Introduction to Computer Logical Organization

1
Computer Logical Organization

1. Computer Logical Organization : Introduction
 In the modern world of electronics, the term Digital is generally associated with
a computer because the term Digital is derived from the way computers perform
operation, by counting digits.
 For many years, the application of digital electronics was only in the computer system.
 But now-a-days, digital electronics is used in many other applications.
 Following are some of the examples in which Digital electronics is heavily used.

Industrial process control

Military system

Television

Communication system

Medical equipment

Radar

Navigation

1.1 Signal
 Signal can be defined as a physical quantity, which contains some
information.
 It is a function of one or more than one independent variables.
 Signals are of two types.
1) Analog Signal
2) Digital Signal

1.1.1 Analog Signal
 An analog signal is defined as the signal having continuous values.
 Analog signal can have infinite number of different values.
 In real world scenario, most of the things observed in nature are analog.
 Examples of the analog signals are following.

Temperature

Pressure

Distance

Sound

Voltage

Current

Power

1.1.1 Graphical representation of Analog Signal (Temperature)
 The circuits that process the analog signals are called as analog circuits or
system.
 Examples of the analog system are following.

Filter

Amplifiers

Television receiver

Motor speed controller

1.1.1 Disadvantage of Analog Systems
i. Less accuracy
ii. Less versatility (No capability to carry multiple tasks)
iii. More noise effect
iv. More distortion (that occurs when the output signal
does not have a linear relation to the input signal.)
v. More effect of weather

1.1.2 Digital Signal
 A digital signal is defined as the signal which has only a finite number of
distinct values.
 Digital signals are not continuous signals.
 In the digital electronic calculator, the input is given with the help of switches.
 This input is converted into electrical signal which have two discrete values or levels.
 One of these may be called low level and another is called high level.
 The signal will always be one of the two levels.
 This type of signal is called digital signal.
 Examples of the digital signal are following.

Binary Signal

Octal Signal

Hexadecimal Signal

1.1.2 Graphical representation of the Digital Signal (Binary)
 The circuits that process the digital signals are called digital
systems or digital circuits.
 Examples of the digital systems are following.

Registers

Flip-flop

Counters

Microprocessors

1.1.2 Advantage of Digital Systems
i. More accuracy
ii. More versatility
iii. Less distortion
iv. Easy communicate
v. Possible storage of information

1.1.3 Comparison of Analog and Digital Signal
S.N. Analog Signal Digital Signal
1. Analog signal has infinite values.
Digital signal has a finite number of
values.
2. Analog signal has a continuous nature. Digital signal has a discrete nature.
3.
Analog signal is generated by
transducers and signal generators.
Digital signal is generated by A to D
converter.
4.
Example of analog signal − sine wave,
triangular waves.
Example of digital signal − binary signal.
Note :
• A transducer is a device that converts one form of energy to another form of energy.
• Energy types include/(but are not limited to) electrical, mechanical, electromagnetic
(including light), chemical, acoustic, and thermal energy.

1.2 Binary Codes
 In the coding, when numbers, letters or words are represented by a
specific group of symbols, it is said that the number, letter or word
is being encoded.
 The group of symbols is called as a code.
 The digital data is represented, stored and transmitted as group of
binary bits.
 This group is also called as binary code.
 The binary code is represented by the number as well as
alphanumeric letter.

1.2.1 Advantages of Binary Codes
 Following is the list of advantages that binary code offers.
1) Binary codes are suitable for the computer applications.
2) Binary codes are suitable for the digital communications.
3) Binary codes make the analysis and designing of digital
circuits if we use the binary codes.
4) Since only 0 & 1 are being used, implementation becomes easy.

1.2.2 Classification of Binary Codes
 The codes are broadly categorized into following four categories.
1) Weighted Codes
2) Non-Weighted Codes
3) Binary Coded Decimal Code
4) Alphanumeric Codes
5) Error Detecting Codes
6) Error Correcting Codes

1.2.2.1 Weighted Codes
 Weighted binary codes are those binary codes which obey the
positional weight principle.
 Each position of the number represents a specific weight.
 Several systems of the codes are used to express the decimal digits 0
through 9.
 In these codes each decimal digit is represented by a group of four bits.

