Mathematical logic: propositional and predicate logic, propositional equivalences, normal forms, predicates and quantifiers, nested quantifiers, rules of inference. - Question Bank

1. Consider the statement: 'For every integer x, if x is even, then x+1 is odd.' This is an example of:
A) An existential statement
B) A universal statement
C) A conditional statement within a universal quantification
D) A contradiction
2. The principle of resolution is primarily used in:
A) Proving unsatisfiability of a set of clauses
B) Deriving tautologies
C) Simplifying logical expressions
D) Converting to DNF
3. Which rule of inference is used to prove statements of the form P → Q?
A) Modus Ponens
B) Modus Tollens
C) Conditional Proof
D) Resolution
4. A well-formed formula (WFF) in propositional logic is:
A) Any string of symbols
B) A formula that follows the formation rules of the logic
C) A formula that is always true
D) A formula that is always false
5. What is the truth value of 'p XOR q' (exclusive or) when both p and q are true?
A) True
B) False
C) Undetermined
D) Depends on the context
6. The statement 'If it rains, the ground gets wet' is an example of:
A) A biconditional statement
B) An implication
C) A conjunction
D) A disjunction
7. Which logical equivalence is known as the law of excluded middle?
A) p ∨ ¬p
B) p ∧ ¬p
C) p → p
D) ¬(p ∧ ¬p)
8. In the rule of inference Universal Generalization (UG), the constant 'c' must be:
A) Arbitrary and not have any special assumptions
B) A specific, well-known constant
C) The only element in the domain
D) A variable itself
9. Universal Generalization (UG) states that if P(c) is true for an arbitrary constant 'c', then:
A) ∃x P(x) is true
B) ¬∃x P(x) is true
C) ∀x P(x) is true
D) ¬∀x P(x) is true
10. Existential Instantiation (EI) states that if ∃x P(x) is true, then:
A) P(c) is true for some arbitrary constant 'c'
B) P(c) is true for all constants 'c'
C) P(c) is true for a specific, but not yet determined, constant 'c'
D) P(c) is false for some constant 'c'
11. Existential Generalization (EG) states that if P(c) is true for some specific term 'c', then:
A) ∀x P(x) is true
B) ∃x P(x) is true
C) ¬∃x P(x) is true
D) ¬∀x P(x) is true
12. Universal Instantiation (UI) states that if ∀x P(x) is true, then:
A) P(a) is true for any specific term 'a'
B) P(a) is true for some specific term 'a'
C) P(a) is false for any specific term 'a'
D) P(a) is false for some specific term 'a'
13. Disjunctive Syllogism allows us to conclude Q from:
A) P ∨ Q and P
B) P ∨ Q and ¬P
C) P → Q and P
D) P ∧ Q and P
14. Hypothetical Syllogism follows the pattern:
A) P → Q, Q → R ⊢ P → R
B) P → Q, ¬Q ⊢ ¬P
C) P ∨ Q, ¬P ⊢ Q
D) P, P → Q ⊢ Q
15. Modus Tollens is represented by:
A) If P → Q and P, then Q
B) If P → Q and ¬Q, then ¬P
C) If P → Q and Q → R, then P → R
D) If P ∨ Q and ¬P, then Q
16. Which rule of inference is represented by 'If P is true and P → Q is true, then Q must be true'?
A) Modus Tollens
B) Hypothetical Syllogism
C) Modus Ponens
D) Disjunctive Syllogism
17. The statement 'For every number x, there exists a number y such that y > x' is:
A) False
B) True
C) Undetermined
D) Contradictory
18. The statement 'There is a number that is greater than every other number' is:
A) True
B) False
C) Undetermined
D) Paradoxical
19. Consider the statement 'Every student has a unique student ID'. Which of the following best represents this?
A) ∀x ∃y (Student(x) ∧ ID(y) ∧ hasID(x,y))
B) ∃y ∀x (Student(x) ∧ ID(y) ∧ hasID(x,y))
C) ∀x ∃y (Student(x) ∧ ID(y) ∧ hasID(x,y) ∧ ∀z (z≠y → ¬hasID(x,z)))
D) ∃x ∀y (Student(x) ∧ ID(y) ∧ hasID(x,y))
20. What does 'nested quantifiers' refer to?
A) Quantifiers applied to the same variable
B) One quantifier inside the scope of another
C) Multiple quantifiers with different scopes
D) Quantifiers appearing consecutively
21. Which statement is equivalent to '¬∃x P(x)'?
A) ∃x ¬P(x)
B) ∀x ¬P(x)
C) ¬∀x P(x)
D) ∀x P(x)
