Test Generation for Analog and Mixed-Signal Circuits Using Hybrid System Models
Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0...
Transcript of Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0...
![Page 1: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/1.jpg)
Digital Circuits: Part 1
M. B. [email protected]
www.ee.iitb.ac.in/~sequel
Department of Electrical EngineeringIndian Institute of Technology Bombay
M. B. Patil, IIT Bombay
![Page 2: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/2.jpg)
Digital circuits
t t
analog signal
0
1high
low
digital signal
* An analog signal x(t) is represented by a real number at a given time point.
* A digital signal is “binary” in nature, i.e., it takes on only two values: low (0) or high (1).
* Although we have shown 0 and 1 as constant levels, in reality, that is not required. Any value in the low(high) band will be interpreted as 0 (1) by digital circuits.
* The definition of low and high bands depends on the technology used, e.g.,
TTL (Transistor-Transistor Logic)
CMOS (Complementary MOS)
ECL (Emitter-Coupled Logic)
M. B. Patil, IIT Bombay
![Page 3: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/3.jpg)
Digital circuits
t t
analog signal
0
1high
low
digital signal
* An analog signal x(t) is represented by a real number at a given time point.
* A digital signal is “binary” in nature, i.e., it takes on only two values: low (0) or high (1).
* Although we have shown 0 and 1 as constant levels, in reality, that is not required. Any value in the low(high) band will be interpreted as 0 (1) by digital circuits.
* The definition of low and high bands depends on the technology used, e.g.,
TTL (Transistor-Transistor Logic)
CMOS (Complementary MOS)
ECL (Emitter-Coupled Logic)
M. B. Patil, IIT Bombay
![Page 4: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/4.jpg)
Digital circuits
t t
analog signal
0
1high
low
digital signal
* An analog signal x(t) is represented by a real number at a given time point.
* A digital signal is “binary” in nature, i.e., it takes on only two values: low (0) or high (1).
* Although we have shown 0 and 1 as constant levels, in reality, that is not required. Any value in the low(high) band will be interpreted as 0 (1) by digital circuits.
* The definition of low and high bands depends on the technology used, e.g.,
TTL (Transistor-Transistor Logic)
CMOS (Complementary MOS)
ECL (Emitter-Coupled Logic)
M. B. Patil, IIT Bombay
![Page 5: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/5.jpg)
Digital circuits
t t
analog signal
0
1high
low
digital signal
* An analog signal x(t) is represented by a real number at a given time point.
* A digital signal is “binary” in nature, i.e., it takes on only two values: low (0) or high (1).
* Although we have shown 0 and 1 as constant levels, in reality, that is not required. Any value in the low(high) band will be interpreted as 0 (1) by digital circuits.
* The definition of low and high bands depends on the technology used, e.g.,
TTL (Transistor-Transistor Logic)
CMOS (Complementary MOS)
ECL (Emitter-Coupled Logic)
M. B. Patil, IIT Bombay
![Page 6: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/6.jpg)
Digital circuits
t t
analog signal
0
1high
low
digital signal
* An analog signal x(t) is represented by a real number at a given time point.
* A digital signal is “binary” in nature, i.e., it takes on only two values: low (0) or high (1).
* Although we have shown 0 and 1 as constant levels, in reality, that is not required. Any value in the low(high) band will be interpreted as 0 (1) by digital circuits.
* The definition of low and high bands depends on the technology used, e.g.,
TTL (Transistor-Transistor Logic)
CMOS (Complementary MOS)
ECL (Emitter-Coupled Logic)
M. B. Patil, IIT Bombay
![Page 7: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/7.jpg)
A simple digital circuit
0 1 2 3 4 5
1
2
3
4
5
0
RB
RC
Vo
VCC
Vi
Vo(Volts)
Vi (Volts)
5 V
* If Vi is low (“0”), Vo is high (“1”).If Vi is high (“1”), Vo is low (“0”).
* The circuit is called an “inverter” because it inverts the logic level of the input. If the input is 0, it makesthe output 1, and vice versa.
* Digital circuits are made using a variety of devices. The simple BJT inverter is only an illustration.
* Most of the VLSI circuits today employ the MOS technology because of the high packing density, highspeed, and low power consumption it offers.
M. B. Patil, IIT Bombay
![Page 8: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/8.jpg)
A simple digital circuit
0 1 2 3 4 5
1
2
3
4
5
0
RB
RC
Vo
VCC
Vi
Vo(Volts)
Vi (Volts)
5 V
* If Vi is low (“0”), Vo is high (“1”).If Vi is high (“1”), Vo is low (“0”).
* The circuit is called an “inverter” because it inverts the logic level of the input. If the input is 0, it makesthe output 1, and vice versa.
* Digital circuits are made using a variety of devices. The simple BJT inverter is only an illustration.
* Most of the VLSI circuits today employ the MOS technology because of the high packing density, highspeed, and low power consumption it offers.
