1. 3 – Boolean Logic and Logic Gates 4 – Binary Numbers
CS 1 Introduction to Computers and Computer Technology
Rick Graziani
Fall 2017

2. BIT – BInary digiT ON OFF
Bit (Binary Digit) = Basic unit of information, representing one of two discrete states. The smallest unit of information within the computer.
The only thing a computer understands.
Abbreviation: b
Bit has one of two values:
0 (off) or 1 (on)
0 (False) or 1 (True) ON
OFF
Rick Graziani

3. Bits
The boxes illustrate a position where magnetism may be set and sensed; pluses (red) indicate magnetism of positive polarity (1 bit), interpreted as “present” and minuses (blue) (0 bit). 1 1 1 1 1 1 1 1
Two patterns are known as the state of the bit.
For example, magnetic encoding of information on tapes, floppy disks, and hard disks are done with positive or negative polarity.
Rick Graziani

4. Bits Bits are really only symbols.
Used to display the one of two different, discrete states.
Bits are used as:
Storing data
Numbers
Text characters
Images
Sound
Etc.
Processing data
Rick Graziani

5. Boolean Operations
Integrated Circuits (microchips) are used to store and manipulate (process) bits.
This is done using Boolean operations (in honor of mathematician George Boole, ).
Boolean Operation: An operation that manipulates one or more true/false values
Specific operations
AND OR
XOR (exclusive or)
NOT
Using Truth Tables we can uses different sets of logic operations to store, add, subtract, and more complicated operations with bit.
Rick Graziani

6. Boolean Algebra and logical expressions (Addendum)
Boolean algebra (due to George Boole) - The mathematics of digital logic
Useful in dealing with binary system of numbers.
Used in the analysis and synthesis of logical expressions.
Logical expressions – Expressions constructed using logical-variables and operators.
Result is: True or False
Boolean algebra – In mathematics a variable uses one of the two possible values: 1 or 0
May also be represented as:
Truth or Falsehood of a statement
On or Off states of a switch
High (5V) or low (0V) of a voltage level
Rick Graziani

7. Used in electronics (Addendum)
Electrical circuits are designed to represent logical expressions
Known as logic circuits.
Used to make important logical decisions in household appliances, computers, communication devices, traffic signals and microprocessors.
Three basic logic operations as listed below:
OR operation
AND operation
NOT operation
Rick Graziani

8. Logic gates
A logic gate is an electronic circuit/device which makes the logical decisions based on these operations.
Logic gates have:
one or more inputs
only one output
The output is active only for certain input combinations.
Logic gates are the building blocks of any digital circuit.
Rick Graziani

9. Boolean Operations - AND
TRUE
TRUE
AND
= TRUE
Truth tables (simple ones)
AND operation
Both input values must be TRUE for output to be TRUE
Kermit is a frog AND Miss Piggy is an actress
Inputs to AND operation represent truth of falseness of the compound statement.
Rick Graziani

10. Boolean Operations Gate: A device that computes a Boolean operation
A device that produces the output of a Boolean operation when given the operation’s input values.
Gates can be:
Gears
Relays
Optic devices
Electronic circuits (microchips)
Rick Graziani

11. Boolean Operations – AND Gate
Truth Table
Inputs Output 1 1 1
0 = FALSE
1 = TRUE
AND operation
Both input values must be TRUE for output to be TRUE 1 1 1
Rick Graziani

12. Off (False) Off (False) On (True)
To build an AND gate: Two transistors connected together
Two inputs (transistors A and B) and one output
Transistor A: Off (False)
Transistor B: On (True)
Output: Off (False)
Rick Graziani

13. On (True) On (True) On (True) Transistor A: On (True)
Transistor B: On (True)
Output: On (True)
Rick Graziani

14. Boolean Operations - OR
TRUE OR
True
= TRUE
Truth tables (simple ones)
OR operation
Only one input values must be TRUE for output to be TRUE
In Rick likes to surf OR Rick likes to go dancing.
Taking both courses will also TRUE.
Rick Graziani

15. Boolean Operations – OR Gate
Truth Table
Inputs Output 1 1 1 1 1 1
0 = FALSE
1 = TRUE
OR operation
At least one input value must be TRUE for output to be TRUE 1 1 1 1
Rick Graziani

