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

30:00

1. What is the basic building block of propositional logic?

A) Predicate B) Quantifier C) Proposition D) Connective

2. Which logical connective represents 'and'?

A) ∨ B) ¬ C) → D) ∧

3. The symbol '∨' in propositional logic represents which logical operation?

A) Conjunction B) Disjunction C) Negation D) Implication

4. What does the symbol '→' represent in propositional logic?

A) Biconditional B) Disjunction C) Implication D) Conjunction

5. Which connective is used to express 'if and only if'?

A) → B) ↔ C) ∧ D) ∨

6. What is the truth value of the proposition '2 + 2 = 4'?

A) False B) True C) Undetermined D) Contradictory

7. A proposition that is always false is called a:

A) Tautology B) Contingency C) Contradiction D) Equivalence

8. A proposition that is always true is called a:

A) Contradiction B) Contingency C) Tautology D) Satisfiable

9. Which of the following is a tautology?

A) p ∧ ¬p B) p ∨ ¬p C) p → ¬p D) p ↔ ¬p

10. Which of the following is a contradiction?

A) p ∨ ¬p B) p → p C) p ∧ ¬p D) ¬(p → p)

11. What is the logical equivalence for 'p → q'?

A) ¬p ∨ q B) p ∨ ¬q C) ¬p ∧ q D) p ∧ ¬q

12. De Morgan's Law states that ¬(p ∧ q) is equivalent to:

A) ¬p ∧ ¬q B) ¬p ∨ ¬q C) p ∨ ¬q D) ¬p ∧ q

13. De Morgan's Law states that ¬(p ∨ q) is equivalent to:

A) ¬p ∨ ¬q B) ¬p ∧ ¬q C) p ∧ ¬q D) ¬p ∨ q

14. 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)

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. In predicate logic, P(x) represents a:

A) Proposition B) Predicate C) Quantifier D) Connective

22. What does the universal quantifier '∀' mean?

A) There exists B) For all C) At least one D) None

23. What does the existential quantifier '∃' mean?

A) For all B) There exists C) Every D) No

24. 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))

25. 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))

26. 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

27. 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

28. 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

29. Which statement is equivalent to '¬∀x P(x)'?

A) ∀x ¬P(x) B) ∃x ¬P(x) C) ¬∃x P(x) D) ∃x P(x)

30. Which statement is equivalent to '¬∃x P(x)'?

A) ∃x ¬P(x) B) ∀x ¬P(x) C) ¬∀x P(x) D) ∀x P(x)

31. 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

32. 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))

33. The statement 'There is a number that is greater than every other number' is:

A) True B) False C) Undetermined D) Paradoxical

34. The statement 'For every number x, there exists a number y such that y > x' is:

A) False B) True C) Undetermined D) Contradictory

35. 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

36. 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

37. 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

38. 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

39. 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'

40. 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

41. 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'

42. 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

43. 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

44. Which logical equivalence is known as the law of excluded middle?

A) p ∨ ¬p B) p ∧ ¬p C) p → p D) ¬(p ∧ ¬p)

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

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