1. Which of the following statements is an example of a tautology?

A. p ∧ ¬p B. p ∨ ¬p C. p → q ∧ ¬q D. p ∧ q → r

2. The rule of hypothetical syllogism (chaining) allows you to deduce:

A. From p → q and q → r, infer p → r B. From p → q and p, infer q C. From p or q and not p, infer q D. From p and q, infer p ∨ q

3. Given the hypothesis "If someone is intelligent, then he/she can count from 1 to 10" and the fact "Gary can only count from 1 to 3," what can you conclude about Gary's intelligence?

A. Gary is intelligent B. Gary is not intelligent C. Gary can count from 4 to 10 D. Nothing can be concluded about Gary's intelligence

4. Which of the following is NOT typically part of a valid logical argument?

A. Hypotheses or premises B. Conclusion C. Statement of the truth values of propositions outside the proof D. Rules of inference justifying steps

5. What is the truth value of the statement \(p \wedge q\) when \(p\) is true and \(q\) is false?

A. True B. False C. Unknown D. Depends on \(p\)

6. What is the purpose of using rules of inference in proofs?

A. To assume the conclusion without justification B. To justify the step-by-step reasoning leading to a conclusion C. To list all possible propositions in an argument D. To contradict hypotheses

7. What does the rule of addition state in propositional logic?

A. From p, infer p ∧ q B. From p, infer p ∨ q C. From p ∧ q, infer p D. From p → q and p, infer q

8. When is an argument considered valid?

A. When the conclusion logically follows from the premises B. When all premises are true C. When the conclusion is true D. When the premises are false

9. Which logical operator represents negation?

A. \(\wedge\) B. \(\vee\) C. \(\neg\) D. \(\rightarrow\)

10. What does universal instantiation allow you to conclude from a statement like ∀x P(x)?

A. P(c) for some arbitrary element c in the domain B. ∃x P(x) C. P(c) for all elements c in the domain D. ¬P(c) for some element c

11. What does the conjunction operator \(\wedge\) represent?

A. Logical OR B. Logical AND C. Logical NOT D. Logical implication

12. What operation does the "addition" rule of inference perform?

A. From \(p\), infer \(p \vee q\) B. From \(p\), infer \(\neg p\) C. From \(p \wedge q\), infer \(p\) D. From \(p\), infer \(q\)

13. Which of the following best describes a valid argument in logic?

A. An argument with true hypotheses and a false conclusion B. An argument with false hypotheses but a true conclusion C. An argument where if all hypotheses are true, the conclusion must be true D. An argument that uses only universal quantifiers

14. What is the conclusion of the argument: "Gary is either intelligent or a good actor; if intelligent, then he can count to 10; Gary can only count to 3"?

A. Gary is intelligent B. Gary is a good actor C. Gary can count to 10 D. Gary is neither intelligent nor a good actor

15. What does the existential quantifier \(\exists x\) mean?

A. For every \(x\) B. There exists at least one \(x\) C. For no \(x\) D. There exists exactly two \(x\)

16. The statement "There is a rational number between every pair of distinct rational numbers" can be logically expressed using which quantifiers?

A. Existential quantifier only B. Universal quantifier only C. Both universal and existential quantifiers D. Neither universal nor existential quantifiers

17. Which rule of inference allows chaining of implications: from \(p \rightarrow q\) and \(q \rightarrow r\), infer \(p \rightarrow r\)?

A. Hypothetical syllogism B. Disjunctive syllogism C. Modus ponens D. Conjunction

18. What logical operator is used in the statement "it is either raining or snowing, but not both"?

A. Conjunction B. Disjunction C. Exclusive OR D. Implication

19. What is the law of excluded middle?

A. \(p \wedge \neg p\) is always false B. \(p \vee \neg p\) is always true C. \(p \rightarrow q\) is always true D. \(p \leftrightarrow q\) is always false

20. Which logical connective does the symbol \(\oplus\) represent?

A. Conjunction B. Exclusive OR C. Biconditional D. Negation

21. Which of the following represents the contrapositive of the implication \(p \rightarrow q\)?

A. \(\neg p \rightarrow \neg q\) B. \(\neg q \rightarrow \neg p\) C. \(q \rightarrow p\) D. \(p \vee \neg q\)

22. The argument form "If p then q; not q; therefore not p" is known as:

A. Modus ponens B. Modus tollens C. Disjunctive syllogism D. Hypothetical syllogism

23. Which proof technique involves assuming the negation of the conclusion and deriving a contradiction?

A. Direct proof B. Proof by contradiction C. Proof by induction D. Construction proof

24. What is a proposition in propositional logic?

A. A statement that can be either true or false B. A question that cannot be answered C. A command to perform an action D. A mathematical equation

25. Which rule of inference allows deriving p from p and q?

A. Addition B. Simplification C. Conjunction D. Modus tollens

26. In the example involving Gary being either intelligent or a good actor, which inference rule was used to conclude is Gary is a good actor?