1.2.2.2 Non-Weighted Codes
 In this type of binary codes, the positional weights are not assigned.
 The examples of non-weighted codes are Excess-3 code and Gray code.
i) Excess-3 code
 The Excess-3 code is also called as XS-3 code.
 It is non-weighted code used to express decimal numbers.
 The Excess-3 code words are derived from the 8421 BCD code words
adding (0011)2
or (3)10
to each code word in 8421.

1.2.2.2 Non-Weighted Codes : Excess-3 code
 The excess-3 codes are obtained as follows −
 Example:

1.2.2.2 Non-Weighted Codes : Gray Code
ii) Gray Code (or Reflected Binary code)
 It is the non-weighted code and it is not arithmetic codes.
 That means there are no specific weights assigned to the bit
position.
 It has a very special feature that, only one bit will change each time
the decimal number is incremented as shown in fig.
 As only one bit changes at a time, the gray code is called as a unit distance
code.
 The gray code is a cyclic code.
 Gray code cannot be used for arithmetic operation.

1.2.2.2 Non-Weighted Codes : Gray Code …
Application of Gray code:
 Gray code is popularly used in the shaft position encoders.
 A shaft position encoder produces a code word which represents
the angular position of the shaft.

0 0 20 11110 40 111100
1 1 21 11111 41 111101
2 11 22 11101 42 111111
3 10 23 11100 43 111110
4 110 24 10100 44 111010
5 111 25 10101 45 111011
6 101 26 10111 46 111001
7 100 27 10110 47 111000
8 1100 28 10010 48 101000
9 1101 29 10011 49 101001
10 1111 30 10001 50 101011
11 1110 31 10000 51 101010
12 1010 32 110000 52 101110
13 1011 33 110001 53 101111
14 1001 34 110011 54 101101
15 1000 35 110010 55 101100
16 11000 36 110110 56 100100
17 11001 37 110111 57 100101
18 11011 38 110101 58 100111
19 11010 39 110100 59 100110

1.2.2.3 Binary Coded Decimal (BCD) code
 In this code each decimal digit is represented by a 4-bit binary
number.
 BCD is a way to express each of the decimal digits with a binary
code.
 In the BCD, with four bits we can represent sixteen numbers (0000
to 1111).
 But in BCD code only first ten of these are used (0000 to 1001).
 The remaining six code combinations
 i.e. 1010 to 1111 are invalid in BCD.

1.2.2.3 Binary Coded Decimal (BCD) code
Decimal number Binary number Binary Coded Decimal(BCD)
0 0000 0000
1 0001 0001
2 0010 0010
3 0011 0011
4 0100 0100
5 0101 0101
6 0110 0110
7 0111 0111
8 1000 1000
9 1001 1001
10 1010 0001 0000
11 1011 0001 0001
12 1100 0001 0010
13 1101 0001 0011
14 1110 0001 0100
15 1111 0001 0101

1.2.2.3 Binary Coded Decimal (BCD) code …
Advantages of BCD Codes:
 It is very similar to decimal system.
 We need to remember binary equivalent of decimal numbers 0
to 9 only.
Disadvantages of BCD Codes:
 The addition and subtraction of BCD have different rules.
 The BCD arithmetic is little more complicated.
 BCD needs more number of bits than binary to represent the
decimal number.

So BCD is less efficient than binary.

1.2.2.4 Alphanumeric codes
 A binary digit or bit can represent only two symbols as it has only
two states '0' or '1'.
 But this is not enough for communication between two computers because
there we need many more symbols for communication.
 These symbols are required to represent 26 alphabets with capital
and small letters, numbers from 0 to 9, punctuation marks and
other symbols.
 The alphanumeric codes are the codes that represent numbers and
alphabetic characters.
 Mostly such codes also represent other characters such as symbol and
various instructions necessary for conveying information.
 An alphanumeric code should at least represent 10 digits and 26 letters of
alphabet i.e. total 36 items.

1.2.2.4 Alphanumeric codes …
 The following three alphanumeric codes are very commonly used for the data
representation.
i. ASCII :American Standard Code for Information Interchange (7 Bit).
ii. EBCDIC :Extended Binary Coded Decimal Interchange Code (8 Bit).
iii. UNICODE
iv. Baudot (or Baudot-Murray) Code (5 Bit).
 ASCII code is a 7-bit code whereas EBCDIC is an 8-bit code.
 ASCII code is more commonly used worldwide while EBCDIC is used primarily
in large IBM computers.

