Binary Codes Logic Gate

& Boolean algebra

By Ratnesh sir

The first successful system of electrical communication was

the telegraph, which was invented by Samuel F.B. Morse in

the year 1832. The telegraph operators used a code of clicks

to send the messages. If the key pressed for a short time it is

the Morse code called ‘dot’ and if the key pressed for a long-

time it is ‘Dash’. A sample Morse code is shown below:

Introduction

In weighted codes, each digit is assigned a specific weight according

to its position. For example, in 8421BCD code, 1001 the weights of 1, 0,

0, 1 (from left to right) are 8, 4, 2 and 1 respectively.

Suppose W1' W

2' W

3and W

4are the weights of binary digits and X

l' X

2,

X3

and X4

are the corresponding digit values then decimal digit. N =

W1

Xl+ W

2X

2+ W

3X

3+ W

4X

4is represented by binary sequence

X4

X3

X2

X4.

Weighted codes

The codes 8421BCD, 2421BCD, 5211BCD are all weighted codes

The non-weighted codes are not positionally weighted. In other words,

each digit position within the number is not assigned a fixed value ( or

weight ).

Non-weighted codes:

Excess-3 and gray code are non-weighted codes.

A code is reflective when the code is self complementing. In

other words, when the code for 9 is the complement the code

for 0, 8 for 1, 7 for 2, 6 for 3 and 5 for 4.

Reflective codes:

2421BCD, 5421BCD and Excess-3 code are reflective codes.

In sequential codes, each succeeding 'code is one binary

number greater than its preceding code. This property helps

in manipulation of data.

Sequential codes:

8421 BCD and Excess-3 are sequential codes

Codes used to represent numbers, alphabetic characters,

symbols and various instructions necessary for conveying

intelligible information.

Alphanumeric codes:

ASCII, EBCDIC, UNICODE are the most-commonly used alphanumeric

codes.

ASCII--------- American Standard Code for Information Interchange

EBCDIC ------ Extended Binary Coded Decimal Interchange Code

Codes which allow error defection and correction are called

error detecting and' correcting codes.

Error defecting and correcting codes:

Hamming code is the mostly commonly used error detecting

and correcting code

Whenever any binary number appears, its decimal equivalent can be

found easily as follows.

When there is 1 in a digit position, weight of that position should be

added.

When there is 0 in a digit position, weight of that position should be

disregarded.

Binary Weights

For example binary number 1101 has a decimal equivalent of 8 + 4 + 0 + 1 =

13.

The decimal numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 can be expressed in

Binary numbers as shown below. All these binary numbers again

expressed in the last column by expanding into 4 bits. As per the

weighted binary digits, the 4 Bit binary numbers can be expressed

according to their place value from left to right as 8421 (2³ 2² 2¹ 2⁰ =

8421).

8421 Code or BCD Code

DECIMAL NUMBER BINARY NUMBER4 BIT

EXPRESSION(8421)

0 0 0000

1 1 0001

2 10 0010

3 11 0011

4 100 0100

5 101 0101

6 110 0110

7 111 0111

8 1000 1000

9 1001 1001

DECIMAL NUMBER BINARY NUMBER 2421 CODE

0 0 0000

1 1 0001

2 10 0010

3 11 0011

4 100 0100

5 101 1011

6 110 1100

7 111 1101

8 1000 1110

9 1001 1111

This code also a 4 bit application code where the binary weights carry 2, 4, 2, 1 from

left to right.

2421 Code

DECIMAL NUMBER BINARY NUMBER 5211 CODE

0 0 0000

1 1 0001

2 10 0011

3 11 0101

4 100 0111

5 101 1000

6 110 1010

7 111 1100

8 1000 1110

9 1001 1111

This code is also a 4 bit application code where the binary weights

carry 5, 4, 2, 1 from left to right.

It can be observed that in the 2421 and 5211 codes,

the code for decimal 9 is the complement of the code for decimal 0,

the code for decimal 8 is the complement of the code for decimal 1,

the code for decimal 7 is the complement of the code for decimal 2,

the code for decimal 6 is the complement of the code for decimal 3,

the code for decimal 5 is the complement of the code for decimal 4,

these codes are called as reflexive codes.

