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

1. What is the Conjunctive Normal Form (CNF) of the proposition (p ∨ q) ∧ r? — ((p ∨ q) ∧ r)
2. The statement 'There is a number that is even and prime' can be represented as: — ∃x (Even(x) ∧ Prime(x))
3. The statement 'Some students are intelligent' can be expressed in predicate logic as: — ∃x (Student(x) ∧ Intelligent(x))
4. What is the negation of ∃x P(x)? — ∀x ¬P(x)
5. The rule of Existential Generalization states that if P(a) is true for some element 'a', then we can conclude: — ∃x P(x)
6. The statement 'All humans are mortal' can be expressed in predicate logic as: — ∀x (Human(x) → Mortal(x))
7. What is the negation of ∀x P(x)? — ∃x ¬P(x)
8. What is the Disjunctive Normal Form (DNF) of ¬(p ∧ q)? — (¬p ∧ q) ∨ (p ∧ ¬q) ∨ (¬p ∧ ¬q)
9. The statement 'p → q' is logically equivalent to: — ¬p ∨ q
10. Modus Ponens is a rule of inference that states if we have P and P → Q, we can conclude: — Q
11. Which rule of inference allows us to conclude P ∨ Q from P? — Addition
12. What is the domain of discourse for the statement '∀x (x is a dog → x has fur)'? — The set of all entities being considered (e.g., all living beings)
13. What does the symbol '∨' represent in logic? — OR
14. Which logical equivalence states that ¬(p ∧ q) is equivalent to ¬p ∨ ¬q? — De Morgan's Law
15. Which logical equivalence is used to convert an implication into a disjunction? — Implication Law (p → q ≡ ¬p ∨ q)
16. The logical equivalence (p ∧ q) ∨ r ≡ (p ∨ r) ∧ (q ∨ r) is known as the: — Distributive Law
17. 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: — Implication
18. The rule of inference that states if P ∧ Q is true, then P is true, is called: — Simplification
19. If we have P and Q, what rule of inference allows us to conclude P ∧ Q? — Conjunction
20. A proposition that is either a tautology or a contradiction is called: — Boolean
21. A proposition that is true for all possible truth assignments of its propositional variables is called: — Tautology
22. Which logical connective is represented by the symbol '∧'? — Conjunction
23. Which normal form consists of a conjunction of clauses, where each clause is a disjunction of literals? — Conjunctive Normal Form (CNF)
24. What does the symbol '∀' represent in predicate logic? — Universal Quantifier
25. The statement 'All men are mortal. Socrates is a man. Therefore, Socrates is mortal.' is an example of deductive reasoning using: — Universal Instantiation and Modus Ponens
26. What is the meaning of '∀x ∃y P(x, y)'? — For every x, there exists a y such that P(x, y) is true.
27. In the statement '∀x P(x)', the variable 'x' is: — Bound
28. In the statement '∃x P(x)', the variable 'x' is: — Bound
29. 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? — Nothing can be concluded about going to the beach
30. The statement 'The sky is blue if and only if grass is green' is a: — Biconditional
31. Which of the following is NOT a valid rule of inference for propositional logic? — Affirming the Consequent
32. Which rule of inference is used to prove statements of the form P → Q by assuming P and deriving Q? — Conditional Proof (or Assumption)
33. The negation of 'All birds can fly' is: — Some birds cannot fly
34. Disjunctive Syllogism allows us to conclude Q from P ∨ Q and: — ¬P
35. Which of the following is an example of affirming the consequent, a logical fallacy? — P → Q, Q ⊢ P
36. Which of the following is a valid argument form? — p → q, ¬p ⊢ ¬q
37. The rule of inference known as Hypothetical Syllogism states that if P → Q and Q → R, then we can conclude: — P → R
38. Which of the following is a tautology? — p ∨ ¬p
39. The statement 'It is not the case that both p and q are true' is logically equivalent to: — ¬p ∨ ¬q
40. 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: — P(a)
41. In predicate logic, a statement about a single subject and predicate is called a: — Predicate
42. The statement 'x > 5' is an example of a: — Predicate
43. Modus Tollens is a rule of inference that states if we have ¬Q and P → Q, we can conclude: — ¬P
44. A compound proposition that is always false, regardless of the truth values of its components, is called a: — Contradiction
45. What is the fundamental unit of meaning in propositional logic? — Proposition
46. What is the meaning of '∃y ∀x P(x, y)'? — There exists a y such that for all x, P(x, y) is true.
47. What is the main purpose of Normal Forms (CNF and DNF)? — To standardize the representation of logical formulas, especially for automated reasoning.
48. What is the truth value of the proposition '2 + 2 = 4'? — True
49. If proposition P is true and proposition Q is false, what is the truth value of P ∧ Q? — False
50. What is the truth value of P ∨ Q if P is false and Q is false? — False
All One Line Questions
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