M. B. Patil, IIT Bombay
![Page 9: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/9.jpg)
A simple digital circuit
0 1 2 3 4 5
1
2
3
4
5
0
RB
RC
Vo
VCC
Vi
Vo(Volts)
Vi (Volts)
5 V
* If Vi is low (“0”), Vo is high (“1”).If Vi is high (“1”), Vo is low (“0”).
* The circuit is called an “inverter” because it inverts the logic level of the input. If the input is 0, it makesthe output 1, and vice versa.
* Digital circuits are made using a variety of devices. The simple BJT inverter is only an illustration.
* Most of the VLSI circuits today employ the MOS technology because of the high packing density, highspeed, and low power consumption it offers.
M. B. Patil, IIT Bombay
![Page 10: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/10.jpg)
A simple digital circuit
0 1 2 3 4 5
1
2
3
4
5
0
RB
RC
Vo
VCC
Vi
Vo(Volts)
Vi (Volts)
5 V
* If Vi is low (“0”), Vo is high (“1”).If Vi is high (“1”), Vo is low (“0”).
* The circuit is called an “inverter” because it inverts the logic level of the input. If the input is 0, it makesthe output 1, and vice versa.
* Digital circuits are made using a variety of devices. The simple BJT inverter is only an illustration.
* Most of the VLSI circuits today employ the MOS technology because of the high packing density, highspeed, and low power consumption it offers.
M. B. Patil, IIT Bombay
![Page 11: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/11.jpg)
A simple digital circuit
0 1 2 3 4 5
1
2
3
4
5
0
RB
RC
Vo
VCC
Vi
Vo(Volts)
Vi (Volts)
5 V
* If Vi is low (“0”), Vo is high (“1”).If Vi is high (“1”), Vo is low (“0”).
* The circuit is called an “inverter” because it inverts the logic level of the input. If the input is 0, it makesthe output 1, and vice versa.
* Digital circuits are made using a variety of devices. The simple BJT inverter is only an illustration.
* Most of the VLSI circuits today employ the MOS technology because of the high packing density, highspeed, and low power consumption it offers.
M. B. Patil, IIT Bombay
![Page 12: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/12.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 13: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/13.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 14: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/14.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 15: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/15.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 16: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/16.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 17: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/17.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 18: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/18.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.
- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 19: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/19.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 20: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/20.jpg)
Digital circuits
toriginal data
V1
tcorrupted data
V2
comparator
V3
V2
Vref
Vref
t
recovered data
V3
* A major advantage of digital systems is that, even if the original data gets distorted (e.g., in transmittingthrough optical fibre or storing on a CD) due to noise, attenuation, etc., it can be retrieved easily.
* There are several other benefits of using digital representation:
- can use computers to process the data.- can store in a variety of storage media.
- can program the functionality. For example, the behaviour of a digital filter can be changed simply
by changing its coefficients.
M. B. Patil, IIT Bombay
![Page 21: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/21.jpg)
Logical operations
Gate
Truth table
Notation
Operation NOT AND OR
10
01
YA
Y = A
A Y
1
1
1 1 1
0 0 0
0
0 0
0
A B Y
= AB
Y = A · B
B
AY
1
1
1 1 1
0 0 0
0
0
A B Y
1
1
Y = A+ B
A
BY
M. B. Patil, IIT Bombay
![Page 22: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/22.jpg)
Logical operations
Gate
Truth table
Notation
Operation NOT AND OR
10
01
YA
Y = A
A Y
1
1
1 1 1
0 0 0
0
0 0
0
A B Y
= AB
Y = A · B
B
AY
1
1
1 1 1
0 0 0
0
0
A B Y
1
1
Y = A+ B
A
BY
M. B. Patil, IIT Bombay
![Page 23: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/23.jpg)
Logical operations
Gate
Truth table
Notation
Operation NOT AND OR
10
01
YA
Y = A
A Y
1
1
1 1 1
0 0 0
0
0 0
0
A B Y
= AB
Y = A · B
B
AY
1
1
1 1 1
0 0 0
0
0
A B Y
1
1
Y = A+ B
A
BY
M. B. Patil, IIT Bombay
