“Mathematical Logic: Propositional and Predicate Logic, Propositional Equivalences, Normal Forms, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference.” - Question Bank

1. The statement 'It is not the case that both p and q are true' is logically equivalent to:
A) p ∧ q
B) ¬p ∧ ¬q
C) ¬p ∨ ¬q
D) p ∨ q
2. Which of the following is a valid argument form?
A) p → q, q ⊢ p
B) p → q, ¬p ⊢ ¬q
C) p → q, q ⊢ ¬p
D) p → q, ¬q ⊢ p
3. What is the Disjunctive Normal Form (DNF) of ¬(p ∧ q)?
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q
C) (¬p ∧ q) ∨ (p ∧ ¬q) ∨ (¬p ∧ ¬q)
D) (¬p ∨ q) ∧ (p ∨ ¬q)
4. The statement 'All men are mortal. Socrates is a man. Therefore, Socrates is mortal.' is an example of deductive reasoning using:
A) Fallacy of the Undistributed Middle
B) Modus Ponens
C) Universal Instantiation and Modus Ponens
D) Disjunctive Syllogism
5. Which rule of inference is used to prove statements of the form P → Q by assuming P and deriving Q?
A) Modus Tollens
B) Hypothetical Syllogism
C) Conditional Proof (or Assumption)
D) Modus Ponens
6. The statement 'x > 5' is an example of a:
A) Proposition
B) Predicate
C) Quantifier
D) Logical connective
7. What is the domain of discourse for the statement '∀x (x is a dog → x has fur)'?
A) All animals
B) All mammals
C) All dogs
D) The set of all entities being considered (e.g., all living beings)
8. The logical equivalence (p ∧ q) ∨ r ≡ (p ∨ r) ∧ (q ∨ r) is known as the:
A) Commutative Law
B) Associative Law
C) Distributive Law
D) Absorption Law
9. Which of the following is an example of affirming the consequent, a logical fallacy?
A) P → Q, P ⊢ Q
B) P → Q, ¬Q ⊢ ¬P
C) P → Q, Q ⊢ P
D) P → Q, P ⊢ ¬Q
10. The statement 'There is a number that is even and prime' can be represented as:
A) ∀x (Even(x) ∧ Prime(x))
B) ∃x (Even(x) ∨ Prime(x))
C) ∃x (Even(x) ∧ Prime(x))
D) ∀x (Even(x) → Prime(x))
11. A proposition that is true for all possible truth assignments of its propositional variables is called:
A) Contradiction
B) Contingency
C) Tautology
D) Satisfiable
12. What is the main purpose of Normal Forms (CNF and DNF)?
A) To simplify complex logical expressions.
B) To standardize the representation of logical formulas, especially for automated reasoning.
C) To increase the number of variables in a formula.
D) To introduce more quantifiers into a formula.
13. Which logical equivalence is used to convert an implication into a disjunction?
A) Commutative Law
B) Implication Law (p → q ≡ ¬p ∨ q)
C) Associative Law
D) De Morgan's Law
14. Consider the statement: 'If it is sunny (S), then I will go to the beach (B)'. Symbolically: S → B. If it is not sunny (¬S), what can we conclude?
A) I will go to the beach (B)
B) I will not go to the beach (¬B)
C) Nothing can be concluded about going to the beach
D) I will definitely go to the beach
15. What is the negation of ∃x P(x)?
A) ∀x ¬P(x)
B) ∃x ¬P(x)
C) ¬∃x P(x)
D) ¬∀x P(x)
16. What is the negation of ∀x P(x)?
A) ∃x ¬P(x)
B) ∀x ¬P(x)
C) ¬∀x P(x)
D) ¬∃x P(x)
17. The rule of Existential Generalization states that if P(a) is true for some element 'a', then we can conclude:
A) ∀x P(x)
B) ∃x P(x)
C) ¬∃x P(x)
D) ¬∀x P(x)
18. In predicate logic, the rule of Universal Instantiation states that if ∀x P(x) is true, then for any specific element 'a' in the domain, we can conclude:
A) P(a)
B) ∃x P(x)
C) ¬P(a)
D) ∀x ¬P(x)
19. If we have P and Q, what rule of inference allows us to conclude P ∧ Q?
A) Conjunction
B) Simplification
C) Addition
D) Modus Ponens
20. The rule of inference that states if P ∧ Q is true, then P is true, is called:
A) Conjunction
B) Simplification
C) Addition
D) Modus Ponens
21. Disjunctive Syllogism allows us to conclude Q from P ∨ Q and:
A) P
B) ¬P
C) Q
D) ¬Q
22. Which rule of inference allows us to conclude P ∨ Q from P?
A) Addition
B) Simplification
C) Disjunctive Syllogism
D) Resolution
23. The rule of inference known as Hypothetical Syllogism states that if P → Q and Q → R, then we can conclude:
A) P → R
B) ¬P → R
C) P → ¬R
D) ¬P → ¬R
24. Modus Tollens is a rule of inference that states if we have ¬Q and P → Q, we can conclude:
A) Q
B) P
C) ¬P
D) ¬Q
25. Modus Ponens is a rule of inference that states if we have P and P → Q, we can conclude:
A) ¬Q
B) Q
C) ¬P
D) P ∧ Q
26. Which of the following is NOT a valid rule of inference for propositional logic?
A) Modus Ponens
B) Modus Tollens
C) Hypothetical Syllogism
D) Affirming the Consequent
27. What is the meaning of '∃y ∀x P(x, y)'?
A) There exists a y such that for all x, P(x, y) is true.
