Truth Table Flashcards: Connectives & Logical Equivalence

这套卡组中的卡片

1. 卡片 1

   问题

   In classical propositional logic, what is a proposition?

   答案

   A statement with a truth value: true or false. A question or command is not a proposition in this setting.

2. 卡片 2

   问题

   What does negation (¬p) do to the truth value of p?

   答案

   It reverses it: true becomes false, and false becomes true.

3. 卡片 3

   问题

   When is the conjunction (p ∧ q) true?

   答案

   Only when p and q are both true.

4. 卡片 4

   问题

   When is the inclusive disjunction (p ∨ q) true?

   答案

   When at least one of p and q is true, including when both are true.

5. 卡片 5

   问题

   What does one valuation assign in a propositional truth table?

   答案

   One truth value to each proposition letter. The assignment stays fixed throughout that row.

6. 卡片 6

   问题

   When is the material conditional (p → q) false?

   答案

   Only when p is true and q is false. Here p is the antecedent and q is the consequent.

7. 卡片 7

   问题

   When is the biconditional (p ↔ q) true?

   答案

   When p and q have the same truth value: both true or both false.

8. 卡片 8

   问题

   When is exclusive OR (p ⊕ q) true?

   答案

   When exactly one of p and q is true. It is false when their truth values match.

9. 卡片 9

   问题

   What is the main connective in ((¬p) ∧ q)?

   答案

   ∧ (AND). It combines the whole left part, (¬p), with q.

10. 卡片 10

    问题

    How many rows does a complete truth table with three distinct proposition letters need?

    答案

    8 rows: each of the three letters has two choices, so 2³ = 8.

11. 卡片 11

    问题

    If p = F, what is (¬p)?

    答案

    T. Negation reverses F to T.

12. 卡片 12

    问题

    If p = T and q = F, what is (p ∧ q)?

    答案

    F. AND needs both inputs to be true.

13. 卡片 13

    问题

    If p = T and q = T, what is inclusive OR (p ∨ q)?

    答案

    T. Inclusive OR allows both inputs to be true.

14. 卡片 14

    问题

    If p = T and q = T, what is the material conditional (p → q)?

    答案

    T. A true antecedent with a true consequent does not make the conditional false.

15. 卡片 15

    问题

    If p = F and q = F, what is (p ↔ q)?

    答案

    T. The two truth values match, even though neither is true.

16. 卡片 16

    问题

    If p = T and q = T, what is exclusive OR (p ⊕ q)?

    答案

    F. XOR requires exactly one true input.

17. 卡片 17

    问题

    Does a true material conditional (p → q) establish that p causes q?

    答案

    No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.

18. 卡片 18

    问题

    What assignments are listed by the two-letter row order TT, TF, FT, FF?

    答案

    (p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.

19. 卡片 19

    问题

    In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?

    答案

    The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.

20. 卡片 20

    问题

    If p = F and q = T, what is the material conditional (p → q)?

    答案

    T. A material conditional with a false antecedent is true.

21. 卡片 21

    问题

    If p = T and q = F, what is (p ↔ q)?

    答案

    F. The two truth values differ.

22. 卡片 22

    问题

    If p = T and q = F, what is exclusive OR (p ⊕ q)?

    答案

    T. Exactly one input is true.

23. 卡片 23

    问题

    Which standard connective is true exactly when both inputs are true?

    答案

    Conjunction (AND), written ∧.

24. 卡片 24

    问题

    Which standard connective is false exactly when both inputs are false?

    答案

    Inclusive disjunction (OR), written ∨. The both-true case is true.

25. 卡片 25

    问题

    Which standard connective takes one input and reverses its truth value?

    答案

    Negation (NOT), written ¬.

26. 卡片 26

    问题

    If p = F and q = F, what is the material conditional (p → q)?

    答案

    T. Its only false case requires a true antecedent and a false consequent.

27. 卡片 27

    问题

    Which standard connective is true exactly when its two inputs have matching truth values?

    答案

    The biconditional (if and only if), written ↔.

28. 卡片 28

    问题

    Which standard connective is true exactly when its two inputs have different truth values?

    答案

    Exclusive OR (XOR), written ⊕.

29. 卡片 29

    问题

    What is the main connective in ((p ∨ q) → (¬r))?

    答案

    → (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.

30. 卡片 30

    问题

    Which standard connective is false exactly when its first input is true and its second input is false?

    答案

    The material conditional, written →. Input order matters.

31. 卡片 31

    问题

    Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.