![Page 24: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/24.jpg)
Logical operations
Gate
Truth table
Notation
Operation NOT AND OR
10
01
YA
Y = A
A Y
1
1
1 1 1
0 0 0
0
0 0
0
A B Y
= AB
Y = A · B
B
AY
1
1
1 1 1
0 0 0
0
0
A B Y
1
1
Y = A+ B
A
BY
M. B. Patil, IIT Bombay
![Page 25: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/25.jpg)
Logical operations
Gate
Truth table
Notation
Operation NORNAND XOR
1
1
1 1
0 0
0
0
A B Y
1
1
1
0
Y = A · B= AB
YA
B
1
1
1 1
0 0
0
0 0
0
A B Y
1
0
Y = A+ B
B
AY
1
1
1 1
0 0 0
0
0
A B Y
1
1
0
Y = A⊕ B
= AB+ AB
A
BY
M. B. Patil, IIT Bombay
![Page 26: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/26.jpg)
Logical operations
Gate
Truth table
Notation
Operation NORNAND XOR
1
1
1 1
0 0
0
0
A B Y
1
1
1
0
Y = A · B= AB
YA
B
1
1
1 1
0 0
0
0 0
0
A B Y
1
0
Y = A+ B
B
AY
1
1
1 1
0 0 0
0
0
A B Y
1
1
0
Y = A⊕ B
= AB+ AB
A
BY
M. B. Patil, IIT Bombay
![Page 27: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/27.jpg)
Logical operations
Gate
Truth table
Notation
Operation NORNAND XOR
1
1
1 1
0 0
0
0
A B Y
1
1
1
0
Y = A · B= AB
YA
B
1
1
1 1
0 0
0
0 0
0
A B Y
1
0
Y = A+ B
B
AY
1
1
1 1
0 0 0
0
0
A B Y
1
1
0
Y = A⊕ B
= AB+ AB
A
BY
M. B. Patil, IIT Bombay
![Page 28: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/28.jpg)
Logical operations
Gate
Truth table
Notation
Operation NORNAND XOR
1
1
1 1
0 0
0
0
A B Y
1
1
1
0
Y = A · B= AB
YA
B
1
1
1 1
0 0
0
0 0
0
A B Y
1
0
Y = A+ B
B
AY
1
1
1 1
0 0 0
0
0
A B Y
1
1
0
Y = A⊕ B
= AB+ AB
A
BY
M. B. Patil, IIT Bombay
![Page 29: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/29.jpg)
Logical operations
* The AND operation is commutative.
→ A · B = B · A.
* The AND operation is associative.
→ (A · B) · C = A · (B · C).
* The OR operation is commutative.
→ A + B = B + A.
* The OR operation is associative.
→ (A + B) + C = A + (B + C).
M. B. Patil, IIT Bombay
![Page 30: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/30.jpg)
Logical operations
* The AND operation is commutative.
→ A · B = B · A.
* The AND operation is associative.
→ (A · B) · C = A · (B · C).
* The OR operation is commutative.
→ A + B = B + A.
* The OR operation is associative.
→ (A + B) + C = A + (B + C).
M. B. Patil, IIT Bombay
![Page 31: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/31.jpg)
Logical operations
* The AND operation is commutative.
→ A · B = B · A.
* The AND operation is associative.
→ (A · B) · C = A · (B · C).
* The OR operation is commutative.
→ A + B = B + A.
* The OR operation is associative.
→ (A + B) + C = A + (B + C).
M. B. Patil, IIT Bombay
![Page 32: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/32.jpg)
Logical operations
* The AND operation is commutative.
→ A · B = B · A.
* The AND operation is associative.
→ (A · B) · C = A · (B · C).
* The OR operation is commutative.
→ A + B = B + A.
* The OR operation is associative.
→ (A + B) + C = A + (B + C).
M. B. Patil, IIT Bombay
![Page 33: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/33.jpg)
Boolean algebra (George Boole, 1815-1864)
* Theorem: A = A.
The theorem can be proved by constructing a truth table:
A A A
0 1 0
1 0 1
Therefore, for all possible values that A can take (i.e., 0 and 1), A is the same as A.
⇒ A = A.
* Similarly, the following theorems can be proved:
A + 0 = A A · 1 = A
A + 1 = 1 A · 0 = 0
A + A = A A · A = A
A + A = 1 A · A = 0
Note the duality: (+←→ ·) and (1←→ 0).
M. B. Patil, IIT Bombay
![Page 34: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/34.jpg)
Boolean algebra (George Boole, 1815-1864)
* Theorem: A = A.
The theorem can be proved by constructing a truth table:
A A A
0 1 0
1 0 1
Therefore, for all possible values that A can take (i.e., 0 and 1), A is the same as A.
⇒ A = A.
* Similarly, the following theorems can be proved:
A + 0 = A A · 1 = A
A + 1 = 1 A · 0 = 0
A + A = A A · A = A
A + A = 1 A · A = 0
Note the duality: (+←→ ·) and (1←→ 0).
M. B. Patil, IIT Bombay
![Page 35: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/35.jpg)
Boolean algebra (George Boole, 1815-1864)
* Theorem: A = A.
The theorem can be proved by constructing a truth table:
A A A
0 1 0
1 0 1
Therefore, for all possible values that A can take (i.e., 0 and 1), A is the same as A.
⇒ A = A.
* Similarly, the following theorems can be proved:
A + 0 = A A · 1 = A
A + 1 = 1 A · 0 = 0
A + A = A A · A = A
A + A = 1 A · A = 0
Note the duality: (+←→ ·) and (1←→ 0).
M. B. Patil, IIT Bombay
![Page 36: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/36.jpg)
Boolean algebra (George Boole, 1815-1864)
* Theorem: A = A.
