1. Logic Gates & Boolean Algebra
Chin-Sung Lin
Eleanor Roosevelt High School

2. Logic Gates & Boolean Algebra
System Concept
Systems & Subsystems
Analog Systems vs. Digital Systems
Combinational vs. Sequential Circuits
Truth Tables & Basic Logic Gates
Logic Simulation
Digital Logic Circuits
Equivalent Logic Circuits
Digital Building Blocks

3. System Concept

4. What Are They In Common?
VS.
Computer
Onion

5. System Concept
System
(System Function)
Input
Output

6. System Concept System (System Function) Input Output Output

7. System Concept
System
(System Function)
Input
Output
A convenient way to view and understand both the nature and man-made worlds.
Hide details and complexity from the viewers by encapsulating this detail information into a “system”.
Only the inputs/outputs (I/O) of the system, and the function of the system (called system function) are important.

8. Systems & Subsystems

9. Systems & Subsystems System
Input
Output
System
(System Function)
Input
Output
Systems can be further divided down to subsystems.
Subsystems are connected together through inputs and outputs to form the larger system.

10. Systems & Subsystems
Subsystem
Sub-system
Input
Output
Input
Output
Input
Output
Subsystem themselves can also be further divided down to “Sub-subsystems”.
This process can be continued until we reach the most basic elements of the digital logic world— basic logic gates.
A system has a layered hierarchical structure like an onion.

11. Systems & Subsystems

12. System & Subsystem Example

13. System & Subsystem Example

14. Analog System vs. Digital Systems

15. Analog System vs. Digital Systems
Input
Output
Digital
System
Input
Output
Two types of systems: Analog system and digital system.

16. Analog System vs. Digital Systems
Input
Output
The values of input/output/internal signals of an analog system can vary over a continuous range of values.

17. Analog System vs. Digital Systems
Input
Output 1 1
The values of input/output/internal signals of a digital system can only be 1’s and 0’s.

18. Analog System vs. Digital Systems
Digital I/P 1
A/D
Converter
Analog I/P
Lots of real-world signals are analog in nature.
Analog-to-Digital (A/D) converter has been used to process (digitization/quantization) the incoming analog signals and change them to digital (binary) signals.
The digital signals can now be processed by the digital system (e.g., microprocessor).

19. Analog System vs. Digital Systems
Digital I/P
Digital O/P 1
D/A
Converter
A/D
Analog I/P
Analog O/P
The output digital signals from the digital system are digital (binary) in nature.
Digital-to-Analog (D/A) converter has been used to process the outgoing digital signals and change them to analog (continuous) signals.
Most of the computers are the mix of analog and digital systems.

20. Analog System vs. Digital Systems
Digital I/P
Digital O/P 1
D/A
Converter
A/D
Analog I/P
Analog O/P

21. Digital Systems
Digital
System
Digital I/P
Digital O/P 1
D/A
Converter
A/D
Analog I/P
Analog O/P
In this unit, we are going to focus ONLY on the digital system.

22. Combinational vs. Sequential Systems

23. Combinational vs. Sequential Systems
Input
Output 1
Sequential
System
Input
Output 1
Two types of digital systems/circuits— combinational and sequential.

24. Combinational vs. Sequential Systems
Input
Output 1
Combinational
System
Input
Output 1
Combinational
System
Input
Output 1
The outputs of a combinational digital system/circuits are solely decided by its inputs.

25. Combinational vs. Sequential Systems
Input
Output 1
Sequential
System
Input
Output 1
Sequential
System
Input
Output 1
The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system.

26. Combinational vs. Sequential Systems
Input
Sequential
System
State: 101
State: 110
State: 001
Output 1 1 1 1 1
Input
Output 1 1 1 1 1
Input
Output 1 1 1 1
The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system.
Sequential system/circuits have some sort of “memory” in it, so it can “memorize” the “current state” of the system and behave accordingly.

27. Combinational vs. Sequential Systems
Input
Output 1
Sequential
System
Input
Output 1

28. Combinational Systems
Input
Output 1
Sequential
System
Input
Output 1
In this unit, we are going to focus MAINLY on the combinational digital system.

29. Truth Tables & Basic Logic Gates

30. Truth Tables p q
pq 1
A truth table shows how a logic circuit's output responds to various combinations of the inputs.
A truth table describe the system function of a logic system.
Use “1” and “0” to represent “T” and “F” respectively.
Use a logic gate symbol to represent the function.

31. Truth Tables & Basic Logic Gates
X = AB
There are eight basic logic gates. However, only AND, OR and NOT are the most fundamental ones.

32. Chapter 1 - Essential of Geometry
Negations (NOT)
The negation of a statement always has the opposite truth value of the original statement and is usually formed by adding the word not to the given statement.
Statement Right angle is 90o
TRUE
Negation Right angle is not 90o
FALSE
Statement Triangle has 4 sides
FALSE
Negation Triangle does not have 4 sides
TRUE

33. Truth Table – Negation (NOT)
Chapter 1 - Essential of Geometry
Truth Table – Negation (NOT)
The relationship between a statement p and its negation ~p can be summarized in a truth table.
A statement p and its negation ~p have opposite truth values. p ~p 1

34. Chapter 1 - Essential of Geometry
Conjunctions (AND)
A compound statement formed by combining two simple statements using the word and.
Statement: p, q
Conjunction: p and q
Symbols: p ^ q

35. Truth Table – Conjunctions (AND)
Chapter 1 - Essential of Geometry
Truth Table – Conjunctions (AND)
A conjunction is true when both statements are true.
When one or both statements are false, the conjunction is false. p q
p ^ q 1

36. Chapter 1 - Essential of Geometry
Disjunctions (OR)
A compound statement formed by combining two simple statements using the word or.
Statement: p, q
Disjunction p or q
Symbols: p V q

37. Truth Table – Disjunctions (OR)
Chapter 1 - Essential of Geometry
Truth Table – Disjunctions (OR)
A disjunction is true when one or both statements are true.
When both statements are false, the disjunction is false. p q
p V q 1

38. Truth Tables & Basic Logic Gates
System Function
Input X
Output Y
0/1
1/0 X Y 1
The truth table summarizes all the possible values of input signals and their corresponding output signal values.
The truth table is an effective way to describe the system function of a digital system.

