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21 Discrete structures questions which includes true or false questions and multiple choice questions. The attached file contains the questions.

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Section I: True (T) or False (F) – Circle Your Answer

1. Propositional well-formed formula may be proven true with an algorithm same as a predicate

well-formed formula may be proven valid.

a. True

b. False

2. The key to a proof by the first principle of induction is to see how the truth of P at the value

k + 1 depends on the truth of P at value k.

a. True

b. False

3. Recursive algorithms are valuable primarily because they run more efficiently than iterative

algorithms.

a. True

b. False

4. The predicate quantifier, , indicates that there exists 1, a few or some objects in the

domain.

a. True

b. False

5. Modus Ponens is probably the most intuitive inference rule.

a. True

b. False

6. Is the following expression true or false? (D U B’)’ = D’ ꓵ B

a. True

b. False

7. Given the truth values A (true), B (true), C (false), D (True) what is the truth value of the

following wff? (A’ V B’) Λ (C’ Λ A) → D

a. True

b. False

8. The expression: A’ → (B V C’) is a tautology.

a. True

b. False

9. A recurrence relation is homogenous if g(n) has a constant coefficient.

a. True

b. False

10. Your professor has 4 pairs of shoes, 12 pairs of socks, 10 pairs of pants, 15 different shirts, 5 suit

jackets and 3 “UTA” lapel pins he may or may not pin to his jacket. Given these options, he

therefore has 108,000 different combinations he may wear.

a. True

b. False

11. The following expression is a well-formed formula: ((A V B) Λ C) → (B V C’))

a. True

b. False

12. The disjunction of any propositional wff with a tautology has a truth value of true.

a. True

b. False

13. (A Λ B) Λ C Λ D → E → F ↔ A Λ B Λ C Λ D Λ E → F

a. True

b. False

Section II: Multiple Choice – Select the best answer by circling your answer

14. ________ is the justification for step 4 in the following proof sequence?

a. Modus Tollens

b. Modus Ponens

c. Contraposition

d. Hypothetical Syllogism

e. None of the above

Proof: 1. A’

2. B

3. B→ (A V C)

4. A V C

15. Which of the following results is a counter example for the following statement: if n is an even

integer, then n2-1 is an odd integer.

a. 3

b. 15

c. 8

d. 35

e. None of the above

16. The number of elements in set C is which of the following:

|C| = |A B’| – |A B|, Where A and B are subsets of super set S, and

S = {j, k, l, m, n, o, 1, 3, 5, 7, !) A = {k, m, n, 1, 5, 7}

B = {n, o, 1, 3, 5, !)

a. 11

b. 6

c. 12

d. 5

e. 3

17. What is the justification for the following statement?

a. Implication

b. DeMorgan’s Law

c. Modus Tollens

d. Modus Ponens

e. None of the above

Statement:

If the business declares bankruptcy, then all assets must be confiscated. The

business declared bankruptcy. It follows that all assets must be confiscated.

18. Select the predicate expression that is equivalent to “Not every mystery is a spy novel” given the

following: Note: the Domain is the whole world

S(x): x is a spy novel

a.

b.

c.

d.

e.

L(x): x is long

M(x): x is a mystery

B(x,y): x is better than y

(x)(M(x) Λ [S(x)]’)

(y)(x)(S(x))’

(y)(S(x))’ Λ M(x)

(x)(M(x) Λ [S(x)])’

None of the above

19. The elements of the set describe by the following statement is/are: {x│x N and x2 – 5x + 6 = 0}

a. {-2, -3, 2, 3}

b. {-2, -3}

c. {2, 3}

d. {-2, -3, 0, 2, 3)

e. None of the above

20. Which of the following statements is true?

a. A

b. {} = {{}}

c. If A B and B C, then A C

d. If A ≠ B and B ≠ C, then A ≠ C

e. If A B and B C, then A C

For the following questions show all work for partial credit.