The theorem can be proved by constructing a truth table:
A A A
0 1 0
1 0 1
Therefore, for all possible values that A can take (i.e., 0 and 1), A is the same as A.
⇒ A = A.
* Similarly, the following theorems can be proved:
A + 0 = A A · 1 = A
A + 1 = 1 A · 0 = 0
A + A = A A · A = A
A + A = 1 A · A = 0
Note the duality: (+←→ ·) and (1←→ 0).
M. B. Patil, IIT Bombay
![Page 37: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/37.jpg)
Boolean algebra (George Boole, 1815-1864)
* Theorem: A = A.
The theorem can be proved by constructing a truth table:
A A A
0 1 0
1 0 1
Therefore, for all possible values that A can take (i.e., 0 and 1), A is the same as A.
⇒ A = A.
* Similarly, the following theorems can be proved:
A + 0 = A A · 1 = A
A + 1 = 1 A · 0 = 0
A + A = A A · A = A
A + A = 1 A · A = 0
Note the duality: (+←→ ·) and (1←→ 0).
M. B. Patil, IIT Bombay
![Page 38: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/38.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0
0 1 1 1 1 0 1 1
0 1
1 0 1 0 0 0 1 1
1 0
1 0 0 1 0 0 1 1
1 1
1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 39: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/39.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0
1 1 1 1 0 1 1
0 1 1
0 1 0 0 0 1 1
1 0 1
0 0 1 0 0 1 1
1 1 1
0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 40: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/40.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1
1 1 1 0 1 1
0 1 1 0
1 0 0 0 1 1
1 0 1 0
0 1 0 0 1 1
1 1 1 0
0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 41: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/41.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1
1 1 0 1 1
0 1 1 0 1
0 0 0 1 1
1 0 1 0 0
1 0 0 1 1
1 1 1 0 0
0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 42: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/42.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1
1 0 1 1
0 1 1 0 1 0
0 0 1 1
1 0 1 0 0 1
0 0 1 1
1 1 1 0 0 0
0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 43: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/43.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1
0 1 1
0 1 1 0 1 0 0
0 1 1
1 0 1 0 0 1 0
0 1 1
1 1 1 0 0 0 0
1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 44: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/44.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0
1 1
0 1 1 0 1 0 0 0
1 1
1 0 1 0 0 1 0 0
1 1
1 1 1 0 0 0 0 1
0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 45: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/45.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1
1
0 1 1 0 1 0 0 0 1
1
1 0 1 0 0 1 0 0 1
1
1 1 1 0 0 0 0 1 0
0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 46: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/46.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 47: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/47.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 48: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/48.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 49: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/49.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 50: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/50.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 51: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/51.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 52: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/52.jpg)
De Morgan’s theorems
A B A + B A + B A B A · B A · B A · B A + B
0 0 0 1 1 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 1 0 0 1 0 0 1 1
1 1 1 0 0 0 0 1 0 0
* Comparing the truth tables for A + B and AB, we conclude that A + B = AB.
* Similarly, A · B = A + B.
* Similar relations hold for more than two variables, e.g.,
A · B · C = A + B + C ,
A + B + C + D = A · B · C · D,
(A + B) · C = (A + B) + C = A · B + C .
M. B. Patil, IIT Bombay
![Page 53: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/53.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0
0 0 0 0 0
0 0 1
1 0 0 0 0
0 1 0
1 0 0 0 0
0 1 1
1 0 0 0 0
1 0 0
0 0 0 0 0
1 0 1
1 1 0 1 1
1 1 0
1 1 1 0 1
1 1 1
1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 54: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/54.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0
0 0 0 0 0
0 0 1
1 0 0 0 0
0 1 0
1 0 0 0 0
0 1 1
1 0 0 0 0
1 0 0
0 0 0 0 0
1 0 1
1 1 0 1 1
1 1 0
1 1 1 0 1
1 1 1
1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 55: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/55.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0
0 0 0 0
0 0 1 1
0 0 0 0
0 1 0 1
0 0 0 0
0 1 1 1
0 0 0 0
1 0 0 0
0 0 0 0
1 0 1 1
1 0 1 1
1 1 0 1
1 1 0 1
1 1 1 1
1 1 1 1
M. B. Patil, IIT Bombay
![Page 56: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/56.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0 0
0 0 0
0 0 1 1 0
0 0 0
0 1 0 1 0
0 0 0
0 1 1 1 0
0 0 0
1 0 0 0 0
0 0 0
1 0 1 1 1
0 1 1
1 1 0 1 1
1 0 1
1 1 1 1 1
1 1 1
M. B. Patil, IIT Bombay
![Page 57: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/57.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0 0 0
0 0
0 0 1 1 0 0
0 0
0 1 0 1 0 0
0 0
0 1 1 1 0 0
0 0
1 0 0 0 0 0
0 0
1 0 1 1 1 0
1 1
1 1 0 1 1 1
0 1
1 1 1 1 1 1
1 1
M. B. Patil, IIT Bombay
![Page 58: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/58.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0 0 0 0
0
0 0 1 1 0 0 0
0
0 1 0 1 0 0 0
0
0 1 1 1 0 0 0
0
1 0 0 0 0 0 0
0
1 0 1 1 1 0 1
1
1 1 0 1 1 1 0
1
1 1 1 1 1 1 1
1
M. B. Patil, IIT Bombay
![Page 59: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/59.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 60: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/60.jpg)