1.2.2.4 Alphanumeric codes : ASCII
 The full form of ASCII code is American Standard Code for
Information Interchange.
 It is a 7-bit code based on the English alphabet.
 In 1967 this code was first published and since then it is being
modified and updated.
 ASCII code has 128 characters some of which are enlisted below
to get familiar with the code.

DEC OCT HEX BIN Symbol Description
0 000 00 00000000 NUL Null char
1 001 01 00000001 SOH Start of Heading
2 002 02 00000010 STX Start of Text
3 003 03 00000011 ETX End of Text
4 004 04 00000100 EOT End of Transmission
5 005 05 00000101 ENQ Enquiry
6 006 06 00000110 ACK Acknowledgment
7 007 07 00000111 BEL Bell
8 010 08 00001000 BS Back Space
9 011 09 00001001 HT Horizontal Tab
10 012 0A 00001010 LF Line Feed
11 013 0B 00001011 VT Vertical Tab
12 014 0C 00001100 FF Form Feed
13 015 0D 00001101 CR Carriage Return
14 016 0E 00001110 SO Shift Out / X-On
15 017 0F 00001111 SI Shift In / X-O

1.2.2.4 Alphanumeric codes : EBCDIC
 The EBCDIC stands for Extended Binary Coded
Decimal Interchange Code.
 IBM invented this code to extend the Binary Coded
Decimal which existed at that time.
 All the IBM computers and peripherals use this code.
 It is an 8 bit code and therefore can accommodate 256
characters.
 Below is given some characters of EBCDIC code to get familiar with
it.

Char EBCDIC HEX Char EBCDIC HEX Char EBCDIC HEX
A 1100 0001 C1 P 1101 0111 D7 4 1111 0100 F4
B 1100 0010 C2 Q 1101 1000 D8 5 1111 0101 F5
C 1100 0011 C3 R 1101 1001 D9 6 1111 0110 F6
D 1100 0100 C4 S 1110 0010 E2 7 1111 0111 F7
E 1100 0101 C5 T 1110 0011 E3 8 1111 1000 F8
F 1100 0110 C6 U 1110 0100 E4 9 1111 1001 F9
G 1100 0111 C7 V 1110 0101 E5 blank ... ...
H 1100 1000 C8 W 1110 0110 E6 . ... ...
I 1100 1001 C9 X 1110 0111 E7 ( ... ...
J 1101 0001 D1 Y 1110 1000 E8 + ... ...
K 1101 0010 D2 Z 1110 1001 E9 $ ... ...
L 1101 0011 D3 0 1111 0000 F0 * ... ...
M 1101 0100 D4 1 1111 0001 F1 ) ... ...
N 1101 0101 D5 2 1111 0010 F2 - ... ...
O 1101 0110 D6 3 1111 0011 F3 /

1.2.2.4 Alphanumeric codes : UNICODE
 Unicode is the newest concept in digital coding. In
Unicode every number has a unique character.
 Leading technological giants have adopted this code for
its uniqueness.
 A part of the Unicode table is given below.