16. Two inputs (transistors A and B) and one output
Transistor A: Off (False)
Transistor B: Off (False)
Output: Off (False)
Rick Graziani

17. Two inputs (transistors A and B) and one output
Transistor A: Off (False)
Transistor B: On (True)
Output: On (True)
Rick Graziani

18. Two inputs (transistors A and B) and one output
Transistor A: On (True)
Transistor B: On (True)
Output: On (True)
Rick Graziani

19. Boolean Operations - XOR
TRUE
XOR
False
= TRUE
Truth tables (simple ones)
XOR operation
One and ONLY one input value can be TRUE for output to be TRUE
At noon Rick is going to surf the Hook XOR surf Liquor Stores (this is a surf spot)
Both cannot be true, as I cannot surf both spots at the same time.
Rick Graziani

20. Boolean Operations – XOR Gate
Truth Table
Inputs Output 1 1 1 1 1 1
0 = FALSE
1 = TRUE
XOR operation
Only one input value is TRUE for output to be TRUE 1 1
Rick Graziani

21. Rick Graziani graziani@cabrillo.edu

22. Rick Graziani graziani@cabrillo.edu

23. Rick Graziani graziani@cabrillo.edu

24. Boolean Operations – NOT Gate
Truth Table 1
Inputs Output 1 1 1
0 = FALSE
1 = TRUE
NOT operation
Only one input
Opposite of input
NOT FALSE = TRUE
NOT TRUE = FALSE
Rick Graziani

25. Current To build an NOT gate: One transistor One input and one output
Transistor A: On (True)
Current flows to ground wire and none to output, so output is Off (False)
Rick Graziani

26. Current Transistor A: Off (False)
Current flows to ground wire and none to output, so output is Off (False)
Rick Graziani

27.
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28. Another way to write it…
0 = FALSE; 1 = TRUE
Rick Graziani

29. Binary Numbers

30. Binary = Of two states
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31. Binary Math
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32. Base 10 (Decimal) Number System
Digits (10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Number of:
10,000’s 1,000’s 100’s 10’s 1’s 1 2 3 9
Rick Graziani

33. Base 10 (Decimal) Number System
Digits (10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Number of:
10,000’s 1,000’s 100’s 10’s 1’s
Rick Graziani

34. Rick’s Number System Rules
All digits start with 0
A Base-n number system has n number of digits:
Decimal: Base-10 has 10 digits
Binary: Base-2 has 2 digits
Hexadecimal: Base-16 has 16 digits
The first column is always the number of 1’s
Each of the following columns is n times the previous column (n = Base-n)
Base 10: 10, ,
Base 2:
Base 16: 65, ,
Rick Graziani

35. Counting in Decimal (0,1,2,3,4,5,6,7,8,9) 1,000’s 100’s 10’s 1’s 1 2 31 2 3
... 9
1 0
1 1
1 8
1 9
2 0
2 1
2 2
1,000’s 100’s 10’s 1’s
. . .
2 9
3 0
3 1
...
9 9
Rick Graziani

36. Counting in Binary (0, 1)
8’s 4’s 2’s 1’s 1
1 0
1 1
Dec
8’s 4’s 2’s 1’s
Dec 9 1 10 2 3 11 4 12 5 6 13 7 14 8 15
Rick Graziani

37. Binary Math (more later)
>
1000 ……
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38. Base 2 (Binary) Number System
Digits (2): 0, 1
Number of:
128’s 64’s 32’s 16’s 8’s 4’s 2’s 1’s
Dec. 17 70
130
255
Rick Graziani

39. Base 2 (Binary) Number System
Digits (2): 0, 1
Number of:
128’s 64’s 32’s 16’s 8’s 4’s 2’s 1’s
Dec.
Rick Graziani

40. Converting between Decimal and Binary
Digits (2): 0, 1
Number of:
128’s 64’s 32’s 16’s 8’s 4’s 2’s 1’s
Dec.
172
192
Rick Graziani

41. Converting between Decimal and Binary
Digits (2): 0, 1
Number of:
128’s 64’s 32’s 16’s 8’s 4’s 2’s 1’s
Dec.
Rick Graziani