Reflective Code

DECIMAL

NUMBER

DECIMAL

NUMBER2421 CODE 5211 CODE

0 0 0000 0000

1 1 0001 0001

2 10 0010 0011

3 11 0011 0101

4 100 0100 0111

5 101 1011 1000

6 110 1100 1010

7 111 1101 1100

8 1000 1110 1110

9 1001 1111 1111

In sequential codes, each succeeding 'code is one binary

number greater than its preceding code. This property helps

in manipulation of data

Sequential codes:

DECIMAL

NUMBERBINARY NUMBER 8421 CODE EXCESS-3

0 0 0000 0011

1 1 0001 0100

2 10 0010 0101

3 11 0011 0110

4 100 100 0111

5 101 0101 1000

6 110 0110 1001

7 111 0111 1010

8 1000 1000 1011

9 1001 1001 1100

Some of the codes will not follow the weights of the sequence binary

numbers these are called as non-weighted codes. ASCII code ,Grey

code and XS-3 are some of the examples where they are coded for

some special purpose applications and they do not follow the weighted

binary number calculations.

Non-Weighted Codes

Some of the codes will not follow the binary weights, Excee-3 code is

an example of it and it is an important 4 bit code. The excess – 3 code

of a decimal number is achieved by adding the number 3 to the 8421

code.

Excess-3 Code

Some of the codes will not follow the weights of the sequence binary

numbers these are called as non-weighted codes. ASCII code ,Grey

code and XS-3 are some of the examples where they are coded for

some special purpose applications and they do not follow the weighted

binary number calculations.

Non-Weighted Codes

Some of the codes will not follow the binary weights, Excee-3 code is

an example of it and it is an important 4 bit code. The excess – 3 code

of a decimal number is achieved by adding the number 3 to the 8421

code.

Excess-3 Code

Decimal BCD + 0011 Excee-3

Decimal Digit BCD Code Excess-3 Code

0 0000 0011

1 0001 0100

2 0010 0101

3 0011 0110

4 0100 0111

5 0101 1000

6 0110 1001

7 0111 1010

8 1000 1011

9 1001 1100

These are following advantages of Excess-3 codes,

These are unweighted binary decimal codes.

These are self-complementary codes.

The codes 0000 and 1111 are not used for any digit which is an

advantage for memory organization as these codes can cause fault in

transmission line.

It has no limitation, and it considerably simplifies arithmetic

operations.

It is particularly significant for arithmetic operations as it overcomes

shortcoming encountered while using 8421 BCD code to add two

decimal digits whose sum exceeds 9.

Advantages of Excess-3 Codes:

The gray code is the code where one bit will be differed to the

preceding number.

Gray code is not weighted that means it does not depends on positional

value of digit.

This cyclic variable code that means every transition from one value to

the next value involves only one bit change.

Gray Code

Gray codes are used in rotary and optical encoders, Karnaugh maps,

and error detection..

Conversion of Binary to Gray Code:

The hamming distance of two neighbours Gray codes is always 1 and

also first Gray code and last Gray code also has Hamming distance is

always 1, so it is also called Cyclic codes.

Using Karnaugh (K) - map:

Decimal Binary Gray Code

0 000 000

1 001 001

2 010 011

3 011 010

4 100 110

5 101 111

6 110 101

7 111 100

These are following steps for n-bit binary numbers:

The most significant bit (MSB) of the Gray code is always equal to the

MSB of the given Binary code.

Other bits of the output Gray code can be obtained by XORing binary

code bit at the index and previous index.

Using Exclusive-Or (⊕) operation:

Binary

b2b

1b

0

Gray Code

g2g

1g

0

000 000

001 001

010 011

011 010

100 110

101 111

110 101

111 100

Error Detection & Correction Codes- Hamming Code

We know that the bits 0 and 1 corresponding to two different range of

analog voltages. So, during transmission of binary data from one

system to the other, the noise may also be added. Due to this, there

may be errors in the received data at other system i.e. a bit 0 may

change to 1 or a bit 1 may change to 0

We can get back the original data first by detecting whether any

error(s) present and then correcting those errors. For this purpose, we

can use the following codes.

Error detection codes

Error correction codes

This method are used to detect the error(s) present in the received

data (bit stream). These codes contain some bit(s), which are included

(appended) to the original bit stream.

Error detection codes

Example − Parity code, Hamming code.

This method are used to correct the error(s) present in the received

data (bit stream) so that, we will get the original data.

Error correction codes −

Example − Hamming code

Parity Code

It is easy to include (append) one parity bit either to the left of MSB or

to the right of LSB of original bit stream. There are two types of parity

codes, namely even parity code and odd parity code based on the type

of parity being chosen.

The value of even parity bit should be zero, if even number of ones

present in the binary code. Otherwise, it should be one. So that, even

number of ones present in even parity code. Even parity code

contains the data bits and even parity bit.

