Data Representation and Boolean Logic

Both A and R are true but R is not the correct explanation of A.

Explanation
Data inside a computer gets stored and manipulated in digital (binary) form (0s and 1s). Number systems are the techniques to represent numbers in the computer system architecture. Every number system includes a set of unique characters or literals. It is a way to represent a number in different forms.

Both A and R are true and R is the correct explanation of A.

Explanation
The base of binary number system is 2 because it has only two digits, i.e., 1 and 0. Every number can be represented with 0 and 1 in binary number system.

A is true but R is false.

Explanation
The base of the octal number system is 8 because it has only 8 digits (0 to 7). The number 342289 is not an octal number because the octal number system only allows digits from 0 to 7, and 342289 contains digits outside this range (8 and 9).

Both A and R are true and R is the correct explanation of A.

Explanation
The dual of the Boolean expression A+B=1 is A⋅B=0. This is derived by applying the duality principle, which states that the dual of any boolean expression is obtained by interchanging the operations (+) and (.), and swapping the constants 1 and 0.

Both A and R are true and R is the correct explanation of A.

Explanation
Unicode represents a single encoding scheme for all languages and characters. Unicode provides a unique number for every character irrespective of the platform, program and the language.

Both A and R are true and R is the correct explanation of A.

Explanation
The number of symbols (digits and alphabets) used in a number system defines its radix or base. The base value of a number system helps in distinguishing numbers from different number systems, such as binary, octal, decimal, and hexadecimal.

Processed data

Reason — Information is defined as processed data organized in a particular manner to generate meaningful piece of data.

2

Reason — The base of binary number system is 2 because it has only two digits.

10

Reason — The decimal number system uses 10 digits (from 0 to 9) hence its has a base of 10.

8

Reason — The base of octal number system is 8 because it has only 8 digits.

(101011)₂

Reason — To convert (43)₁₀ from decimal to binary, we perform the following calculation:

Therefore, (43)₁₀ = (101011)₂

18

Reason —

Equivalent decimal number = 2 + 16 = 18

Therefore, (10010)₂ = (18)₁₀.

Indian Standard Code for Information Interchange

Reason — The full form of ISCII is Indian Standard Code for Information Interchange.

16 symbols

Reason — A hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F.

11E

Reason — A binary number system only includes the digits 0 and 1. The number "11E" contains the letter 'E', which is not a valid digit in the binary system. Therefore, "11E" is not a binary number.

Hexadecimal

Reason — A hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F.

Should I wear the mask or not?

Reason — A logical or Boolean statement is one that can be evaluated as either true or false. The statement "Should I wear the mask or not?" poses a question that requires a true or false answer, thus making it a logical or Boolean statement.

A.B.C+(B'+C').A

Reason — The above logic circuit is equivalent to the boolean expression A.B.C + (B'+C').A.

AND

Reason — The above logic gate is an AND operator.

(a) 52

Therefore, (52)₁₀ = (110100)₂.

(b) 44

Therefore, (44)₁₀ = (101100)₂.

(c) 25.80

Let us first convert 25 into binary as shown below:

The binary equivalent of integer part (25)₁₀ = (11001)₂.

Now let us convert (0.80)₁₀ into binary as shown below:

The binary equivalent of fractional part (0.8)₁₀ = (11001)₂.

Therefore, (25.80)₁₀ = (11001.11001)₂.

(d) 62.325

Let us first convert 62 into binary as shown below:

The binary equivalent of integer part (62)₁₀ = (111110)₂.

Now let us convert (0.325)₁₀ into binary as shown below:

The binary equivalent of fractional part (0.325)₁₀ = (0101001)₂.

Therefore, (62.325)₁₀ = (111110.0101001)₂.

(a) 911

Therefore, (911)₁₀ = (1617)₈.

(b) 540

Therefore, (540)₁₀ = (1034)₈.

(c) 61

Therefore, (61)₁₀ = (75)₈.

(d) 132

Therefore, (132)₁₀ = (204)₈.

(a) 132

Therefore, (132)₁₀ = (84)₁₆.

(b) 3619

Therefore, (3619)₁₀ = (E23)₁₆.

(c) 206

Therefore, (206)₁₀ = (CE)₁₆.

(d) 752

Therefore, (752)₁₀ = (2F0)₁₆

(a) 10111

Equivalent decimal number = 1 + 2 + 4 + 16 = 23

Therefore, (10111)₂ = (23)₁₀.