42. 0 1 Computers do Binary Bits have two values: OFF and ON
0 1
Bits have two values: OFF and ON
The Binary number system (Base-2) can represent OFF and ON very well since it has two values, 0 and 1
0 = OFF
1 = ON
Understanding Binary to Decimal conversion is critical in computer science, computer networking, digital media, etc.
Rick Graziani

43. Rick’s Program
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44. Rick’s Program
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45. Rick’s Program
Rick Graziani

46. Decimal Math - Addition
10,000’s 1,000’s 100’s 10’s 1’s 1 1 1 3 3 1 5
Rick Graziani

47. Binary Math - Addition 1 1 1 1 1 1 1 1 Double check using Decimal.
64’s 32’s 16’s 8’s 4’s 2’s 1’s
Dec 1 1 1 1 58 + 27
----- 1 1 1 1 85
Double check using Decimal.
Rick Graziani

48. Half Adder Gate – Adding two bits
XOR
Inputs: A, B
S = Sum
C = Carry
AND
A + B = 2’s 1’s
Rick Graziani

49. Half Adder Gate – Adding two bits
XOR
Inputs: A, B
S = Sum
C = Carry
AND C S
+ 0
----
A + B = 2’s 1’s =
Rick Graziani

50. Half Adder Gate – Adding two bits
XOR
Inputs: A, B
S = Sum
C = Carry 1 1
AND C S
+ 1
----
A + B = 2’s 1’s = 1 1
Rick Graziani

51. Half Adder Gate – Adding two bits
XOR 1
Inputs: A, B
S = Sum
C = Carry 1
AND C S 1
+ 0
----
A + B = 2’s 1’s = 1 1
Rick Graziani

52. Half Adder Gate – Adding two bits
XOR 1
Inputs: A, B
S = Sum
C = Carry 1 1
AND C S 1
+ 1
----
A + B = 2’s 1’s = 1
1 0
Rick Graziani

53. Marble Adding Machine
Rick Graziani

54. Rick Graziani graziani@cabrillo.edu

55. Rick Graziani graziani@cabrillo.edu

56. Rick Graziani graziani@cabrillo.edu

57. Text

58. Digitizing Text
Earliest uses of PandA (Presence and Absence) was to digitize text (keyboard characters).
We will look at digitizing images and video later.
Assigning Symbols in United States:
26 upper case letters
26 lower case letters
10 numerals
20 punctuation characters
10 typical arithmetic characters
3 non-printable characters (enter, tab, backspace)
95 symbols needed
Rick Graziani

59. ASCII-7
In the early days, a 7 bit code was used, with 128 combinations of 0’s and 1’s, enough for a typical keyboard.
The standard was developed by ASCII (American Standard Code for Information Interchange)
Each group of 7 bits was mapped to a single keyboard character.
0 =
1 =
2 =
3 =
… 127 =
Rick Graziani

60. Byte
Byte = A collection of bits (usually 7 or 8 bits) which represents a character, a number, or other information.
More common: 8 bits = 1 byte
Abbreviation: B
Rick Graziani

61. Bytes 1 byte (B) Kilobyte (KB) = 1,024 bytes (210)
“one thousand bytes”
1,024 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2
Megabyte (MB) = 1,048,576 bytes (220)
“one million bytes”
Gigabyte (GB) = 1,073,741,824 bytes (230)
“one billion bytes”
Rick Graziani

62. Wikipedia
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63. ASCII-8 IBM later extended the standard, using 8 bits per byte.
This was known as Extended ASCII or ASCII-8
This gave 256 unique combinations of 0’s and 1’s.
0 =
1 =
2 =
3 =
… 255 = 1
Rick Graziani

64. ASCII-8
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65. Try it! 1
Write out Cabrillo College (Upper and Lower case) in bits (binary) using the chart above. …
C a
Rick Graziani

66. The answer! 1
C a b r i l
l o space C o l
l e g e
Rick Graziani

67. Unicode
Although ASCII works fine for English, many other languages need more than 256 characters, including numbers and punctuation.
Unicode uses a 16 bit representation, with 65,536 possible symbols.
Unicode can handle all languages.
Rick Graziani

68. 3 – Boolean Logic and Logic Gates 4 – Binary Numbers
CS 1 Introduction to Computers and Computer Technology
Rick Graziani