Y
X



 Prove with the
truth tables…
'
'
2
'
1
'
2
1
'
'
2
'
1
'
2
1
...
)
...
(
...
)
...
(
n
n
n
n
X
X
X
X
X
X
X
X
X
X
X
X








 The complement of the product is the sum of the
The complement of the product is the sum of the
complements
complements
 The complement of the sum is the product of the
The complement of the sum is the product of the
complements
complements

Examples
Examples
 Find the complements of
Find the complements of
  ?
)
)
((
?
]
)
[(
?
]
)
[(
'
'
'
'
'
'
'
'
'
'
'








D
BC
A
E
D
C
AB
C
B
A

Study Problems
Study Problems
1. Draw a network to realize the following by using
only one AND gate and one OR gate
only one AND gate and one OR gate
ABCF
ABCE
ABCD
Y 


two OR gates and two AND gates
two OR gates and two AND gates
)
)(
)(
( Z
V
Y
X
V
X
W
V
F 





3. Prove the following equations using truth table
3. Prove the following equations using truth table
'
'
'
'
)
)(
(
)
)(
(
AC
AB
C
AB
C
A
XY
W
Z
W
WZ
XY
W









Solution of problem 2
Solution of problem 2
L.H.S.=(V+X+W)(V+X+Y)(V+Z)
L.H.S.=(V+X+W)(V+X+Y)(V+Z)
=[(V+X)+W(V+X)+Y(V+X)+WY](V+Z)
=[(V+X)+W(V+X)+Y(V+X)+WY](V+Z)
=[(V+X)(1+W+Y)+WY](V+Z)
=[(V+X)(1+W+Y)+WY](V+Z)
=(V+X+WY)(V+Z)
=(V+X+WY)(V+Z)
This can be implemented by two OR
This can be implemented by two OR
gates and two AND gate.
gates and two AND gate.

Minterms
Minterms
 Consider variables
Consider variables A
A and
and B
B
 Assume that they are somehow combined
Assume that they are somehow combined
with AND operator
with AND operator
 There are 4 possible combinations
There are 4 possible combinations
 Each of those terms is called a minterm
Each of those terms is called a minterm
(standard product)
(standard product)
 In general, if there are n variables, there are
In general, if there are n variables, there are
minterms
minterms
'
'
'
'
,
,
, B
A
AB
B
A
AB
n
2

Exercise
Exercise
 List the minterms for 3 variables
List the minterms for 3 variables
A
A B
B C
C Minterm
Minterm Designation
Designation
0
0 0
0 0
0 A
A '
' B
B '
' C
C'
'
0
0 0
0 1
1 A
A '
' B
B '
' C
C
0
0 1
1 0
0 A
A '
' B C
B C'
'
0
0 1
1 1
1 A
A '
' B C
B C
1
1 0
0 0
0 A B
A B '
' C
C'
'
1
1 0
0 1
1 A B
A B '
' C
C
1
1 1
1 0
0 A B C
A B C'
'
1
1 1
1 1
1 A B C
A B C
0
m
1
m
2
m
3
m
4
m
5
m
6
m
7
m

Maxterms
Maxterms
 Consider variables
Consider variables A
A and
and B
B
 Assume that they are somehow combined with
Assume that they are somehow combined with
OR operator
OR operator
 There are 4 possible combinations
There are 4 possible combinations
 Each of those terms is called a maxterm
Each of those terms is called a maxterm
(standard sums)
(standard sums)
 In general, if there are n variables, there are
In general, if there are n variables, there are
maxterms
maxterms
'
'
'
'
,
,
, B
A
B
A
B
A
B
A 



n
2

Exercise
Exercise
 List the maxterm for 3 variables
List the maxterm for 3 variables
A
A B
B C
C Maxterm
Maxterm Designation
Designation
0
0 0
0 0
0 A+B
A+B+
+C
C
0
0 0
0 1
1 A
A+
+B
B+
+C
C'
'
0
0 1
1 0
0 A
A+
+B
B'
'+C
+C
0
0 1
1 1
1 A+B
A+B'+
'+C
C'
'
1
1 0
0 0
0 A
A'+
'+B+C
B+C
1
1 0
0 1
1 A
A'
'+B+C
+B+C'
'
1
1 1
1 0
0 A
A'+
'+B
B'+
'+C
C
1
1 1
1 1
1 A
A'+
'+B
B'+
'+C
C'
'
0
M
1
M
2
M
3
M
4
M
5
M
6
M
7
M