Distributive laws
1. A · (B + C) = AB + AC .
A B C B + C A · (B + C) AB AC AB + AC
0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 0 0 0 0
1 0 0 0 0 0 0 0
1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 61: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/61.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0
0 0 0 0 0
0 0 1
0 0 0 1 0
0 1 0
0 0 1 0 0
0 1 1
1 1 1 1 1
1 0 0
0 1 1 1 1
1 0 1
0 1 1 1 1
1 1 0
0 1 1 1 1
1 1 1
1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 62: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/62.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0
0 0 0 0 0
0 0 1
0 0 0 1 0
0 1 0
0 0 1 0 0
0 1 1
1 1 1 1 1
1 0 0
0 1 1 1 1
1 0 1
0 1 1 1 1
1 1 0
0 1 1 1 1
1 1 1
1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 63: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/63.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0
0 0 0 0
0 0 1 0
0 0 1 0
0 1 0 0
0 1 0 0
0 1 1 1
1 1 1 1
1 0 0 0
1 1 1 1
1 0 1 0
1 1 1 1
1 1 0 0
1 1 1 1
1 1 1 1
1 1 1 1
M. B. Patil, IIT Bombay
![Page 64: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/64.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0 0
0 0 0
0 0 1 0 0
0 1 0
0 1 0 0 0
1 0 0
0 1 1 1 1
1 1 1
1 0 0 0 1
1 1 1
1 0 1 0 1
1 1 1
1 1 0 0 1
1 1 1
1 1 1 1 1
1 1 1
M. B. Patil, IIT Bombay
![Page 65: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/65.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0 0 0
0 0
0 0 1 0 0 0
1 0
0 1 0 0 0 1
0 0
0 1 1 1 1 1
1 1
1 0 0 0 1 1
1 1
1 0 1 0 1 1
1 1
1 1 0 0 1 1
1 1
1 1 1 1 1 1
1 1
M. B. Patil, IIT Bombay
![Page 66: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/66.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0 0 0 0
0
0 0 1 0 0 0 1
0
0 1 0 0 0 1 0
0
0 1 1 1 1 1 1
1
1 0 0 0 1 1 1
1
1 0 1 0 1 1 1
1
1 1 0 0 1 1 1
1
1 1 1 1 1 1 1
1
M. B. Patil, IIT Bombay
![Page 67: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/67.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0
0 1 1 1 1 1 1 1
1 0 0 0 1 1 1 1
1 0 1 0 1 1 1 1
1 1 0 0 1 1 1 1
1 1 1 1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 68: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/68.jpg)
Distributive laws
2. A + B · C = (A + B) · (A + C).
A B C B C A + B C A + B A + C (A + B) (A + C)
0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0
0 1 1 1 1 1 1 1
1 0 0 0 1 1 1 1
1 0 1 0 1 1 1 1
1 1 0 0 1 1 1 1
1 1 1 1 1 1 1 1
M. B. Patil, IIT Bombay
![Page 69: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/69.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 70: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/70.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 71: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/71.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 72: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/72.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 73: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/73.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 74: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/74.jpg)
Useful theorems
* A + AB = A.
To prove this theorem, we can follow two approaches:
(a) Construct truth tables for LHS and RHS for all possible input combinations, and show that they arethe same.
(b) Use identities and theorems stated earlier to show that LHS=RHS.
A + AB = A · 1 + A · B= A · (1 + B)= A · (1)= A
* A · (A + B) = A.
Proof: A · (A + B) = A · A + A · B= A + AB= A
M. B. Patil, IIT Bombay
![Page 75: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/75.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 76: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/76.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 77: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/77.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 78: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/78.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 79: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/79.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 80: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/80.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 81: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/81.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 82: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/82.jpg)
Duality
A + AB = A ←→ A · (A + B) = A.
Note the duality between OR and AND.
Dual of A + (AB) (LHS): AB → A + BA + AB → A · (A + B).
Dual of A (RHS) = A (since there are no operations involved).
⇒ A · (A + B) = A.
Similarly, consider A + A = 1, with (+←→ .) and (1←→ 0).
Dual of LHS = A · A.
Dual of RHS = 0.
⇒ A · A = 0.
M. B. Patil, IIT Bombay
![Page 83: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/83.jpg)
Useful theorems
* A + AB = A + B.