1.2.2.4 Alphanumeric codes : UNICODE
0 1 2 3 4 5 6 7 8 9 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
0000 ‑ Symbols
0020 ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? Number
0040 @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ ] ^ _ Alphabet
0060 ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~
0080 € ‚ ƒ „ … † ‡ ˆ ‰ Š ‹ Œ Ž ‘ ’ “ ” • – — ˜ ™ š › œ ž Ÿ
00A0 ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿
00C0 À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß Latin
000E à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ
0100 Ā ā Ă ă Ą ą Ć ć Ĉ ĉ Ċ ċ Č č Ď ď Đ đ Ē ē Ĕ ĕ Ė ė Ę ę Ě ě Ĝ ĝ Ğ ğ
0120 Ġ ġ Ģ ģ Ĥ ĥ Ħ ħ Ĩ ĩ Ī ī Ĭ ĭ Į į İ ı Ĳ ĳ Ĵ ĵ Ķ ķ ĸ Ĺ ĺ Ļ ļ Ľ ľ Ŀ
0140 ŀ Ł ł Ń ń Ņ ņ Ň ň ŉ Ŋ ŋ Ō ō Ŏ ŏ Ő ő Œ œ Ŕ ŕ Ŗ ŗ Ř ř Ś ś Ŝ ŝ Ş ş
0160 Š š Ţ ţ Ť ť Ŧ ŧ Ũ ũ Ū ū Ŭ ŭ Ů ů Ű ű Ų ų Ŵ ŵ Ŷ ŷ Ÿ Ź ź Ż ż Ž ž ſ
0180 ƀ Ɓ Ƃ ƃ Ƅ ƅ Ɔ Ƈ ƈ Ɖ Ɗ Ƌ ƌ ƍ Ǝ Ə Ɛ Ƒ ƒ Ɠ Ɣ ƕ Ɩ Ɨ Ƙ ƙ ƚ ƛ Ɯ Ɲ ƞ Ɵ
01A0 Ơ ơ Ƣ ƣ Ƥ ƥ Ʀ Ƨ ƨ Ʃ ƪ ƫ Ƭ ƭ Ʈ Ư ư Ʊ Ʋ Ƴ ƴ Ƶ ƶ Ʒ Ƹ ƹ ƺ ƻ Ƽ ƽ ƾ ƿ
01C0 ǀ ǁ ǂ ǃ Ǆ ǅ ǆ Ǉ ǈ ǉ Ǌ ǋ ǌ Ǎ ǎ Ǐ ǐ Ǒ ǒ Ǔ ǔ Ǖ ǖ Ǘ ǘ Ǚ ǚ Ǜ ǜ ǝ Ǟ ǟ
100E Ǡ ǡ Ǣ ǣ Ǥ ǥ Ǧ ǧ Ǩ ǩ Ǫ ǫ Ǭ ǭ Ǯ ǯ ǰ Ǳ ǲ ǳ Ǵ ǵ Ƕ Ƿ Ǹ ǹ Ǻ ǻ Ǽ ǽ Ǿ ǿ
200 Ȁ ȁ Ȃ ȃ Ȅ ȅ Ȇ ȇ Ȉ ȉ Ȋ ȋ Ȍ ȍ Ȏ ȏ Ȑ ȑ Ȓ ȓ Ȕ ȕ Ȗ ȗ Ș ș Ț ț Ȝ ȝ Ȟ ȟ
0220 Ƞ ȡ Ȣ ȣ Ȥ ȥ Ȧ ȧ Ȩ ȩ Ȫ ȫ Ȭ ȭ Ȯ ȯ Ȱ ȱ Ȳ ȳ ȴ ȵ ȶ ȷ ȸ ȹ Ⱥ Ȼ ȼ Ƚ Ⱦ ȿ
0240 ɀ Ɂ ɂ Ƀ Ʉ Ʌ Ɇ ɇ Ɉ ɉ Ɋ ɋ Ɍ ɍ Ɏ ɏ ɐ ɑ ɒ ɓ ɔ ɕ ɖ ɗ ɘ ə ɚ ɛ ɜ ɝ ɞ ɟ
0260 ɠ ɡ ɢ ɣ ɤ ɥ ɦ ɧ ɨ ɩ ɪ ɫ ɬ ɭ ɮ ɯ ɰ ɱ ɲ ɳ ɴ ɵ ɶ ɷ ɸ ɹ ɺ ɻ ɼ ɽ ɾ ɿ
0280 ʀ ʁ ʂ ʃ ʄ ʅ ʆ ʇ ʈ ʉ ʊ ʋ ʌ ʍ ʎ ʏ ʐ ʑ ʒ ʓ ʔ ʕ ʖ ʗ ʘ ʙ ʚ ʛ ʜ ʝ ʞ ʟ
02A0 ʠ ʡ ʢ ʣ ʤ ʥ ʦ ʧ ʨ ʩ ʪ ʫ ʬ ʭ ʮ ʯ ʰ ʱ ʲ ʳ ʴ ʵ ʶ ʷ ʸ ʹ ʺ ʻ ʼ ʽ ʾ ʿ
02C0 ˀ ˁ ˂ ˃ ˄ ˅ ˆ ˇ ˈ ˉ ˊ ˋ ˌ ˍ ˎ ˏ ː ˑ ˒ ˓ ˔ ˕ ˖ ˗ ˘ ˙ ˚ ˛ ˜ ˝ ˞ ˟
200E ˠ ˡ ˢ ˣ ˤ ˥ ˦ ˧ ˨ ˩ ˪ ˫ ˬ ˭ ˮ ˯ ˰ ˱ ˲ ˳ ˴ ˵ ˶ ˷ ˸ ˹ ˺ ˻ ˼ ˽ ˾ ˿
0300 ̀ ́ ̂ ̃ ̄ ̅ ̆ ̇ ̈ ̉ ̊ ̋ ̌ ̍ ̎ ̏ ̐ ̑ ̒ ̓ ̔ ̕ ̖ ̗ ̘ ̙ ̚ ̛ ̜ ̝ ̞ ̟
0320 ̠ ̡ ̢ ̣ ̤ ̥ ̦ ̧ ̨ ̩ ̪ ̫ ̬ ̭ ̮ ̯ ̰ ̱ ̲ ̳ ̴ ̵ ̶ ̷ ̸ ̹ ̺ ̻ ̼ ̽ ̾ ̿
0340 ̀ ́ ͂ ̓ ̈́ ͅ ͆ ͇ ͈ ͉ ͊ ͋ ͌ ͍ ͎ ͏ ͐ ͑ ͒ ͓ ͔ ͕ ͖ ͗ ͘ ͙ ͚ ͛ ͜ ͝ ͞ ͟
0360 ͠ ͡ ͢ ͣ ͤ ͥ ͦ ͧ ͨ ͩ ͪ ͫ ͬ ͭ ͮ ͯ Ͱ ͱ Ͳ ͳ ʹ ͵ Ͷ ͷ ͸ ͹ ͺ ͻ ͼ ͽ ; Ϳ