Even Parity Code

Binary Code Even Parity bit Even Parity Code

000 0 0000

001 1 0011

010 1 0101

011 0 0110

100 1 1001

101 0 1010

110 0 1100

111 1 1111

The value of odd parity bit should be zero, if odd number of ones

present in the binary code. Otherwise, it should be one. So that, odd

number of ones present in odd parity code. Odd parity code contains

the data bits and odd parity bit.

Odd Parity Code

Binary Code Odd Parity bit Odd Parity Code

000 1 0001

001 0 0010

010 0 0100

011 1 0111

100 0 1000

101 1 1011

110 1 1101

111 0 1110

Hamming Code

Hamming code is useful for both detection and correction of error

present in the received data. This code uses multiple parity bits and we

have to place these parity bits in the positions of powers of 2.

The minimum value of 'k' for which the following relation is correct

(valid) is nothing but the required number of parity bits.

2k ≥ n+k+1

Where,

‘n’ is the number of bits in the binary code (information)

‘k’ is the number of parity bits

Laws of Boolean Algebra

Idempotent Law

A * A = A

A + A = A

Associative Law

(A * B) * C = A * (B * C)

(A + B) + C = A + (B + C)

Commutative Law

A * B = B * A

A + B = B + A

Distributive Law

A * (B + C) = A * B + A * C

A + (B * C) = (A + B) * (A + C)

Identity Law

A * 0 = 0 A * 1 = A

A + 1 = 1 A + 0 = A

Complement Law

A * ~A = 0

A + ~A = 1

Involution Law

~(~A) = A

Absorption

A + (A * B) = A

A * (A + B) = A

In boolean algebra, the OR operation is performed by which

properties?

a) Associative properties

b) Commutative properties

c) Distributive properties

d) All of the Mentioned

The expression for Absorption law is given by _________

a) A + AB = A

b) A + AB = B

c) AB + AA’ = A

d) A + B = B + A

According to boolean law: A + 1 = ?

a) 1

b) A

c) 0

d) A’

The involution of A is equal to _________

a) A

b) A’

c) 1

d) 0

A(A + B) = ?

a) AB

b) 1

c) (1 + AB)

d) A

DeMorgan’s theorem states that _________

a) (AB)’ = A’ + B’

b) (A + B)’ = A’ * B

c) A’ + B’ = A’B’

d) (AB)’ = A’ + B

(A + B)(A’ * B’) = ?

a) 1

b) 0

c) AB

d) AB’

Complement of the expression A’B + CD’ is _________

a) (A’ + B)(C’ + D)

b) (A + B’)(C’ + D)

c) (A’ + B)(C’ + D)

d) (A + B’)(C + D’)

The boolean function A + BC is a reduced form of ____________

a) AB + BC

b) (A + B)(A + C)

c) A’B + AB’C

d) (A + C)B

The idempotence law of Boolean algebra says that

a) x + y = 1

b) x + x = x

c) x + xy = x

d) x(x+y) = x

The expression Y=AB+BC+AC shows the _________ operation.

a) EX-OR

b) SOP

c) POS

d) NOR

The expression Y=(A+B)(B+C)(C+A) shows the _____________

operation.

a) AND

b) POS

c) SOP

d) NAND

The NOR gate output will be high if the two inputs are _______

a) 00

b) 01

c) 10

d) 11

A universal logic gate is one which can be used to generate

any logic function. Which of the following is a universal logic

gate?

a) OR

b) AND

c) XOR

d) NAND

Which of following are known as universal gates?

a) NAND & NOR

b) AND & OR

c) XOR & OR

d) EX-NOR & XOR

A single transistor can be used to build which of the

following digital logic gates?

a) AND gates

b) OR gates

c) NOT gates

d) NAND gates

How many truth table entries are necessary for a four-input

circuit?

a) 4

b) 8

c) 12

d) 16

Which input values will cause an AND logic gate to produce a

HIGH output?

a) At least one input is HIGH

b) At least one input is LOW

c) All inputs are HIGH

d) All inputs are LOW

Exclusive-OR (XOR) logic gates can be constructed from

what other logic gates?

a) OR gates only

b) AND gates and NOT gates

c) AND gates, OR gates, and NOT gates

d) OR gates and NOT gates

The basic logic gate whose output is the complement of the

input is the ___________

a) OR gate

b) AND gate

c) INVERTER gate

d) XOR gate

If we use an AND gate to inhibit a signal from passing one of

the inputs must be ___________

a) LOW

b) HIGH

c) Inverted

d) Floating