Proof: A + AB = (A + A) · (A + B) (by distributive law)
= 1 · (A + B)
= A + B
Dual theorem: A · (A + B) = AB.
* AB + AB = A.
Proof: AB + AB = A · (B + B) (by distributive law)
= A · 1= A
Dual theorem: (A + B) · (A + B) = A.
M. B. Patil, IIT Bombay
![Page 84: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/84.jpg)
Useful theorems
* A + AB = A + B.
Proof: A + AB = (A + A) · (A + B) (by distributive law)
= 1 · (A + B)
= A + B
Dual theorem: A · (A + B) = AB.
* AB + AB = A.
Proof: AB + AB = A · (B + B) (by distributive law)
= A · 1= A
Dual theorem: (A + B) · (A + B) = A.
M. B. Patil, IIT Bombay
![Page 85: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/85.jpg)
Useful theorems
* A + AB = A + B.
Proof: A + AB = (A + A) · (A + B) (by distributive law)
= 1 · (A + B)
= A + B
Dual theorem: A · (A + B) = AB.
* AB + AB = A.
Proof: AB + AB = A · (B + B) (by distributive law)
= A · 1= A
Dual theorem: (A + B) · (A + B) = A.
M. B. Patil, IIT Bombay
![Page 86: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/86.jpg)
Useful theorems
* A + AB = A + B.
Proof: A + AB = (A + A) · (A + B) (by distributive law)
= 1 · (A + B)
= A + B
Dual theorem: A · (A + B) = AB.
* AB + AB = A.
Proof: AB + AB = A · (B + B) (by distributive law)
= A · 1= A
Dual theorem: (A + B) · (A + B) = A.
M. B. Patil, IIT Bombay
![Page 87: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/87.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 88: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/88.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 89: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/89.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 90: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/90.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 91: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/91.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 92: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/92.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 93: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/93.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 94: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/94.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 95: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/95.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 96: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/96.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 97: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/97.jpg)
A game of words
In an India-Australia match, India will win if one or more of the following conditions are met:
(a) Tendulkar scores a century.
(b) Tendulkar does not score a century AND Warne fails (to get wickets).
(c) Tendulkar does not score a century AND Sehwag scores a century.
Let T ≡ Tendulkar scores a century.
S ≡ Sehwag scores a century.
W ≡ Warne fails.
I ≡ India wins.
I = T + T W + T S
= T + T + T W + T S
= (T + T W ) + (T + T S)
= (T + T ) · (T + W ) + (T + T ) · (T + S)
= T + W + T + S
= T + W + S
i.e., India will win if one or more of the following hold:
(a) Tendulkar strikes, (b) Warne fails, (c) Sehwag strikes.
M. B. Patil, IIT Bombay
![Page 98: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/98.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 99: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/99.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 100: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/100.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 101: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/101.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 102: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/102.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 103: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/103.jpg)
Logical functions in standard forms
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C + AB C
≡ X1 + X2 + X3 + X4
This form is called the “sum of products” form (“sum” corresponding to ORand “product” corresponding to AND).
We can construct the truth table for X in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate X1 = AB C , etc. Note that X1 is 1 only if A=B =C = 1 (i.e., A= 0, B = 1, C = 0),and 0 otherwise.
(3) Since X = X1 + X2 + X3 + X4,X is 1 if any of X1, X2, X3, X4 is 1; else X is 0.→ tabulate X .
M. B. Patil, IIT Bombay
![Page 104: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/104.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 105: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/105.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 106: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/106.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 107: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/107.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 108: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/108.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 109: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/109.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 110: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/110.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 111: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/111.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 112: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/112.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 113: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/113.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 114: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/114.jpg)
“Sum of products” form
XCBA X4X3X2X1
X = X1 + X2 + X3 + X4 = ABC+ ABC+ ABC+ ABC
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
M. B. Patil, IIT Bombay
![Page 115: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/115.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 116: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/116.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 117: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/117.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 118: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/118.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 119: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/119.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 120: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/120.jpg)
Logical functions in standard forms
Consider a function Y of three variables A, B, C :
Y = (A + B + C) · (A + B + C) · (A + B + C) · (A + B + C)
≡ Y1 · Y2 · Y3 · Y4
This form is called the “product of sums” form (“sum” corresponding to OR,and “product” corresponding to AND).
We can construct the truth table for Y in a systematic manner:
(1) Enumerate all possible combinations of A, B, C .Since each of A, B, C can take two values (0 or 1), we have 23 possibilities.
(2) Tabulate Y1 = A + B + C , etc. Note that Y1 is 0 only if A=B =C = 0;Y1 is 1 otherwise.
(3) Since Y = Y1 Y2 Y3 Y4,Y is 0 if any of Y1, Y2, Y3, Y4 is 0; else Y is 1.→ tabulate Y .