    答案

    T, F, F, F. Only the both-true row satisfies AND.

32. 卡片 32

    问题

    If p = T and q = F, what is ¬(p ∧ q)?

    答案

    T. First (p ∧ q) is F; negating it gives T.

33. 卡片 33

    问题

    Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.

    答案

    (a ∧ b). Both statements are asserted.

34. 卡片 34

    问题

    Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.

    答案

    T, F, T, T. Only the true-antecedent, false-consequent row fails.

35. 卡片 35

    问题

    What makes a formula a tautology in classical propositional logic?

    答案

    It is true on every possible valuation, not just the row currently being checked.

36. 卡片 36

    问题

    Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.

    答案

    T, T, T, F. Only the both-false row fails inclusive OR.

37. 卡片 37

    问题

    When are two propositional formulas logically equivalent?

    答案

    When their final truth values match on every valuation of their combined proposition letters.

38. 卡片 38

    问题

    Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.

    答案

    T, F, F, T. The first and last rows have matching truth values.

39. 卡片 39

    问题

    If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?

    答案

    F. Both p and (¬q) are F.

40. 卡片 40

    问题

    If p = T and q = F, what is the material conditional (p → q)?

    答案

    F. This is its only false input combination.

41. 卡片 41

    问题

    Give the output column for (¬p), with p rows T, F.

    答案

    F, T. Negation reverses each row.

42. 卡片 42

    问题

    Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.

    答案

    (s → a). The scan statement is the antecedent; the alert statement is the consequent.

43. 卡片 43

    问题

    Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.

    答案

    F, T, T, F. Exactly one input is true in the middle two rows.

44. 卡片 44

    问题

    What makes a formula a contradiction in classical propositional logic?

    答案

    It is false on every possible valuation.

45. 卡片 45

    问题

    If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?

    答案

    T. The conjunction is F, but r is T, so the disjunction is T.

46. 卡片 46

    问题

    Simplify ¬(¬p) without changing its truth value.

    答案

    p. Two negations restore the original truth value.

47. 卡片 47

    问题

    Let d mean “the door is unlocked” and c mean “the code is accepted”. Symbolize “the door is unlocked if and only if the code is accepted”.

    答案

    (d ↔ c). Both directions of the conditional are required.

48. 卡片 48

    问题

    What makes a propositional formula contingent?

    答案

    It is true on at least one valuation and false on at least one other valuation.

49. 卡片 49

    问题

    If p = T and q = T, what is (p → (¬q)), using material implication?

    答案

    F. Its antecedent is T and its consequent (¬q) is F.

50. 卡片 50

    问题

    Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.

    答案

    ((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.

51. 卡片 51

    问题

    Let n mean “the north path is open” and e mean “the east path is open”. Symbolize “at least one of these paths is open, possibly both”.

    答案

    (n ∨ e). This is inclusive OR.

52. 卡片 52

    问题

    Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.

    答案

    Tautology. Whether p is T or F, one disjunct is T.

53. 卡片 53

    问题

    If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?

    答案

    T. The disjunction is F, so its negation is T.

54. 卡片 54

    问题

    Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.

    答案

    ((¬p) ∧ (¬q)). Both disjuncts must be false.

55. 卡片 55

    问题

    Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.

    答案

    If the heater is on, then the window is closed. The formula itself makes no causal claim.

56. 卡片 56

    问题

    Classify (p ∧ (¬p)): tautology, contradiction or contingent?

    答案

    Contradiction. The two conjuncts cannot both be true on any valuation.

57. 卡片 57

    问题

    Rewrite the material conditional (p → q) using only NOT and inclusive OR.

    答案

    ((¬p) ∨ q). It is false exactly when p is T and q is F.

58. 卡片 58

    问题

    If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?

    答案

    T. The XOR is F and r is F, so the biconditional compares matching values.

59. 卡片 59

    问题

    Write an expression with exactly two NOT operators that is equivalent to p.

    答案

    ¬(¬p). Negating twice leaves every truth value unchanged.

60. 卡片 60

    问题

    Classify (p ∧ q): tautology, contradiction or contingent?

    答案

    Contingent. It is T at p = T, q = T and F at p = F, q = T.

61. 卡片 61

    问题

    What is the contrapositive of the material conditional (p → q)?

    答案

    ((¬q) → (¬p)). Swap the two sides and negate both.

62. 卡片 62

    问题

    Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.

    答案

    (p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.

63. 卡片 63

    问题

    Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.

