Discrete Structure Exams By: Ishrat Hayat Malik

1. Which of the following two sets are equal?

A = {1, 2} and B = {1}

A = {1, 2} and B = {1, 2, 3}

A = {1, 2, 3} and B = {2, 1, 3}

) A = {1, 2, 4} and B = {1, 2, 3}

The sets A = {1, 2, 3} and B = {2, 1, 3} are equal because the order of elements does not matter in a set. Both sets contain the same elements: 1, 2, and 3. Therefore, they are considered equal.

Explanation

The sets A = {1, 2, 3} and B = {2, 1, 3} are equal because the order of elements does not matter in a set. Both sets contain the same elements: 1, 2, and 3. Therefore, they are considered equal.

2. Power set of empty set has exactly _________ subset.

1

2

3

4

The power set of the empty set has exactly one subset. This is because the power set includes the empty set itself as a subset, in addition to any other possible subsets. Since the empty set is the only possible subset of the empty set, the power set has exactly one subset.

Explanation

The power set of the empty set has exactly one subset. This is because the power set includes the empty set itself as a subset, in addition to any other possible subsets. Since the empty set is the only possible subset of the empty set, the power set has exactly one subset.

3. The set O of odd positive integers less than 10 can be expressed by

{1, 2, 3}

{1, 3, 5, 7, 9}

{1, 2, 5, 9}

{1, 5, 7, 9, 11}

The set O of odd positive integers less than 10 can be expressed as {1, 3, 5, 7, 9} because these are the odd numbers that are less than 10. The other options either include even numbers or numbers greater than 10, which do not fit the criteria of the set O.

Explanation

The set O of odd positive integers less than 10 can be expressed as {1, 3, 5, 7, 9} because these are the odd numbers that are less than 10. The other options either include even numbers or numbers greater than 10, which do not fit the criteria of the set O.

4. A compound proposition that is always ___________ is called a tautology

True

False

None

A compound proposition that is always true regardless of the truth values of its individual components is called a tautology. This means that no matter what the truth values of its components are, the compound proposition will always evaluate to true. In other words, it is impossible for the compound proposition to be false. Therefore, the correct answer is true.

Explanation

A compound proposition that is always true regardless of the truth values of its individual components is called a tautology. This means that no matter what the truth values of its components are, the compound proposition will always evaluate to true. In other words, it is impossible for the compound proposition to be false. Therefore, the correct answer is true.

5. What is the Cartesian product of A = {1, 2} and B = {a, b}?

{(1, a), (1, b), (2, a), (b, b)}

{(1, 1), (2, 2), (a, a), (b, b)}

{(1, a), (2, a), (1, b), (2, b)}

The Cartesian product of two sets A and B is a set of all possible ordered pairs where the first element comes from set A and the second element comes from set B. In this case, the Cartesian product of A = {1, 2} and B = {a, b} would be {(1, a), (1, b), (2, a), (2, b)}. This is because we can pair each element of set A with each element of set B to form the ordered pairs.

Explanation

The Cartesian product of two sets A and B is a set of all possible ordered pairs where the first element comes from set A and the second element comes from set B. In this case, the Cartesian product of A = {1, 2} and B = {a, b} would be {(1, a), (1, b), (2, a), (2, b)}. This is because we can pair each element of set A with each element of set B to form the ordered pairs.

6. All of the x values or inputs are called what?

Range

Domain

Image

The correct answer is "domain." In mathematics, the domain refers to the set of all possible input values or x-values in a function. It represents the independent variable in a function and determines the valid inputs for the function. In this context, since the question asks for the term used to describe all the x-values or inputs, the correct answer is "domain." The range refers to the set of all possible output values or y-values in a function, while the image refers to the set of all actual output values.

Explanation

The correct answer is "domain." In mathematics, the domain refers to the set of all possible input values or x-values in a function. It represents the independent variable in a function and determines the valid inputs for the function. In this context, since the question asks for the term used to describe all the x-values or inputs, the correct answer is "domain." The range refers to the set of all possible output values or y-values in a function, while the image refers to the set of all actual output values.

7. What is the cardinality of the set of odd positive integers less than 10?

10

5

3

20

The cardinality of a set refers to the number of elements in that set. In this case, the set of odd positive integers less than 10 includes the numbers 1, 3, 5, 7, and 9. Therefore, the cardinality of this set is 5.

Explanation

The cardinality of a set refers to the number of elements in that set. In this case, the set of odd positive integers less than 10 includes the numbers 1, 3, 5, 7, and 9. Therefore, the cardinality of this set is 5.

8. What is the Cardinality of the Power set of the set {0, 1, 2}.

8

6

7

9

The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself. In this case, the set {0, 1, 2} has 3 elements. The power set will have 2^3 = 8 subsets, including the empty set and the set itself. Therefore, the cardinality of the power set is 8.

Explanation

The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself. In this case, the set {0, 1, 2} has 3 elements. The power set will have 2^3 = 8 subsets, including the empty set and the set itself. Therefore, the cardinality of the power set is 8.

9. 3. If A is any statement, then which of the following is a tautology?

A ∧ F

A ∨ F

A ∨ ¬A

The statement "A ∨ ¬A" is a tautology because it represents the logical disjunction (OR) between any statement A and its negation (¬A). In other words, it is always true regardless of the truth value of A. This is because either A is true, making the whole statement true, or A is false, making the negation of A true and the whole statement true.

Explanation

The statement "A ∨ ¬A" is a tautology because it represents the logical disjunction (OR) between any statement A and its negation (¬A). In other words, it is always true regardless of the truth value of A. This is because either A is true, making the whole statement true, or A is false, making the negation of A true and the whole statement true.

10. Is the relation a function? Why.

Yes, because the x-value 11 has two y-values pair with it.

Yes, because each x-value has only one y-value paired with it.

No, because the x-value 11 has two y-values pair with it

No, because each x-value has only one y-value paired with it.

The answer is "No, because the x-value 11 has two y-values paired with it." This is because in a function, each x-value should have only one corresponding y-value. In this case, the x-value 11 violates this rule by having two y-values paired with it. Therefore, the relation is not a function.

Explanation

The answer is "No, because the x-value 11 has two y-values paired with it." This is because in a function, each x-value should have only one corresponding y-value. In this case, the x-value 11 violates this rule by having two y-values paired with it. Therefore, the relation is not a function.

11. The Cartesian Product B x A is equal to the Cartesian product A x B. Is it True or False?

True

False

None

The statement "The Cartesian Product B x A is equal to the Cartesian product A x B" is false. The Cartesian product of two sets A and B is the set of all possible ordered pairs where the first element comes from A and the second element comes from B. Switching the order of the sets in the Cartesian product will result in a different set of ordered pairs. Therefore, A x B is not equal to B x A.

Explanation

The statement "The Cartesian Product B x A is equal to the Cartesian product A x B" is false. The Cartesian product of two sets A and B is the set of all possible ordered pairs where the first element comes from A and the second element comes from B. Switching the order of the sets in the Cartesian product will result in a different set of ordered pairs. Therefore, A x B is not equal to B x A.

12. In the given relation, what domain value corresponds to the range value -2? {(-1,2), (-2,4), (2,5), (0,-2), (2,0)}

-2

2

4

0

not-available-via-ai

Explanation

not-available-via-ai

13. The universal quantifier is used to assert a property of _____ values of a variable in a particular domain

The universal quantifier is used to assert a property of "all" values of a variable in a particular domain. It indicates that the property holds true for every single value in the domain.

Explanation

The universal quantifier is used to assert a property of "all" values of a variable in a particular domain. It indicates that the property holds true for every single value in the domain.

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