M. B. Patil, IIT Bombay
![Page 121: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/121.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 122: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/122.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 123: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/123.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 124: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/124.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 125: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/125.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 126: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/126.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 127: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/127.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 128: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/128.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 129: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/129.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 130: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/130.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 131: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/131.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 132: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/132.jpg)
“Product of sums” form
YCBA Y4Y3Y2Y1
Y = Y1 Y2 Y3 Y4 = (A+ B+ C) (A+ B+ C) (A+ B+ C) (A+ B+ C)
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1 0
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
Note that Y is identical to X (seen two slides back). This is an example of how the same function can be written in two
seemingly different forms (in this case, the sum-of-products form and the product-of-sums form).
M. B. Patil, IIT Bombay
![Page 133: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/133.jpg)
Standard sum-of-products form
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C
This form is called the standard sum-of-products form, and each individual term (consisting of all threevariables) is called a “minterm.”
In the truth table for X , the numbers of 1s is the same as the number of minterms, as we have seen in anexample.
X can be rewritten as,
X = AB C + AB (C + C)
= AB C + AB.
This is also a sum-of-products form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 134: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/134.jpg)
Standard sum-of-products form
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C
This form is called the standard sum-of-products form, and each individual term (consisting of all threevariables) is called a “minterm.”
In the truth table for X , the numbers of 1s is the same as the number of minterms, as we have seen in anexample.
X can be rewritten as,
X = AB C + AB (C + C)
= AB C + AB.
This is also a sum-of-products form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 135: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/135.jpg)
Standard sum-of-products form
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C
This form is called the standard sum-of-products form, and each individual term (consisting of all threevariables) is called a “minterm.”
In the truth table for X , the numbers of 1s is the same as the number of minterms, as we have seen in anexample.
X can be rewritten as,
X = AB C + AB (C + C)
= AB C + AB.
This is also a sum-of-products form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 136: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/136.jpg)
Standard sum-of-products form
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C
This form is called the standard sum-of-products form, and each individual term (consisting of all threevariables) is called a “minterm.”
In the truth table for X , the numbers of 1s is the same as the number of minterms, as we have seen in anexample.
X can be rewritten as,
X = AB C + AB (C + C)
= AB C + AB.
This is also a sum-of-products form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 137: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/137.jpg)
Standard sum-of-products form
Consider a function X of three variables A, B, C :
X = AB C + AB C + AB C
This form is called the standard sum-of-products form, and each individual term (consisting of all threevariables) is called a “minterm.”
In the truth table for X , the numbers of 1s is the same as the number of minterms, as we have seen in anexample.
X can be rewritten as,
X = AB C + AB (C + C)
= AB C + AB.
This is also a sum-of-products form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 138: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/138.jpg)
Standard product-of-sums form
Consider a function X of three variables A, B, C :
X = (A + B + C) (A + B + C) (A + B + C)
This form is called the standard product-of-sums form, and each individual term (consisting of all threevariables) is called a “maxterm.”
In the truth table for X , the numbers of 0s is the same as the number of maxterms, as we have seen in anexample.
X can be rewritten as,
X = (A + B + C) (A + B + C) (A + B + C)
= (A + B + C) (A + C + B) (A + C + B)
= (A + B + C) (A + C + B B)
= (A + B + C) (A + C).
This is also a product-of-sums form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 139: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/139.jpg)
Standard product-of-sums form
Consider a function X of three variables A, B, C :
X = (A + B + C) (A + B + C) (A + B + C)
This form is called the standard product-of-sums form, and each individual term (consisting of all threevariables) is called a “maxterm.”
In the truth table for X , the numbers of 0s is the same as the number of maxterms, as we have seen in anexample.
X can be rewritten as,
X = (A + B + C) (A + B + C) (A + B + C)
= (A + B + C) (A + C + B) (A + C + B)
= (A + B + C) (A + C + B B)
= (A + B + C) (A + C).
This is also a product-of-sums form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 140: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/140.jpg)
Standard product-of-sums form
Consider a function X of three variables A, B, C :
X = (A + B + C) (A + B + C) (A + B + C)
This form is called the standard product-of-sums form, and each individual term (consisting of all threevariables) is called a “maxterm.”
In the truth table for X , the numbers of 0s is the same as the number of maxterms, as we have seen in anexample.
X can be rewritten as,
X = (A + B + C) (A + B + C) (A + B + C)
= (A + B + C) (A + C + B) (A + C + B)
= (A + B + C) (A + C + B B)
= (A + B + C) (A + C).
This is also a product-of-sums form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 141: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/141.jpg)
Standard product-of-sums form
Consider a function X of three variables A, B, C :
X = (A + B + C) (A + B + C) (A + B + C)
This form is called the standard product-of-sums form, and each individual term (consisting of all threevariables) is called a “maxterm.”
In the truth table for X , the numbers of 0s is the same as the number of maxterms, as we have seen in anexample.
X can be rewritten as,
X = (A + B + C) (A + B + C) (A + B + C)
= (A + B + C) (A + C + B) (A + C + B)
= (A + B + C) (A + C + B B)
= (A + B + C) (A + C).