(b) 111101

Equivalent decimal number = 1 + 4 + 8 + 16 + 32 = 61

Therefore, (111101)₂ = (61)₁₀.

(c) 101010.011

For the integer part (101010):

Equivalent decimal number = 2 + 8 + 32 = 42

For the fractional part (0.011):

Equivalent decimal number = 0.25 + 0.125 = 0.375

Therefore, (101010.011)₂ = (42.375)₁₀.

(d) 101101

Equivalent decimal number = 1 + 4 + 8 + 32 = 45

Therefore, (101101)₂ = (45)₁₀.

(a) 75

Equivalent decimal number = 5 + 56 = 61

Therefore, (75)₈ = (61)₁₀.

(b) 321

Equivalent decimal number = 1 + 16 + 192 = 209

Therefore, (321)₈ = (209)₁₀.

(c) 142

Equivalent decimal number = 2 + 32 + 64 = 98

Therefore, (142)₈ = (98)₁₀.

(d) 205

Equivalent decimal number = 5 + 128 = 133

Therefore, (205)₈ = (133)₁₀.

(a) A2

Equivalent decimal number = 2 + 160 = 162

Therefore, (A2)₁₆ = (162)₁₀.

(b) 13B

Equivalent decimal number = 11 + 48 + 256 = 315

Therefore, (13B)₁₆ = (315)₁₀.

(c) 271

Equivalent decimal number = 1 + 112 + 512 = 625

Therefore, (271)₁₆ = (625)₁₀.

(d) 132

Equivalent decimal number = 2 + 48 + 256 = 306

Therefore, (132)₁₆ = (306)₁₀.

UTF-8 is a variable-width encoding that can represent every character in the Unicode character set. The code unit of UTF-8 is 8 bits (an octet). It uses 1 to 4 octets to represent code points, depending on their size. For example, it uses 1 byte for ASCII characters and up to 4 bytes for others. This makes UTF-8 a multi-byte encoding that efficiently handles different character sets.

UTF-32, in contrast, is a fixed-length encoding scheme that always uses exactly 4 bytes (32 bits) to represent all Unicode code points, regardless of their size, making it less space-efficient than UTF-8.

The values which are stored in binary variables are known as Boolean Constants. Therefore, the values true/false, yes/no or 1/0 are Boolean constants.

Operators used in Boolean algebra are known as Boolean/Logical operators.

1. AND Operator — AND operator is a binary operator that operates on two variables and the operation performed by AND operator is known as logical multiplication. The symbol used for logical multiplication is dot(.) operator. The truth table for AND operator is as follows:

Truth Table

The AND operation will result in true value (1) when both inputs are 1 (true/high) and for all other values it results in 0 (false/low).

2. OR Operator — The OR operator is a binary operator that operates on two variables and the operation performed by OR operator is known as logical addition. The symbol used for logical addition is plus (+) operator. The truth table for OR operator is as follows:

Truth Table

The OR operation results in true value (1) when either of the inputs is 1 (true) or both the inputs are 1 (true/high), and for all other values of inputs it results in 0 (false/low).

3. NOT Operator — The NOT operator is a unary operator that operates on one variable and the operation performed by NOT operator is known as negation or complementation. The truth table for NOT operator is as follows:

Truth Table

It means that the logical statements A and A' are opposite to each other. If the value of a variable A is 0, its complement would be 1, and if the value of the variable A is 1, its complement would be 0.

The use of the hexadecimal number system in computers includes the following:

1. The Hexadecimal number system is used in computers to specify memory addresses (which are 16-bit or 32-bit long). For example, a memory address 1101011010101111 is a big binary address but with hex it is D6AF which is easier to remember.
2. The Hexadecimal number system is used to represent color codes. For example, FFFFFF represents White, FF0000 represents Red, etc.

(i) a.(a+b) = a

Truth Table

As columns "a.(a+b)" and "a" have same values, hence the expression is proved.

(ii) X.(Y+Z) = X.Y + X.Z

Truth Table

As columns X.(Y+Z) and X.Y+X.Z have same values, hence the expression is proved.

The boolean expression for the above logic circuit is $\overline{\overline{x^n.y} + \overline{x .y^n}}$.

The first law states that when two (or more) input variables are OR'ed and negated, they are equivalent to the AND of the complements of the individual variables.

$\overline{A+B}$ = $\overline{A}.\overline{B}$

Proof using truth table:

The second law states that when two (or more) input variables are AND'ed and negated, they are equivalent to the OR of the complements of the individual variables.