39. Truth Tables & Basic Logic GatesX
0/1
1/0
A logic gate is a device performing a logical operation on one or more logical inputs, and produces a single logical output.

40. Boolean Algebra & Boolean Functions
Logic Gate X
0/1
1/0
Boolean algebra is a branch of algebra in which the values of the variables are the truth values true (1) and false (0).
The main operations of Boolean algebra are the conjunction (and, ), the disjunction (or, +), and the negation (not, ).

41. Boolean Algebra & Boolean Functions
Y = X
Y = A  B
Y = A + B
(NOT)
(AND)
(OR)
A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs.

42. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = X X Y 1 X
NOT
(Inverter)

43. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A  B A B Y 1
AND

44. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A + B A B Y 1 OR

45. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = X X Y 1 X
Buffer

46. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A  B A B Y 1
NAND

47. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A + B A B Y 1
NOR

48. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A ⊕ B A B Y 1
XOR

49. Truth Tables & Basic Logic Gates
Boolean Function
Truth Table
Logic Gate
Y = A  B A B Y 1
Y = A ⊕ B
XNOR

50. Truth Tables & Basic Logic GatesX X

51. Truth Tables & Basic Logic GatesX X

52. Truth Tables & Basic Logic GatesX X

53. Truth Tables & Basic Logic Gates
X = AB

54. Digital Logic Circuits

55. Digital Logic CircuitsY
Logic gates can be connected and cascaded to form logic circuits.
Every logic circuits can be treated as a system, and can be described by a system function— truth table.
We can derive the truth table of the circuit by evaluating the logic circuit stage by stage.

56. Digital Logic CircuitsE Y F D A B C D E F Y 1

57. Digital Logic CircuitsE Y F D A B C D E F Y 1

58. Digital Logic CircuitsE Y F D A B C D E F Y 1

59. Digital Logic CircuitsE Y F D A B C D E F Y 1

60. Digital Logic CircuitsE Y F D A B C D E F Y 1

61. Digital Logic CircuitsE Y F D A B C D E F Y 1

62. Digital Logic CircuitsE
Y = A  B + F D A B C D E F Y 1

63. Digital Logic Circuits ExerciseF G A B C D E F G Q 1

64. Digital Logic Circuits ExerciseF G A B C D E F G Q 1

65. Logic Simulation

66. Logic Simulation
Logic simulation is the use of simulation software (Logisim) to predict the behavior of digital circuits.

67. Equivalent Logic Circuits

68. Equivalent Logic Circuits
Logic circuits of different gates and forms can have an identical truth table. These circuits are called equivalent logic circuits.
This implies that a digital system with certain system function (truth table) can have many different implementations.

69. Laws of Boolean Algebra
One variable
NOT:
x = x
AND:
x · x = x
x · x = 0
OR:
x + x = x
x + x = 1
XOR:
x ⊕ x = 0
x ⊕ x = 1

70. Laws of Boolean Algebra
Commutativity
AND: x · y = y · x
OR: x + y = y + x
XOR: x ⊕ y = y ⊕ x
Associativity
AND: (x · y) · z = x · (y · z)
OR: (x + y) + z = x + (y + z)
XOR: (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z)

71. Laws of Boolean Algebra
Distributivity
x · (y + z) = (x · y) + (x · z)
x + (y · z) = (x + y) · (x + z)
x · (y ⊕ z) = (x · y) ⊕ (x · z)
De Morgan’s laws
NAND: x · y = x + y
NOR: x + y = x · y

72. Logic Simplification
We can apply the laws of Boolean algebra to reduce the expression to its simplest form (simplest defined as requiring the fewest gates to implement)

73. Equivalent Logic Circuits
Different implementations are chosen to meet different design considerations
Less number of gates (area)
less kind of gates (gate type), and
less stages of the circuits (speed).

74. Equivalent Logic Circuits
For example, what are the following logic circuits?

75. Equivalent Logic Circuits
For example, all of the following four logic circuits have the same truth table, and implement the XOR gate functionality.

76. Digital Building Blocks

77. Digital Building Blocks
All these logic circuits can be encapsulated into a block (XOR gate) and treated as a system.

78. Digital Building Blocks
This XOR gate can be further used as a building block to build larger and more complicated logic circuit such as a “full adder”.

79. Digital Building Blocks
A full adder can again be treated as a building block to build a larger logic circuit called Arithmetic-Logic Unit (ALU).

80. Digital Building Blocks
ALU can again be treated as a building block to build a Central Processing Unit (CPU).

81. Digital Building Blocks
This process can go on and on to build an Intel® Intel® Core™ i7 Processor with these digital logic building blocks.
Intel Core i7 Processor

82. Digital Building Blocks
The processor is so complicated at this level that it can contain hundreds of millions of basic logic gates.

83. Logic Gates & Boolean AlgebraX X