This is also a product-of-sums form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 142: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/142.jpg)
Standard product-of-sums form
Consider a function X of three variables A, B, C :
X = (A + B + C) (A + B + C) (A + B + C)
This form is called the standard product-of-sums form, and each individual term (consisting of all threevariables) is called a “maxterm.”
In the truth table for X , the numbers of 0s is the same as the number of maxterms, as we have seen in anexample.
X can be rewritten as,
X = (A + B + C) (A + B + C) (A + B + C)
= (A + B + C) (A + C + B) (A + C + B)
= (A + B + C) (A + C + B B)
= (A + B + C) (A + C).
This is also a product-of-sums form, but not the standard one.
M. B. Patil, IIT Bombay
![Page 143: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/143.jpg)
The “don’t care” condition
I want to design a box (with inputs A, B, C , and output S) which will help in scheduling my appointments.
A ≡ I am in town, and the time slot being suggested for the appointment is free.
B ≡ My favourite player is scheduled to play a match (which I can watch on TV).
C ≡ The appointment is crucial for my business.
S ≡ Schedule the appointment.
The following truth table summarizes the expected functioning of the box.
A B C S
0 X X 0
1 0 X 1
1 1 0 0
1 1 1 1
Note that we have a new entity called X in the truth table.
X can be 0 or 1 (it does not matter) and is therefore called the “don’t care” condition.
Don’t care conditions can often be used to get a more efficient implementation of a logical function.
M. B. Patil, IIT Bombay
![Page 144: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/144.jpg)
The “don’t care” condition
I want to design a box (with inputs A, B, C , and output S) which will help in scheduling my appointments.
A ≡ I am in town, and the time slot being suggested for the appointment is free.
B ≡ My favourite player is scheduled to play a match (which I can watch on TV).
C ≡ The appointment is crucial for my business.
S ≡ Schedule the appointment.
The following truth table summarizes the expected functioning of the box.
A B C S
0 X X 0
1 0 X 1
1 1 0 0
1 1 1 1
Note that we have a new entity called X in the truth table.
X can be 0 or 1 (it does not matter) and is therefore called the “don’t care” condition.
Don’t care conditions can often be used to get a more efficient implementation of a logical function.
M. B. Patil, IIT Bombay
![Page 145: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/145.jpg)
The “don’t care” condition
I want to design a box (with inputs A, B, C , and output S) which will help in scheduling my appointments.
A ≡ I am in town, and the time slot being suggested for the appointment is free.
B ≡ My favourite player is scheduled to play a match (which I can watch on TV).
C ≡ The appointment is crucial for my business.
S ≡ Schedule the appointment.
The following truth table summarizes the expected functioning of the box.
A B C S
0 X X 0
1 0 X 1
1 1 0 0
1 1 1 1
Note that we have a new entity called X in the truth table.
X can be 0 or 1 (it does not matter) and is therefore called the “don’t care” condition.
Don’t care conditions can often be used to get a more efficient implementation of a logical function.
M. B. Patil, IIT Bombay
![Page 146: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/146.jpg)
The “don’t care” condition
I want to design a box (with inputs A, B, C , and output S) which will help in scheduling my appointments.
A ≡ I am in town, and the time slot being suggested for the appointment is free.
B ≡ My favourite player is scheduled to play a match (which I can watch on TV).
C ≡ The appointment is crucial for my business.
S ≡ Schedule the appointment.
The following truth table summarizes the expected functioning of the box.
A B C S
0 X X 0
1 0 X 1
1 1 0 0
1 1 1 1
Note that we have a new entity called X in the truth table.
X can be 0 or 1 (it does not matter) and is therefore called the “don’t care” condition.
Don’t care conditions can often be used to get a more efficient implementation of a logical function.
M. B. Patil, IIT Bombay
![Page 147: Digital Circuits: Part 1 - IIT Bombaysequel/ee101/mc_dgtl_1.pdfDigital circuits t t analog signal 0 1 high low digital signal * An analog signal x(t) is represented by a real number](https://reader030.fdocuments.in/reader030/viewer/2022040116/5ea4ff90c6f0d4107f6ba239/html5/thumbnails/147.jpg)
The “don’t care” condition
I want to design a box (with inputs A, B, C , and output S) which will help in scheduling my appointments.
A ≡ I am in town, and the time slot being suggested for the appointment is free.
B ≡ My favourite player is scheduled to play a match (which I can watch on TV).
C ≡ The appointment is crucial for my business.
S ≡ Schedule the appointment.
The following truth table summarizes the expected functioning of the box.
A B C S
0 X X 0
1 0 X 1
1 1 0 0
1 1 1 1
Note that we have a new entity called X in the truth table.
X can be 0 or 1 (it does not matter) and is therefore called the “don’t care” condition.
Don’t care conditions can often be used to get a more efficient implementation of a logical function.
M. B. Patil, IIT Bombay