B) For all x, there exists a y such that P(x, y) is true.
C) There exists an x such that for all y, P(x, y) is true.
D) For all y, there exists an x such that P(x, y) is true.
28. What is the meaning of '∀x ∃y P(x, y)'?
A) For every x, there exists a y such that P(x, y) is true.
B) There exists an x such that for every y, P(x, y) is true.
C) For all x and all y, P(x, y) is true.
D) There exists an x and there exists a y such that P(x, y) is true.
29. In the statement '∃x P(x)', the variable 'x' is:
A) Free
B) Bound
C) Quantified
D) Constant
30. In the statement '∀x P(x)', the variable 'x' is:
A) Free
B) Bound
C) Undefined
D) Quantified
31. The statement 'Some students are intelligent' can be expressed in predicate logic as:
A) ∀x (Student(x) → Intelligent(x))
B) ∃x (Student(x) ∧ Intelligent(x))
C) ∀x (Student(x) ∨ Intelligent(x))
D) ∃x (Student(x) ∨ Intelligent(x))
32. What does the symbol '∃' represent in predicate logic?
A) Universal Quantifier
B) Existential Quantifier
C) Conjunction
D) Disjunction
33. The statement 'All humans are mortal' can be expressed in predicate logic as:
A) ∃x (Human(x) ∧ Mortal(x))
B) ∀x (Human(x) → Mortal(x))
C) ∀x (Human(x) ∨ Mortal(x))
D) ∃x (Human(x) → Mortal(x))
34. What does the symbol '∀' represent in predicate logic?
A) Existential Quantifier
B) Universal Quantifier
C) Negation
D) Implication
35. In predicate logic, a statement about a single subject and predicate is called a:
A) Proposition
B) Quantifier
C) Predicate
D) Formula
36. Which normal form consists of a conjunction of clauses, where each clause is a disjunction of literals?
A) Disjunctive Normal Form (DNF)
B) Conjunctive Normal Form (CNF)
C) Principal Conjunctive Normal Form
D) Principal Disjunctive Normal Form
37. A proposition that is either a tautology or a contradiction is called:
A) Contingent
B) Equivalence
C) Boolean
D) Boolean Expression
38. What is the Conjunctive Normal Form (CNF) of the proposition (p ∨ q) ∧ r?
A) ((p ∨ q) ∧ r)
B) (p ∧ r) ∨ (q ∧ r)
C) (p ∨ r) ∧ (q ∨ r)
D) ¬(¬p ∧ ¬q) ∧ r
39. The statement 'p → q' is logically equivalent to:
A) ¬p ∨ q
B) p ∨ ¬q
C) ¬p ∧ q
D) p ∧ ¬q
40. Which logical equivalence states that ¬(p ∧ q) is equivalent to ¬p ∨ ¬q?
A) Commutative Law
B) Associative Law
C) De Morgan's Law
D) Distributive Law
41. The negation of 'All birds can fly' is:
A) No birds can fly
B) Some birds cannot fly
C) All birds cannot fly
D) Some birds can fly
42. What is the truth value of P ∨ Q if P is false and Q is false?
A) True
B) False
C) Undefined
D) True only if P is true
43. If proposition P is true and proposition Q is false, what is the truth value of P ∧ Q?
A) True
B) False
C) Cannot be determined
D) True if P or Q is true
44. What is the truth value of the proposition '2 + 2 = 4'?
A) True
B) False
C) Undefined
D) Contingent
45. The statement 'The sky is blue if and only if grass is green' is a:
A) Implication
B) Conjunction
C) Disjunction
D) Biconditional
46. Which of the following is a tautology?
A) p ∧ ¬p
B) p → p
C) p ∨ ¬p
D) p ⊕ p
47. A compound proposition that is always false, regardless of the truth values of its components, is called a:
A) Tautology
B) Contradiction
C) Contingency
D) Equivalence
48. What does the symbol '∨' represent in logic?
A) AND
B) OR
C) NOT
D) IF AND ONLY IF
49. The statement 'If it is raining, then the ground is wet' can be represented symbolically as P → Q, where P is 'It is raining' and Q is 'The ground is wet'. This is an example of:
A) Conjunction
B) Disjunction
C) Implication
D) Biconditional
50. Which logical connective is represented by the symbol '∧'?
A) Disjunction
B) Conjunction
C) Implication
D) Negation
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