21. Let A = {2, 4, 5, 6, 8}, B = {1, 4, 5, 9}, C = {x | x ε Domain Z and 2 =1 (Partial Credit Possible)

Section I: True (T) or False (F) – Circle Your Answer

1. Propositional well-formed formula may be proven true with an algorithm same as a predicate

well-formed formula may be proven valid.

a. True

b. False

2. The key to a proof by the first principle of induction is to see how the truth of P at the value

k + 1 depends on the truth of P at value k.

a. True

b. False

3. Recursive algorithms are valuable primarily because they run more efficiently than iterative

algorithms.

a. True

b. False

4. The predicate quantifier, , indicates that there exists 1, a few or some objects in the

domain.

a. True

b. False

5. Modus Ponens is probably the most intuitive inference rule.

a. True

b. False

6. Is the following expression true or false? (D U B’)’ = D’ ꓵ B

a. True

b. False

7. Given the truth values A (true), B (true), C (false), D (True) what is the truth value of the

following wff? (A’ V B’) Λ (C’ Λ A) → D

a. True

b. False

8. The expression: A’ → (B V C’) is a tautology.

a. True

b. False

9. A recurrence relation is homogenous if g(n) has a constant coefficient.

a. True

b. False

10. Your professor has 4 pairs of shoes, 12 pairs of socks, 10 pairs of pants, 15 different shirts, 5 suit

jackets and 3 “UTA” lapel pins he may or may not pin to his jacket. Given these options, he

therefore has 108,000 different combinations he may wear.

a. True

b. False

11. The following expression is a well-formed formula: ((A V B) Λ C) → (B V C’))

a. True

b. False

12. The disjunction of any propositional wff with a tautology has a truth value of true.

a. True

b. False

13. (A Λ B) Λ C Λ D → E → F ↔ A Λ B Λ C Λ D Λ E → F

a. True

b. False

Section II: Multiple Choice – Select the best answer by circling your answer

14. ________ is the justification for step 4 in the following proof sequence?

a. Modus Tollens

b. Modus Ponens

c. Contraposition

d. Hypothetical Syllogism

e. None of the above

Proof: 1. A’

2. B

3. B→ (A V C)

4. A V C

15. Which of the following results is a counter example for the following statement: if n is an even

integer, then n2-1 is an odd integer.

a. 3

b. 15

c. 8

d. 35

e. None of the above

16. The number of elements in set C is which of the following:

|C| = |A B’| – |A B|, Where A and B are subsets of super set S, and

S = {j, k, l, m, n, o, 1, 3, 5, 7, !) A = {k, m, n, 1, 5, 7}

B = {n, o, 1, 3, 5, !)

a. 11

b. 6

c. 12

d. 5

e. 3

17. What is the justification for the following statement?

a. Implication

b. DeMorgan’s Law

c. Modus Tollens

d. Modus Ponens

e. None of the above

Statement:

If the business declares bankruptcy, then all assets must be confiscated. The

business declared bankruptcy. It follows that all assets must be confiscated.

18. Select the predicate expression that is equivalent to “Not every mystery is a spy novel” given the

following: Note: the Domain is the whole world

S(x): x is a spy novel

a.

b.

c.

d.

e.

L(x): x is long

M(x): x is a mystery

B(x,y): x is better than y

(x)(M(x) Λ [S(x)]’)

(y)(x)(S(x))’

(y)(S(x))’ Λ M(x)

(x)(M(x) Λ [S(x)])’

None of the above

19. The elements of the set describe by the following statement is/are: {x│x N and x2 – 5x + 6 = 0}

a. {-2, -3, 2, 3}

b. {-2, -3}

c. {2, 3}

d. {-2, -3, 0, 2, 3)

e. None of the above

20. Which of the following statements is true?

a. A

b. {} = {{}}

c. If A B and B C, then A C

d. If A ≠ B and B ≠ C, then A ≠ C

e. If A B and B C, then A C

For the following questions show all work for partial credit.

21. Let A = {2, 4, 5, 6, 8}, B = {1, 4, 5, 9}, C = {x | x ε Domain Z and 2

Purchase answer to see full

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Tags:

Modus Ponens

Recursive Algorithms

principle of induction

Predicate well formed formula

Predicate quantifier

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