1.2.2.4 Alphanumeric codes : Baudot-Murray code
 In digital telegraphy (teleprinter, telex) a standard 5-bit code is
commonly used to represent a character (letter, number or
punctuation mark).
 Although this code is generally known as the Baudot code, this
name is actually wrong.
 The official name for the latest telegraphy standard is ITA2 (International Telegraph
Alphabet No 2).
 It was superceeded by ASCII in 1963, but is still used by amateurs today.
 The most common 'Baudot' code is also known as Murray code, or
as Baudot-Murray code.
 The ITA2 standard is used widely with historical cipher machines.

ITA2 Standard:
 Normal text consists of over 50 different characters (26 letters, 10 numbers, 10
punctuation marks and some control codes).
 In the ITA2 standard, 5 bits are used to represent a character, which means that we can create
only 32 different codes (25).
 As this would not be sufficient to create normal text, most codes are used twice (i.e. have two
different meanings) and special codes are used to 'shift' between the two code sets known as
Letters (LTRS) and Figures (FIGS).
 The table below shows the interpretation of the commonly used ITA2 standard.
 Although this code has officially been superseded by ASCII, it is still in use on some old telex
networks and by Radio Amateurs.
 Some of the cipher machines described on this website, use 5-bit digital telegraphy and many
of these (if they support the Latin character set) follow the ITA2 standard.

 LTRS-shift is represented by 111·11 (5 holes), so that it can be used to wipe part of
a paper tape, without affecting the rest.
 Normally, a paper tape would start with two LTRS characters, to ensure that the
teletype is in Letters-mode. By convention, the holes (bits) in the tape are called
channels or tracks and the tape is shown as it would run through the reader
away from you.