22. Which statement is equivalent to '¬∀x P(x)'?
A) ∀x ¬P(x)
B) ∃x ¬P(x)
C) ¬∃x P(x)
D) ∃x P(x)
23. A variable is free if it is:
A) Introduced by a quantifier
B) Not bound by any quantifier
C) Always universally quantified
D) Always existentially quantified
24. A variable is bound if it is:
A) Introduced by a quantifier
B) Assigned a specific value
C) Used in a predicate
D) Free in the formula
25. What is the scope of a quantifier?
A) The part of the formula where the variable is bound by the quantifier
B) The entire formula
C) The part of the formula following the quantifier
D) The part of the formula preceding the quantifier
26. The statement 'Some students are intelligent' can be represented 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))
27. The statement 'All humans are mortal' can be represented 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))
28. What does the existential quantifier '∃' mean?
A) For all
B) There exists
C) Every
D) No
29. What does the universal quantifier '∀' mean?
A) There exists
B) For all
C) At least one
D) None
30. In predicate logic, P(x) represents a:
A) Proposition
B) Predicate
C) Quantifier
D) Connective
31. What is a predicate?
A) A statement with a truth value
B) A statement involving variables that can be assigned truth values
C) A logical connective
D) A quantifier
32. Which of the following is a valid DNF for the expression p → q?
A) ¬p ∧ ¬q
B) ¬p ∨ q
C) p ∧ q
D) ¬p ∨ ¬q
33. Which of the following is a valid CNF for the expression ¬(p → q)?
A) p ∧ ¬q
B) ¬p ∨ q
C) p ∨ q
D) ¬p ∧ q
34. What is the Disjunctive Normal Form (DNF)?
A) A conjunction of disjunctions of literals
B) A disjunction of conjunctions of literals
C) A conjunction of literals
D) A disjunction of literals
35. What is the Conjunctive Normal Form (CNF)?
A) A conjunction of disjunctions of literals
B) A disjunction of conjunctions of literals
C) A disjunction of literals
D) A conjunction of literals
36. The distributive law states that p ∨ (q ∧ r) is equivalent to:
A) (p ∨ q) ∧ (p ∨ r)
B) (p ∧ q) ∨ (p ∧ r)
C) p ∧ (q ∨ r)
D) (p ∨ q) ∧ r
37. The distributive law states that p ∧ (q ∨ r) is equivalent to:
A) (p ∧ q) ∨ (p ∧ r)
B) (p ∨ q) ∧ (p ∨ r)
C) (p ∧ q) ∨ r
D) p ∨ (q ∧ r)
38. De Morgan's Law states that ¬(p ∨ q) is equivalent to:
A) ¬p ∨ ¬q
B) ¬p ∧ ¬q
C) p ∧ ¬q
D) ¬p ∨ q
39. De Morgan's Law states that ¬(p ∧ q) is equivalent to:
A) ¬p ∧ ¬q
B) ¬p ∨ ¬q
C) p ∨ ¬q
D) ¬p ∧ q
40. What is the logical equivalence for 'p → q'?
A) ¬p ∨ q
B) p ∨ ¬q
C) ¬p ∧ q
D) p ∧ ¬q
41. Which of the following is a contradiction?
A) p ∨ ¬p
B) p → p
C) p ∧ ¬p
D) ¬(p → p)
42. Which of the following is a tautology?
A) p ∧ ¬p
B) p ∨ ¬p
C) p → ¬p
D) p ↔ ¬p
43. A proposition that is always true is called a:
A) Contradiction
B) Contingency
C) Tautology
D) Satisfiable
44. A proposition that is always false is called a:
A) Tautology
B) Contingency
C) Contradiction
D) Equivalence
45. What is the truth value of the proposition '2 + 2 = 4'?
A) False
B) True
C) Undetermined
D) Contradictory
46. Which connective is used to express 'if and only if'?
A) →
B) ↔
C) ∧
D) ∨
47. What does the symbol '→' represent in propositional logic?
A) Biconditional
B) Disjunction
C) Implication
D) Conjunction
48. The symbol '∨' in propositional logic represents which logical operation?
A) Conjunction
B) Disjunction
C) Negation
D) Implication
49. Which logical connective represents 'and'?
A) ∨
B) ¬
C) →
D) ∧
50. What is the basic building block of propositional logic?
A) Predicate
B) Quantifier
C) Proposition
D) Connective
All Question Banks
E-Book Mathematical logic: propositional and predicate logic, propositional equivalences, normal forms, predicates and quantifiers, nested quantifiers, rules of inference. One Line Questions Mathematical logic: propositional and predicate logic, propositional equivalences, normal forms, predicates and quantifiers, nested quantifiers, rules of inference. Online Test Mathematical logic: propositional and predicate logic, propositional equivalences, normal forms, predicates and quantifiers, nested quantifiers, rules of inference.