Chapter 1

Question 1

1. Which of the following is a proposition?
   a) What time is it?
   b) The sky is blue.
   c) Open the door.
   d) x + 2 = 5

Answer 1

b) The sky is blue.

Question 2

3. What is the truth value of the proposition P∧Q, if P=True and Q=False?
   a) True
   b) False
   c) Depends on the context
   d) Undefined

Answer 2

b) False

Question 3

4. Translate the statement “If it rains, then the ground will be wet” into propositional logic.
   a) P∧Q
   b) P∨Q
   c) P→Q
   d) ¬P∧Q

Answer 3

c) P→Q

Question 4

5. Which of the following statements is a tautology?
   a) P∨¬P
   b) P∧Q
   c) P∧¬P
   d) ¬P

Answer 4

a) P∨¬P

Question 5

6. Which of the following is NOT a tautological equivalence?
   a) ¬(P∨Q)≡(¬P∧¬Q)
   b) ¬(P∧Q)≡(¬P∨¬Q)
   c) P∧Q≡P∨Q
   d) (P→Q)≡(¬P∨Q)

Answer 5

c) P∧Q≡P∨Q

Question 6

7. Which of the following is a valid use of De Morgan’s Laws?
   a) ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
   b) ¬P ∧ Q ≡ P ∨ Q
   c) ¬P ∨ Q ≡ P ∧ Q
   d) ¬(P ∨ Q) ≡ P ∧ Q

Answer 6

a) ¬(P ∧ Q) ≡ ¬P ∨ ¬Q

Question 7

8. In a valid argument, if the premises are true, what can be said about the conclusion?
   a) The conclusion must also be true.
   b) The conclusion can be false.
   c) The truth value of the conclusion is unrelated to the premises.
   d) The conclusion must be false.

Answer 7

a) The conclusion must also be true.

Question 8

9. Which of the following is an equivalence rule?
   a) Modus Ponens
   b) Modus Tollens
   c) Double Negation
   d) Hypothetical Syllogism

Answer 8

c) Double Negation

Question 9

10. Using the Deduction Method, what can you infer from P→Q?
    a) Q
    b) ¬Q
    c) P ∧ Q
    d) ¬P

Answer 9

a) Q

Question 10

11. Which of the following is an example of Modus Ponens?
    a) P→Q,¬Q⇒¬P
    b) P→Q,P⇒Q
    c) P ∨ Q,¬P⇒Q
    d) P ∧ Q,¬P⇒Q

Answer 10

b) P→Q,P⇒Q

Question 11

12. Consider the verbal argument: “All humans are mortal. Socrates is human. Therefore, Socrates is mortal.” What type of reasoning does this use?
    a) Deductive reasoning
    b) Inductive reasoning
    c) Abductive reasoning
    d) Analogical reasoning

Answer 11

a) Deductive reasoning

Question 12

1. What is a proposition in logic?
   a) A sentence that is always true
   b) A declarative sentence that is either true or false
   c) A question or command
   d) A statement with no truth value

Answer 12

b) A declarative sentence that is either true or false

Question 13

8. A contradiction is a proposition that is:
   a) Always true
   b) Sometimes true
   c) Always false
   d) Dependent on the context

Answer 13

c) Always false

Question 14

9. Using De Morgan’s Laws, simplify ¬(P∧Q):
   a) ¬P∨¬Q
   b) P∨Q
   c) ¬P∧Q
   d) P∧Q

Answer 14

a) ¬P ∨ ¬Q

Question 15

12. What is an invalid argument called?
    a) Tautology
    b) Fallacy
    c) Contradiction
    d) Equivalence

Answer 15

b) Fallacy

Question 16

11. Identify the form of the argument:
    Premises: P→Q,P
    Conclusion: Q
    a) Modus Ponens
    b) Modus Tollens
    c) Hypothetical Syllogism
    d) Disjunction Elimination

Answer 16

a) Modus Ponens

Question 17

12. What is an invalid argument called?
    a) Tautology
    b) Fallacy
    c) Contradiction
    d) Equivalence

Answer 17

b) Fallacy

Question 18

14. The inference rule “If P→Q and ¬Q, then ¬P” is known as:
    a) Modus Ponens
    b) Modus Tollens
    c) Hypothetical Syllogism
    d) Disjunction Introduction

Answer 18

b) Modus Tollens

Question 19

15. What can be inferred from the premises P→Q and Q→R?
    a) P→R
    b) R→Q
    c) P→Q ∧ R
    d) Q→P

Answer 19

a) P→R

Question 20

18. “Most people in this city are left-handed. John is from this city. Therefore, John is probably left-handed.” This is an example of:
    a) Deductive reasoning
    b) Inductive reasoning
    c) Abductive reasoning
    d) Circular reasoning

Answer 20

b) Inductive reasoning

Question 21

What type of reasoning is this?

Definition: A reasoning process where generalizations are made based on specific observations or evidence. It moves from specific instances to broader generalizations or theories.
Key Feature: The conclusion is probable but not guaranteed to be true, even if the premises are true.

Answer 21

inductive reasoning

Question 22

What type of reasoning is this?

Definition: A reasoning process where a conclusion logically follows from a set of premises. It moves from general principles to specific conclusions.
Key Feature: The conclusion is guaranteed to be true if the premises are true and the reasoning is valid.

Answer 22

Deductive Reasoning

Question 23

For each of the following, indicate whether it is a proposition or not:
(1) The sun sets in the west.
(2) Can you help me with this math problem?
(3) All humans have 12 fingers.
(4) Close the door, please.

A. (1) and (2) are propositions.
B. (1) and (3) are propositions.
C. (2) and (4) are propositions.
D. (3) and (4) are propositions.

Answer 23

B. (1) and (3) are propositions.

Question 24

Truth Table Row Count
How many rows will be in a truth table for a statement containing 4 variables?

A. 8
B. 16
C. 32
D. 4

Answer 24

B. 16

Question 25

Negate A ∧ B:

A. ¬A ∨ ¬B
B. A ∨ B
C. ¬A ∧ ¬B
D. ¬(A ∨ B)

Answer 25

A. ¬A ∨ ¬B

Question 26

Construct the truth table for A ∨ B and determine its final column for the following values of A and B:
A=T, B=F
A=F, B=T
A=T, B=T
A=F, B=F

A. T,T,T,F
B. F,T,T,T
C. T,F,F,F
D. F,F,T,T

Answer 26

A. T,T,T,F

Question 27

Which of the following logical statements is a tautology?

1. A ∨ ¬A
2. A ∧ ¬A
3. A ∨ B ∨ C when A=T, B=F, C=F

A. Only 1
B. Only 2
C. Both 1 and 3
D. None

Answer 27

A. Only 1

Question 28

Which pair of logical statements is equivalent?
1. A ∧ (B ∨ C)
2. (A ∧ B) ∨ (A ∧ C)
3. A ∨ (B ∧ C)
4. (A ∨ B) ∧ (A ∨ C)

A. 1 and 2
B. 1 and 3
C. 2 and 4
D. 3 and 4

Answer 28

A. 1 and 2

Question 29

Simplify the following expression:
!(A ∨ B) ∧ B
A. ¬A ∧ B
B. ¬B
C. B
D. ¬A ∧ ¬B

Answer 29

D. ¬A ∧ ¬B

Question 30

Which of the following statements is always a contradiction?
1. A ∧ ¬A
2. A ∨ ¬A
3. (A ∨ B) ∧ (¬A ∧ ¬B)

A. 1 only
B. 3 only
C. Both 1 and 3
D. None

Answer 30

C. Both 1 and 3