A. Modus ponens and disjunctive syllogism B. Addition and simplification C. Universal instantiation only D. Hypothetical syllogism only

27. What does the biconditional operator \(p \leftrightarrow q\) mean?

A. \(p\) implies \(q\) B. \(p\) if and only if \(q\) C. \(p\) or \(q\) but not both D. \(p\) and \(q\) are false

28. How do you obtain \(P(c)\) from \(\forall x P(x)\)?

A. By universal generalization B. By existential generalization C. By universal instantiation D. By existential instantiation

29. In a disjunctive syllogism, if the statement "p or q" is true and p is false, what conclusion can be drawn?

A. Both p and q are true B. q is true C. p is true D. Nothing can be concluded

30. Which of these is a characteristic of a theorem?

A. It is an assumption made without proof B. It is a statement proven using axioms and previously established theorems C. It is a conjecture that has not yet been proven D. It is a rule of inference

31. What is the meaning of the logical equivalence \((p \wedge q) \equiv (q \wedge p)\)?

A. Conjunction is associative B. Conjunction is commutative C. Conjunction is distributive D. Conjunction is idempotent

32. Which of the following is a tautology?

A. \(p \vee \neg p\) B. \(p \wedge \neg p\) C. \(p \rightarrow \neg p\) D. \(\neg (p \vee q)\)

33. What is the statement format for modus tollens?

A. From \(p\) and \(p \rightarrow q\), infer \(q\) B. From \(\neg q\) and \(p \rightarrow q\), infer \(\neg p\) C. From \(p \vee q\) and \(\neg p\), infer \(q\) D. From \(p\) and \(q\), infer \(p \wedge q\)

34. If an argument has a false hypothesis, can the conclusion still be valid?

A. No, the conclusion must be false B. Yes, but the argument may not be sound C. Only if the conclusion is also false D. Only if modus tollens is used

35. What is a conjecture in mathematical logic?

A. A proven statement B. An accepted definition C. A hypothesis that is yet to be proven D. A false statement

36. What is the outcome of applying the simplification rule on \(p \wedge q\)?

A. \(p \vee q\) B. \(p\) C. \(q \rightarrow p\) D. Neither \(p\) nor \(q\)

37. Which rule of inference allows deriving \(q\) from \(p \vee q\) and \(\neg p\)?

A. Modus ponens B. Disjunctive syllogism C. Hypothetical syllogism D. Conjunction

38. Which of the following is NOT a commonly used rule of inference?

A. Modus ponens B. Disjunctive syllogism C. Pigeonhole principle D. Hypothetical syllogism

39. Which proof technique will be introduced in later chapters, as mentioned in the summary?

A. Direct proof only B. Disjunctive argument method C. Induction and pigeonhole principle D. Proof by universal generalization

40. What distinguishes an axiom from a theorem?

A. Axioms are assumed true without proof; theorems require proof B. Axioms are proven statements; theorems are assumptions C. Axioms use quantifiers; theorems do not D. There is no difference

41. In a truth table, what is the output of \(p \vee q\) when both \(p\) and \(q\) are false?

A. True B. False C. Unknown D. Depends on other variables

42. The implication \(p \rightarrow q\) is false only when:

A. \(p\) is true and \(q\) is false B. \(p\) is false and \(q\) is true C. Both \(p\) and \(q\) are false D. Both \(p\) and \(q\) are true

43. Modus ponens is based on which of the following logical forms?

A. If p then q; p; therefore q B. p or q; not p; therefore q C. If p then q; not q; therefore not p D. p and q; therefore p

44. When using indirect proof (proof by contradiction), what is usually the first step?

A. Assume the conclusion is true B. Assume the negation of the conclusion C. Use modus ponens to derive conclusion D. List all hypotheses as true

45. What type of proof directly establishes the conclusion assuming the premises?

A. Indirect proof B. Direct proof C. Proof by induction D. Contradiction proof

46. Which rule of inference states that from \(p \rightarrow q\) and \(p\), one can infer \(q\)?

A. Modus tollens B. Disjunctive syllogism C. Modus ponens D. Hypothetical syllogism

47. When is a propositional function considered a proposition?

A. When all variables are instantiated or quantified B. When it contains at least one variable C. When it has no logical operators D. When it is always false

48. Which of the following is an example of a valid argument?

A. Premises \(p\), \(p \rightarrow q\); conclusion \(q\) B. Premises \(p\), \(\neg p \rightarrow q\); conclusion \(\neg q\) C. Premises \(p \vee q\), \(q \rightarrow r\); conclusion \(p\) D. Premises \(\neg p\), \(p \rightarrow q\); conclusion \(p\)

49. What is the dual of the universal quantifier \(\forall\)?

A. \(\exists\) B. \(\forall\) C. \(\neg\) D. \(\emptyset\)

50. What does the universal quantifier \(\forall x P(x)\) express?

A. \(P(x)\) is true for some values of \(x\) B. \(P(x)\) is true for exactly one \(x\) C. \(P(x)\) is true for all \(x\) in the universe D. \(P(x)\) is false for all \(x\)