$\overline{A.B}$ = $\overline{A} + \overline{B}$

Proof using truth table:

Following are the advantages of preparing a digital content in Indian language using UNICODE font:

1. Unicode provides a universal character encoding standard, ensuring consistent display and compatibility across different platforms, devices, and applications.
2. It supports a wide range of Indian scripts, allowing seamless text exchange and readability without font issues.
3. Content prepared using Unicode can be easily indexed by search engines, improving accessibility and searchability.

Representing the word 'COMPUTER' in ASCII values of its characters:

COMPUTER → 67 79 77 80 85 84 69 82

Binary Equivalent → 01000011 01001111 01001101 01010000 01010101 01010100 01000101 01010010

Explanation

ASCII value of C is 67 and its equivalent 7-bit binary code = 01000011
ASCII value of O is 79 and its equivalent 7-bit binary code = 01001111
ASCII value of M is 77 and its equivalent 7-bit binary code = 01001101
ASCII value of P is 80 and its equivalent 7-bit binary code = 01010000
ASCII value of U is 85 and its equivalent 7-bit binary code = 01010101
ASCII value of T is 84 and its equivalent 7-bit binary code = 01010100
ASCII value of E is 69 and its equivalent 7-bit binary code = 01000101
ASCII value of R is 82 and its equivalent 7-bit binary code = 01010010

Hence, binary value for the word 'COMPUTER' is 01000011 01001111 01001101 01010000 01010101 01010100 01000101 01010010.

ASCII — American Standard Code for Information Interchange

ISCII — Indian Standard Code for Information Interchange

(i)

(ii)

(a) 2306

Therefore, (2306)₈ = ($\bold{\underlinesegment{010}}\medspace\bold{\underlinesegment{011}}\medspace\bold{\underlinesegment{000}}\medspace\bold{\underlinesegment{110}}$)₂.

(b) 5610

Therefore, (5610)₈ = ($\bold{\underlinesegment{101}}\medspace\bold{\underlinesegment{110}}\medspace\bold{\underlinesegment{001}}\medspace\bold{\underlinesegment{000}}$)₂.

(c) 742

Therefore, (742)₈ = ($\bold{\underlinesegment{111}}\medspace\bold{\underlinesegment{100}}\medspace\bold{\underlinesegment{010}}$)₂.

(d) 65.203

Therefore, (65.203)₈ = ($\bold{\underlinesegment{110}}\medspace\bold{\underlinesegment{101}}.\medspace\bold{\underlinesegment{010}}\medspace\bold{\underlinesegment{000}}\medspace\bold{\underlinesegment{011}}$)₂.

(a) 4026

Therefore, (4026)₁₆ = (0100000000100110)₂.

(b) BCA1

B → 1011
C → 1100
A → 1010
1 → 0001

Therefore, (BCA1)₁₆ = (1011110010100001)₂.

(c) 98E

Therefore, (98E)₁₆ = (100110001110)₂.

(d) 132.45

Therefore, (132.45)₁₆ = (000100110010.01000101)₂.

(a) (54)₁₀

Therefore, (54)₁₀ = (110110)₂.

(b) (120)₁₀

Therefore, (120)₁₀ = (1111000)₂.

(c) (76)₁₀

Therefore, (76)₁₀ = (114)₈.

(d) (889)₁₀

Therefore, (889)₁₀ = (1571)₈.

(e) (789)₁₀

Therefore, (789)₁₀ = (315)₁₆.

(f) (108)₁₀

Therefore, (108)₁₀ = (6C)₁₆.

(a) 100101111

Grouping in bits of 3:

$\underlinesegment{100} \quad \underlinesegment{101} \quad \underlinesegment{111}$

Therefore, (100101111)₂ = (457)₈

(b) 111011010

Grouping in bits of 3:

$\underlinesegment{111} \quad \underlinesegment{011} \quad \underlinesegment{010}$

Therefore, (111011010)₂ = (732)₈

(c) 1010011

Grouping in bits of 3:

$\underlinesegment{001} \quad \underlinesegment{010} \quad \underlinesegment{011}$

Therefore, (1010011)₂ = (123)₈

(d) 10011101

Grouping in bits of 3:

$\underlinesegment{010} \quad \underlinesegment{011} \quad \underlinesegment{101}$

Therefore, (10011101)₂ = (235)₈

(a) 1111101101100011

Grouping in bits of 4:

$\underlinesegment{1111} \quad \underlinesegment{1011} \quad \underlinesegment{0110} \quad \underlinesegment{0011}$

Therefore, (1111101101100011)₂ = (FB63)₁₆.