Example
Example
 Express
Express F
F in the sum of minterms and
in the sum of minterms and
product of maxterms formats
product of maxterms formats
 
 
3
,
1
,
0
7
,
6
,
5
,
4
,
2
)
(
)
)(
(
2
4
5
6
7
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'



























m
m
m
m
m
BC
A
C
AB
C
AB
ABC
ABC
BC
A
ABC
C
AB
C
AB
ABC
ABC
BC
A
A
C
C
B
B
A
BC
A
F
'
BC
A
F 


Sum-of-Products
Sum-of-Products
 All products are the product of single
All products are the product of single
variable only
variable only
EF
CD
B
A
E
D
C
B
A
E
AC
E
CD
AB







)
(
'
'
'
'
'
'
YES
YES
NO

Sum-of-Products
Sum-of-Products
 One or more AND gates feeding a
One or more AND gates feeding a
single OR gate at the output
single OR gate at the output
'
'
'
'
E
AC
E
CD
AB 

A
'
B
C
'
D
E
A
'
C
'
E

Product-of-Sums
Product-of-Sums
 All sums are the sums of single
All sums are the sums of single
variables
variables
EF
D
C
B
A
E
D
C
AB
E
C
A
E
D
C
B
A









)
)(
(
)
(
)
)(
)(
(
'
'
'
'
'
'
YES
YES
NO

Product-of-Sums
Product-of-Sums
 One or more OR gates feeding a single
One or more OR gates feeding a single
AND gate at the output
AND gate at the output
)
)(
)(
( '
'
'
'
E
C
A
E
D
C
B
A 




A
'
B
C
'
D
E
A
'
C
'
E

Logic Gates
Logic Gates
B
A
F 
 B
A
F 


Exclusive-OR
Exclusive-OR
both
not
but
1
or
1
1 



 B
A
B
A
'
'
'
'
'
'
'
)
(
)
(
)
(
)
(
1
0
1
0
B
A
AB
B
A
B
A
B
A
AC
AB
C
B
A
C
B
A
C
B
A
C
B
A
A
B
B
A
A
A
A
A
A
A
A
A






























Equivalence
Equivalence
 Equivalence is the complement of
Equivalence is the complement of
exclusive-OR
exclusive-OR
)
(
)
)(
(
)
(
)
(
'
'
'
'
'
'
'
'
B
A
B
A
AB
B
A
B
A
AB
B
A
B
A












A
B
)
( B
A 
)
(
)
( '
'
AC
B
C
B
A
F 


 Simplify it…

Integrated Circuits
Integrated Circuits
 SSI (Small Scale)
SSI (Small Scale)
 Less than 10 gates in a package
Less than 10 gates in a package
 MSI (Medium Scale)
MSI (Medium Scale)
 10-1000 gates in a package
10-1000 gates in a package
 LSI (Large Scale)
LSI (Large Scale)
 1000s of gates in a single package
1000s of gates in a single package
 VLSI (Very Large Scale)
VLSI (Very Large Scale)
 Hundred of thousands of gates in a single
Hundred of thousands of gates in a single
package
package

Study Problems
Study Problems
 Course Book Chapter – 2 Problems
Course Book Chapter – 2 Problems
 2 – 1
2 – 1
 2 – 3
2 – 3
 2 – 5
2 – 5
 2 – 8
2 – 8
 2 – 12
2 – 12
 2 – 14
2 – 14

Questions
Questions

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