Ltr Letters (A-Z)
Fig Figures (Numbers and punctuation marks)
Ctrl Control characters
Hex Hexadecimal code
Bin 1 Binary, Positions of the holes in the paper
tape
Ltr Fig Hex Bin
0 NUL 00 000·00 NULL, Nothing (blank tape)
1 E 3 01 000·01
2 LF 02 000·10 Line Feed (new line)
3 A - 03 000·11
4 SP 04 001·00 Space
5 S ' 05 001·01
6 I 8 06 001·10
7 U 7 07 001·11
8 CR 08 010·00 Cariage Return
9 D ENC 09 010·01 Enquiry (Who are you?, WRU)
10 R 4 0A 010·10
11 J BEL 0B 010·11 BELL (ring bell at the other end)
12 N , 0C 011·00
13 F ! 0D 011·01
14 C : 0E 011·10
15 K ( 0F 011·11
16 T 5 10 100·00
17 Z + 11 100·01
18 L ) 12 100·10
19 W 2 13 100·11
20 H $ 14 101·00 Currency symbol (can also be £)
21 Y 6 15 101·01
22 P 0 16 101·10
23 Q 1 17 101·11
24 O 9 18 110·00
25 B ? 19 110·01
26 G & 1A 110·10 Can also be @
27 FIGS 1B 110·11 Figures (Shift on)
28 M . 1C 111·00
29 X / 1D 111·01
30 V ; 1E 111·10
31 LTRS 1F 11·111 Letters (Shift off)

1.2.2.5 Error Codes
What is Error?
 Error is a condition when the output information does not match with the
input information.
 During transmission, digital signals suffer from noise that can introduce errors
in the binary bits travelling from one system to other.
 That means a 0 bit may change to 1 or a 1 bit may change to 0.

1.2.2.5 Error Codes …
Error-Detecting codes:
 Whenever a message is transmitted, it may get scrambled by noise or data may
get corrupted.
 To avoid this, we use error-detecting codes which are additional data added to a given
digital message to help us detect if an error occurred during transmission of the
message.
 A simple example of error-detecting code is parity check.
Error-Correcting codes:
 Along with error-detecting code, we can also pass some data to figure out the
original message from the corrupt message that we received.
 This type of code is called an error-correcting code.
 Error-correcting codes also deploy the same strategy as error-detecting codes but
additionally, such codes also detect the exact location of the corrupt bit.
 In error-correcting codes, parity check has a simple way to detect errors along
with a sophisticated mechanism to determine the corrupt bit location.
 Once the corrupt bit is located, its value is reverted (from 0 to 1 or 1 to 0) to get the
original message.

How to Detect and Correct Errors?
 To detect and correct the errors, additional bits are added to the
data bits at the time of transmission.
 The additional bits are called parity bits.
 They allow detection or correction of the errors.
 The data bits along with the parity bits form a code word.

Parity Checking of Error Detection:
 It is the simplest technique for detecting and correcting errors.
 The MSB of an 8-bits word is used as the parity bit and the remaining 7 bits are
used as data or message bits.
 The parity of 8-bits transmitted word can be either even parity or odd parity.
 Even parity : Even parity means the number of 1's in the given word including the
parity bit should be even (2,4,6,....).
 Odd parity : Odd parity means the number of 1's in the given word including the parity
bit should be odd (1,3,5,....).

Use of Parity Bit:
 The parity bit can be set to 0 and 1 depending on the type of the parity
required.
 For even parity, this bit is set to 1 or 0 such that the no. of "1 bits" in the entire
word is even.
 Shown in fig. (a).
 For odd parity, this bit is set to 1 or 0 such that the no. of "1 bits" in the entire
word is odd.
 Shown in fig. (b).

How Does Error Detection Take Place?
 Parity checking at the receiver can detect the presence of an error if the parity
of the receiver signal is different from the expected parity.
 That means, if it is known that the parity of the transmitted signal is always going
to be "even" and if the received signal has an odd parity, then the receiver can
conclude that the received signal is not correct.
 If an error is detected, then the receiver will ignore the received byte and request
for retransmission of the same byte to the transmitter.

1.3 Boolean Algebra
 Boolean Algebra is used to analyze and simplify the digital (logic)
circuits.
 It uses only the binary numbers i.e. 0 and 1.
 It is also called as Binary Algebra or logical Algebra.
 Boolean algebra was invented by George Boole in 1854.

1.3.1 Rule in Boolean Algebra
Following are the important rules used in Boolean algebra.
I. Variable used can have only two values.
 Binary 1 for HIGH and Binary 0 for LOW.
II. Complement of a variable is represented by an overbar (-).
 Thus, complement of variable B is represented as . .
 Thus if B = 0 then = 1 and B = 1 then = 0.
III. ORing of the variables is represented by a plus (+) sign between
them.
 For example ORing of A, B, C is represented as A + B + C.
IV. Logical ANDing of the two or more variable is represented by
writing a dot between them such as A.B.C. Sometime the dot may
be omitted like ABC.

