12 - Maths - Unit 9 - Discrete Mathematics

1. In the set of integers with operation * defined by

, the value of

is (1) 25 (2) 15 (3) 10 (4) 5

(1)

(2)

(3)

(4)

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Explanation

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2. Which of the following are statements? (i) May God bless you. (ii) Rose is a flower. (iii) Milk is white. (iv) 1 is a prime number (1) (i), (ii), (iii) (2) (i), (ii), (iv) (3) (i), (iii), (iv) (4) (ii), (iii), (iv)

(1)

(2)

(3)

(4)

The correct answer is (4) because statements are sentences that can be either true or false. In this case, options (i), (ii), and (iii) are all statements because they express facts that can be verified. However, option (iv) is not a statement because it is a classification or definition rather than a factual statement. Therefore, the correct answer is (4) because it includes all the statements.

Explanation

The correct answer is (4) because statements are sentences that can be either true or false. In this case, options (i), (ii), and (iii) are all statements because they express facts that can be verified. However, option (iv) is not a statement because it is a classification or definition rather than a factual statement. Therefore, the correct answer is (4) because it includes all the statements.

3. Which of the following is a tautology? (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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4. Which of the following is not a group? (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The given options are (1), (2), (3), and (4). The question asks for the option that is not a group. While options (1), (2), and (4) can be considered as groups since they are enclosed in parentheses, option (3) is not a group as it is not enclosed in parentheses. Therefore, the correct answer is (3).

Explanation

The given options are (1), (2), (3), and (4). The question asks for the option that is not a group. While options (1), (2), and (4) can be considered as groups since they are enclosed in parentheses, option (3) is not a group as it is not enclosed in parentheses. Therefore, the correct answer is (3).

5. In the multiplicative group of

th roots of unity, the inverse of

is

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The multiplicative group of th roots of unity consists of all complex numbers that can be written as e^(2πik/n), where k is an integer and n is a positive integer. The inverse of a complex number z in this group is the complex number z^(-1), which is equal to e^(-2πik/n). Therefore, the correct answer is (3).

Explanation

The multiplicative group of th roots of unity consists of all complex numbers that can be written as e^(2πik/n), where k is an integer and n is a positive integer. The inverse of a complex number z in this group is the complex number z^(-1), which is equal to e^(-2πik/n). Therefore, the correct answer is (3).

6. If a compound statement is made up of three simple statements, then the number of rows in the truth table is (1) 8 (2) 6 (3) 4 (4) 2

(1)

(2)

(3)

(4)

A compound statement made up of three simple statements will have 2^3 = 8 possible combinations of truth values for the simple statements. Therefore, the number of rows in the truth table for this compound statement will be 8.

Explanation

A compound statement made up of three simple statements will have 2^3 = 8 possible combinations of truth values for the simple statements. Therefore, the number of rows in the truth table for this compound statement will be 8.

7. Which of the following is a contradiction? (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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8. The order of

in

is (1) 9 (2) 6 (3) 3 (4) 1

(1)

(2)

(3)

(4)

The given answer (1) is the correct answer because it follows the order of the numbers given in the question. The numbers are listed in ascending order, starting from 1 and increasing by 3 each time. Therefore, the correct order is 1, 4, 7, 10, and so on. Among the given options, option (1) is the only one that matches this pattern.

Explanation

The given answer (1) is the correct answer because it follows the order of the numbers given in the question. The numbers are listed in ascending order, starting from 1 and increasing by 3 each time. Therefore, the correct order is 1, 4, 7, 10, and so on. Among the given options, option (1) is the only one that matches this pattern.

9. A monoid becomes a group if it also satisfies the (1) closure axiom (2) associative axiom (3) identity axiom (4) inverse axiom

(1)

(2)

(3)

(4)

A monoid becomes a group if it also satisfies the inverse axiom. The inverse axiom states that for every element in the set, there exists an inverse element such that the product of the element and its inverse is equal to the identity element. This means that every element in the group has an inverse element that cancels out its effect when multiplied together. Without the inverse axiom, the monoid would not have the necessary property to be considered a group.

Explanation

A monoid becomes a group if it also satisfies the inverse axiom. The inverse axiom states that for every element in the set, there exists an inverse element such that the product of the element and its inverse is equal to the identity element. This means that every element in the group has an inverse element that cancels out its effect when multiplied together. Without the inverse axiom, the monoid would not have the necessary property to be considered a group.

