Mathematical logic: propositional and predicate logic, propositional equivalences, normal forms, predicates and quantifiers, nested quantifiers, rules of inference. - One Line Questions

1. The distributive law states that p ∧ (q ∨ r) is equivalent to: — (p ∧ q) ∨ (p ∧ r)
2. The distributive law states that p ∨ (q ∧ r) is equivalent to: — (p ∨ q) ∧ (p ∨ r)
3. Which connective is used to express 'if and only if'? — ↔
4. The statement 'Some students are intelligent' can be represented in predicate logic as: — ∃x (Student(x) ∧ Intelligent(x))
5. Consider the statement 'Every student has a unique student ID'. Which of the following best represents this? — ∀x ∃y (Student(x) ∧ ID(y) ∧ hasID(x,y) ∧ ∀z (z≠y → ¬hasID(x,z)))
6. Which statement is equivalent to '¬∀x P(x)'? — ∃x ¬P(x)
7. Existential Generalization (EG) states that if P(c) is true for some specific term 'c', then: — ∃x P(x) is true
8. The statement 'All humans are mortal' can be represented in predicate logic as: — ∀x (Human(x) → Mortal(x))
9. Which statement is equivalent to '¬∃x P(x)'? — ∀x ¬P(x)
10. Universal Generalization (UG) states that if P(c) is true for an arbitrary constant 'c', then: — ∀x P(x) is true
11. Which of the following is a valid DNF for the expression p → q? — ¬p ∨ q
12. De Morgan's Law states that ¬(p ∧ q) is equivalent to: — ¬p ∨ ¬q
13. De Morgan's Law states that ¬(p ∨ q) is equivalent to: — ¬p ∧ ¬q
14. What is the logical equivalence for 'p → q'? — ¬p ∨ q
15. Which logical connective represents 'and'? — ∧
16. The statement 'If it rains, the ground gets wet' is an example of: — An implication
17. What is the Conjunctive Normal Form (CNF)? — A conjunction of disjunctions of literals
18. What is the Disjunctive Normal Form (DNF)? — A disjunction of conjunctions of literals
19. What is a predicate? — A statement involving variables that can be assigned truth values
20. Consider the statement: 'For every integer x, if x is even, then x+1 is odd.' This is an example of: — A conditional statement within a universal quantification
21. A well-formed formula (WFF) in propositional logic is: — A formula that follows the formation rules of the logic
22. In the rule of inference Universal Generalization (UG), the constant 'c' must be: — Arbitrary and not have any special assumptions
23. What does the symbol '→' represent in propositional logic? — Implication
24. The symbol '∨' in propositional logic represents which logical operation? — Disjunction
25. A proposition that is always true is called a: — Tautology
26. The statement 'For every number x, there exists a number y such that y > x' is: — True
27. What is the truth value of the proposition '2 + 2 = 4'? — True
28. What does the existential quantifier '∃' mean? — There exists
29. Modus Tollens is represented by: — If P → Q and ¬Q, then ¬P
30. A variable is bound if it is: — Introduced by a quantifier
31. A variable is free if it is: — Not bound by any quantifier
32. Which rule of inference is used to prove statements of the form P → Q? — Conditional Proof
33. Which rule of inference is represented by 'If P is true and P → Q is true, then Q must be true'? — Modus Ponens
34. Hypothetical Syllogism follows the pattern: — P → Q, Q → R ⊢ P → R
35. Which of the following is a tautology? — p ∨ ¬p
36. Which of the following is a valid CNF for the expression ¬(p → q)? — p ∧ ¬q
37. Which logical equivalence is known as the law of excluded middle? — p ∨ ¬p
38. Which of the following is a contradiction? — p ∧ ¬p
39. Disjunctive Syllogism allows us to conclude Q from: — P ∨ Q and ¬P
40. Universal Instantiation (UI) states that if ∀x P(x) is true, then: — P(a) is true for any specific term 'a'
41. Existential Instantiation (EI) states that if ∃x P(x) is true, then: — P(c) is true for a specific, but not yet determined, constant 'c'
42. What is the basic building block of propositional logic? — Proposition
43. In predicate logic, P(x) represents a: — Predicate
44. The principle of resolution is primarily used in: — Proving unsatisfiability of a set of clauses
45. What does 'nested quantifiers' refer to? — One quantifier inside the scope of another
46. A proposition that is always false is called a: — Contradiction
47. What is the scope of a quantifier? — The part of the formula where the variable is bound by the quantifier
48. What does the universal quantifier '∀' mean? — For all
49. The statement 'There is a number that is greater than every other number' is: — False
50. What is the truth value of 'p XOR q' (exclusive or) when both p and q are true? — False
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