1.3.2 Boolean Laws
 There are six types of Boolean Laws.
1) Commutative law
2) Associative law
3) Distributive law
4) AND law
5) OR law
6) INVERSION law

1.3.2 Boolean Laws …
1) Commutative law
 Any binary operation which satisfies the following expression is referred to as
commutative operation.
 Commutative law states that changing the sequence of the variables does not
have any effect on the output of a logic circuit.
2) Associative law
 This law states that the order in which the logic operations are performed is
irrelevant as their effect is the same.

1.3.2 Boolean Laws …
3) Distributive law
 Distributive law states the following condition.
4) AND law
 These laws use the AND operation.
 Therefore they are called as AND laws.
5) OR law
 These laws use the OR operation.
 Therefore they are called as OR laws.
6) INVERSION law
 This law uses the NOT operation.
 The inversion law states that double inversion of a variable results in the original variable itself.

1.3.3 Boolean Expression ⁄ Function
 Boolean algebra deals with binary variables and logic operation.
 A Boolean Function is described by an algebraic expression called Boolean
expression which consists of binary variables, the constants 0 and 1, and the
logic operation symbols.
 Consider the following example.
 Here the left side of the equation represents the output Y.
 So we can state equation no. 1

1.3.3 Boolean Expression ⁄ Function …
Truth Table Formation:
 A truth table represents a table having all combinations of inputs and their
corresponding result.

It is possible to convert the switching equation into a truth table.
 For example, consider the following switching equation.
 The output will be high (1) if A = 1 or BC = 1 or both are 1.
 The truth table for this equation is shown by Table (a).
 The number of rows in the truth table is 2n

where n is the number of input variables (n=3 for the given equation).
 Hence there are 23
= 8 possible input combination of inputs.

Methods to simplify the Boolean function:
 The methods used for simplifying the Boolean function are as follows −
1) Karnaugh-map or K-map, and
2) NAND gate method.
1) Karnaugh-map or K-map
 The Boolean theorems and the De-Morgan's theorems are useful in
manipulating the logic expression.
 We can realize the logical expression using gates.
 The number of logic gates required for the realization of a logical expression
should be reduced to a minimum possible value by K-map method.
 This method can be done in two different ways, as discussed below.

a) Sum of Products (SOP) Form:
 It is in the form of sum of three terms AB, AC, BC with each individual term is a
product of two variables.
 Say A.B or A.C etc.
 Therefore such expressions are known as expression in SOP form.
 The sum and products in SOP form are not the actual additions or multiplications.

In fact they are the OR and AND functions.
 In SOP form, 0 represents a bar and 1 represents an unbar.
 SOP form is represented by .
Given below is an example of SOP.

b) Product of Sums (POS) Form:
 It is in the form of product of three terms (A+B), (B+C), or (A+C) with each term
is in the form of a sum of two variables.
 Such expressions are said to be in the product of sums (POS) form.
 In POS form, 0 represents an unbar and 1 represents a bar.
 POS form is represented by .
 Given below is an example of POS.

1.3.3 Boolean Expression ⁄ Function
2) NAND gates Realization:
 NAND gates can be used to simplify Boolean functions as shown in the example
below.

1.3.4 De Morgan's Theorems
 De Morgan has suggested two theorems which are extremely useful in Boolean
Algebra.
 The two theorems are discussed below.
Theorem 1:
 The left hand side (LHS) of this theorem represents a NAND gate with inputs A
and B, whereas the right hand side (RHS) of the theorem represents an OR gate
with inverted inputs.
 This OR gate is called as Bubbled OR.

1.3.4 De Morgan's Theorems …
Theorem 1:
 Table showing verification of the De Morgan's first theorem -

1.3.4 De Morgan's Theorems
Theorem 2:
 The LHS of this theorem represents a NOR gate with inputs A and B, whereas
the RHS represents an AND gate with inverted inputs.
 This AND gate is called as Bubbled AND.

Theorem 2:
 Table showing verification of the De Morgan's second theorem-

 Error

1.2.2.4 Alphanumeric codes : Five bit Baudot Code

Introduction to Computer Logical Organization

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