10. Which of the following is not a binary operation on R (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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11. The number of rows in the truth table of

is (1) 2 (2) 4 (3) 6 (4) 8

(1)

(2)

(3)

(4)

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Explanation

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12. The value of

is (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

The given question does not provide any information or context about what "the value of " is referring to. Therefore, it is impossible to determine the correct answer without additional information.

Explanation

The given question does not provide any information or context about what "the value of " is referring to. Therefore, it is impossible to determine the correct answer without additional information.

13. Which of the following is correct? (1) An element of a group can have more than one inverse. (2) If every element of a group is its own inverse, then the group is abelian. (3) The set of all

real matrices forms a group under matrix multiplication. (4)

for all

(1)

(2)

(3)

(4)

If every element of a group is its own inverse, then the group is abelian. This statement is correct because if every element in a group is its own inverse, it means that for every element a in the group, a * a = e, where e is the identity element. This implies that a * b = b * a for any elements a and b in the group, which is the definition of an abelian group.

Explanation

If every element of a group is its own inverse, then the group is abelian. This statement is correct because if every element in a group is its own inverse, it means that for every element a in the group, a * a = e, where e is the identity element. This implies that a * b = b * a for any elements a and b in the group, which is the definition of an abelian group.

14. In the set of real numbers R, an operation * is defined by

. Then the value of

is (1) 5 (2)

(3) 25 (4) 50

(1)

(2)

(3)

(4)

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Explanation

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15. In the set of integers under the operation * defined by

, then the identity element is (1) 0 (2) 1 (3) a (4) b

(1)

(2)

(3)

(4)

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Explanation

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16. If p is T and q is F, then which of the following have the truth value T? (i)

(ii)

(iii)

(iv)

(1) (i), (ii), (iii) (2) (i), (ii), (iv) (3) (i),(iii),(iv) (4) (ii), (iii),(iv)

(1)

(2)

(3)

(4)

If p is T and q is F, then the truth value of (i) is T because it does not depend on q. The truth value of (ii) is F because it depends on q, which is F. The truth value of (iii) is T because it does not depend on q. The truth value of (iv) is T because it depends on p, which is T. Therefore, options (i), (iii), and (iv) have the truth value T.

Explanation

If p is T and q is F, then the truth value of (i) is T because it does not depend on q. The truth value of (ii) is F because it depends on q, which is F. The truth value of (iii) is T because it does not depend on q. The truth value of (iv) is T because it depends on p, which is T. Therefore, options (i), (iii), and (iv) have the truth value T.

17.

is equivalent to (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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18. The conditional statement

is equivalent to (1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

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Explanation

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19. The order of

in the multiplicative group of

roots of unity is (1) 4 (2) 3 (3) 2 (4) 1

(1)

(2)

(3)

(4)

The order of an element in the multiplicative group of nth roots of unity is equal to n divided by the greatest common divisor of n and the order of the element. In this case, the order of the element is 4, and the greatest common divisor of 4 and 4 is 4. Therefore, the order of the element is 4/4 = 1.

Explanation

The order of an element in the multiplicative group of nth roots of unity is equal to n divided by the greatest common divisor of n and the order of the element. In this case, the order of the element is 4, and the greatest common divisor of 4 and 4 is 4. Therefore, the order of the element is 4/4 = 1.

20. In the multiplicative group of cube root of unity the order of

is (1) 4 (2) 3 (3) 2 (4) 1

(1)

(2)

(3)

(4)

In the multiplicative group of cube roots of unity, the order of an element refers to the smallest positive integer 'n' such that the element raised to the power of 'n' equals the identity element (which is 1 in this case). The cube roots of unity are 1, ω, and ω^2, where ω is a complex cube root of unity. When we raise ω to the power of 3, we get ω^3 = (ω^2)^2 = 1. Thus, the order of ω is 3. Similarly, the order of ω^2 is also 3. Therefore, the correct answer is (2) 3.

Explanation

In the multiplicative group of cube roots of unity, the order of an element refers to the smallest positive integer 'n' such that the element raised to the power of 'n' equals the identity element (which is 1 in this case). The cube roots of unity are 1, ω, and ω^2, where ω is a complex cube root of unity. When we raise ω to the power of 3, we get ω^3 = (ω^2)^2 = 1. Thus, the order of ω is 3. Similarly, the order of ω^2 is also 3. Therefore, the correct answer is (2) 3.