(b) 100000101011100

Grouping in bits of 4:

$\underlinesegment{0100} \quad \underlinesegment{0001} \quad \underlinesegment{0101} \quad \underlinesegment{1100}$

Therefore, (100000101011100)₂ = (415C)₁₆.

(c) 1000111010100011

Grouping in bits of 4:

$\underlinesegment{1000} \quad \underlinesegment{1110} \quad \underlinesegment{1010} \quad \underlinesegment{0011}$

Therefore, (1000111010100011)₂ = (8EA3)₁₆.

(d) 111011111

Grouping in bits of 4:

$\underlinesegment{0001} \quad \underlinesegment{1101} \quad \underlinesegment{1111}$

Therefore, (111011111)₂ = (1DF)₁₆.

(a) 145

Equivalent decimal number = 5 + 32 + 64 = 101

Therefore, (145)₈ = (101)₁₀.

(b) 6760

Equivalent decimal number = 48 + 448 + 3072 = 3568

Therefore, (6760)₈ = (3568)₁₀.

(c) 455

Equivalent decimal number = 5 + 40 + 256 = 301

Therefore, (455)₈ = (301)₁₀.

(d) 10.75

Integer part:

Equivalent decimal number = 8

Fractional part:

Equivalent decimal number = 0.875 + 0.078125 = 0.953125

Therefore, (10.75)₈ = (8.953125)₁₀.

The base value of the hexadecimal number system is 16.

To insert any Indian language character using Unicode in Microsoft Word, the steps are:

1. Click Insert → Symbol → More Symbol
2. From the Symbols dialog that appears, select a Unicode supporting font and Indian language's subset.
3. Choose desired character and click Insert.

(a) (1010100)₁₀

Therefore, (1010100)₁₀ = (11110110100110110100)₂.

​ (b) (3674)₈

Therefore, (3674)₈ = ($\bold{\underlinesegment{011}}\medspace\bold{\underlinesegment{110}}\medspace\bold{\underlinesegment{111}}\medspace\bold{\underlinesegment{100}}$)₂.

(c) (266)₁₀

Therefore, (266)₁₀ = (412)₈.

(c) (9F2)₁₆

Therefore, (9F2)₁₆ = (100111110010)₂.

a(b + c) = ab + ac is distributive law.

Truth Table

As columns "a(b + c)" and "ab + ac" have same values, hence the expression is proved.

The logic circuit for the Boolean expression is shown below:

True

Reason — Binary data representation uses only two symbols, 0 and 1, because it is based on base 2. In base 2, each digit (bit) can only be one of two values: 0 or 1. These digits are combined to represent numbers and data, with each position representing a power of 2, similar to how base 10 uses powers of 10.

True

Reason — ASCII, ISCII, and UNICODE are three types of character encoding schemes used for internal storage representations.

False

Reason — The digit '8' is not valid in the octal system because the octal number system only includes digits from 0 to 7. Therefore, (128)₈ is not a valid octal number.

True

Reason — The first 128 characters (ranging from 0 to 127) in both ASCII-7 and ASCII-8 are identical. ASCII-7 uses 7 bits to represent characters, covering the first 128 characters. ASCII-8 uses 8 bits, allowing for 256 characters in total. However, the first 128 characters remain consistent between both versions.

False

Reason — ASCII-8 uses 8 bits, allowing for a total of 256 characters to be represented.

True

Reason — Unicode provides a unique number for every character irrespective of the platform, program and the language.

True

Reason — A hexadecimal number system has sixteen (16) alphanumeric values from 0-9 and A-F. Therefore, "ABC" is a valid hexadecimal number.

False

Reason — In the decimal number system, a number with both integer and fractional parts has digits raised to positive powers of 10 for the integer part and negative powers of 10 for the fractional part.

False

Reason — To convert (52)₁₆ from hexadecimal to decimal, we calculate:

5×16¹ + 2×16⁰ = 5×16 + 2×1 = 80 + 2 = 82

Thus, (52)₁₆ is equivalent to 82 in the decimal number system.

True

Reason — Octal number system has only eight digits from 0 to 7. Every number can be represented with 0, 1, 2, 3, 4, 5, 6, 7 in this number system.

True

Reason — The NAND and NOR gates are called universal gates because any digital circuits can be implemented using these gates.