    答案

    ¬(p ∧ q). This is De Morgan’s law in the reverse direction.

64. 卡片 64

    问题

    Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.

    答案

    T, T, F, T. The only false row has p = F and q = T.

65. 卡片 65

    问题

    What is the converse of (p → q)?

    答案

    (q → p). Swap the antecedent and consequent without negating either.

66. 卡片 66

    问题

    Rewrite (p ↔ q) as an AND of two material conditionals.

    答案

    ((p → q) ∧ (q → p)). Both directions must hold.

67. 卡片 67

    问题

    Rewrite ((¬p) ∧ (¬q)) as one negation of an inclusive disjunction.

    答案

    ¬(p ∨ q). This is De Morgan’s law in the reverse direction.

68. 卡片 68

    问题

    Give one valuation showing that (p ∨ q) and (p ∧ q) are not equivalent. Use inclusive OR.

    答案

    p = T, q = F gives T for the OR and F for the AND. The swapped assignment also works.

69. 卡片 69

    问题

    What is the inverse of (p → q)?

    答案

    ((¬p) → (¬q)). Negate both sides without swapping them.

70. 卡片 70

    问题

    Rewrite ((¬p) ∨ q) as a single material conditional, using inclusive OR.

    答案

    (p → q). Both expressions fail exactly when p is T and q is F.

71. 卡片 71

    问题

    Rewrite exclusive OR (p ⊕ q) using AND, inclusive OR and NOT.

    答案

    ((p ∨ q) ∧ ¬(p ∧ q)). Require at least one true input and rule out both being true.

72. 卡片 72

    问题

    A formula is true for p = T and q = F. Is that enough to call it a tautology?

    答案

    No. A tautology must be true on every valuation. One true row establishes only that it can be true.

73. 卡片 73

    问题

    Is a material conditional (p → q) logically equivalent to its contrapositive ((¬q) → (¬p))?

    答案

    Yes. Both are false exactly when p is T and q is F.

74. 卡片 74

    问题

    Give one valuation showing that (p → q) and its converse (q → p) are not equivalent. Use material implication.

    答案

    p = T, q = F. Then (p → q) is F and (q → p) is T.

75. 卡片 75

    问题

    Rewrite ((p → q) ∧ (q → p)) using one connective, with both arrows meaning material implication.

    答案

    (p ↔ q). This is the biconditional.

76. 卡片 76

    问题

    What does one valuation with different outputs prove about two formulas?

    答案

    They are not logically equivalent. Equivalence requires agreement on every valuation.

77. 卡片 77

    问题

    Give one valuation showing that (p → q) and its inverse ((¬p) → (¬q)) are not equivalent. Use material implication.

    答案

    p = T, q = F. The original is F, while the inverse has a false antecedent and is T.

78. 卡片 78

    问题

    Give one valuation showing that ¬(p ∧ q) and ((¬p) ∧ (¬q)) are not equivalent.

    答案

    p = T, q = F. The negated conjunction is T; the conjunction of negations is F.

79. 卡片 79

    问题

    Which connective does ((p ∨ q) ∧ ¬(p ∧ q)) express? Here ∨ is inclusive OR.

    答案

    Exclusive OR: (p ⊕ q). Exactly one input must be true.

80. 卡片 80

    问题

    If (p ↔ q) is true on one row, does that show that the formulas p and q are logically equivalent?

    答案

    No. They match on that row only. Logical equivalence requires the biconditional to be true on every valuation.

81. 卡片 81

    问题

    Are the converse (q → p) and inverse ((¬p) → (¬q)) of (p → q) equivalent to each other? Use material implication.

    答案

    Yes. They are contrapositives of each other, and both are false exactly when q is T and p is F.

82. 卡片 82

    问题

    At p = T, q = F and r = F, compare ((p ∨ q) ∧ r) with (p ∨ (q ∧ r)). Use inclusive OR.

    答案

    The first is F; the second is T. Parentheses change which operations combine first.

83. 卡片 83

    问题

    Rewrite (p ∧ (¬q)) as the negation of one material conditional.

    答案

    ¬(p → q). The conjunction describes exactly the conditional’s false case.

84. 卡片 84

    问题

    Among AND (∧), inclusive OR (∨), XOR (⊕), material conditional (→) and biconditional (↔), which has outputs T, F, F, T for (p, q) rows TT, TF, FT, FF?

    答案

    Biconditional (↔). It is true on the two rows where the inputs match.

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Truth Table Flashcards: Connectives & Logical Equivalence

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