Truth Table Flashcards: Connectives & Logical Equivalence

このデッキのカード

1. カード 1

   質問

   In classical propositional logic, what is a proposition?

   回答

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

2. カード 2

   質問

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

   回答

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

3. カード 3

   質問

   When is the conjunction (p ∧ q) true?

   回答

   Only when p and q are both true.

4. カード 4

   質問

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

   回答

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

5. カード 5

   質問

   What does one valuation assign in a propositional truth table?

   回答

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

6. カード 6

   質問

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

   回答

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

7. カード 7

   質問

   When is the biconditional (p ↔ q) true?

   回答

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

8. カード 8

   質問

   When is exclusive OR (p ⊕ q) true?

   回答

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

9. カード 9

   質問

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

   回答

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

10. カード 10

    質問

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

    回答

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

11. カード 11

    質問

    If p = F, what is (¬p)?

    回答

    T. Negation reverses F to T.

12. カード 12

    質問

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

    回答

    F. AND needs both inputs to be true.

13. カード 13

    質問

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

    回答

    T. Inclusive OR allows both inputs to be true.

14. カード 14

    質問

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

    回答

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

15. カード 15

    質問

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

    回答

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

16. カード 16

    質問

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

    回答

    F. XOR requires exactly one true input.

17. カード 17

    質問

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

    回答

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

18. カード 18

    質問

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

    回答

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

19. カード 19

    質問

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

    回答

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

20. カード 20

    質問

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

    回答

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

21. カード 21

    質問

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

    回答

    F. The two truth values differ.

22. カード 22

    質問

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

    回答

    T. Exactly one input is true.

23. カード 23

    質問

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

    回答

    Conjunction (AND), written ∧.

24. カード 24

    質問

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

    回答

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

25. カード 25

    質問

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

    回答

    Negation (NOT), written ¬.

26. カード 26

    質問

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

    回答

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

27. カード 27

    質問

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

    回答

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

28. カード 28

    質問

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

    回答

    Exclusive OR (XOR), written ⊕.

29. カード 29

    質問

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

    回答

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

30. カード 30

    質問

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

    回答

    The material conditional, written →. Input order matters.

31. カード 31

    質問

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

    回答

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

32. カード 32

    質問

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

    回答

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

33. カード 33

    質問

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

    回答

    (a ∧ b). Both statements are asserted.

34. カード 34

    質問

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

    回答

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

35. カード 35

    質問

    What makes a formula a tautology in classical propositional logic?

    回答

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

36. カード 36

    質問

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

    回答

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

37. カード 37

    質問

    When are two propositional formulas logically equivalent?

    回答

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

38. カード 38

    質問

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

    回答

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

39. カード 39

    質問

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

    回答

    F. Both p and (¬q) are F.

40. カード 40

    質問

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

    回答

    F. This is its only false input combination.

41. カード 41

    質問

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

    回答

    F, T. Negation reverses each row.

42. カード 42

    質問

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

    回答

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

43. カード 43

    質問

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

    回答

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

44. カード 44

    質問

    What makes a formula a contradiction in classical propositional logic?

    回答

    It is false on every possible valuation.

45. カード 45

    質問

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

    回答

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

46. カード 46

    質問

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

    回答

    p. Two negations restore the original truth value.

47. カード 47

    質問

    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”.

    回答

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

48. カード 48

    質問

    What makes a propositional formula contingent?

    回答

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

49. カード 49

    質問

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

    回答

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

50. カード 50

    質問

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

    回答

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

51. カード 51

    質問

    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”.

    回答

    (n ∨ e). This is inclusive OR.

52. カード 52

    質問

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

    回答

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

53. カード 53

    質問

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

    回答

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

54. カード 54

    質問

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

    回答

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

55. カード 55

    質問

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

    回答

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

56. カード 56

    質問

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

    回答

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

57. カード 57

    質問

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

    回答

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

58. カード 58

    質問

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

    回答

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

59. カード 59

    質問

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

    回答

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

60. カード 60

    質問

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

    回答

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

61. カード 61

    質問

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

    回答

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

62. カード 62

    質問

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

    回答

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

63. カード 63

    質問

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

    回答

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

64. カード 64

    質問

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

    回答

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

65. カード 65

    質問

    What is the converse of (p → q)?

    回答

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

66. カード 66

    質問

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

    回答

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

67. カード 67

    質問

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

    回答

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

68. カード 68

    質問

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

    回答

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

69. カード 69

    質問

    What is the inverse of (p → q)?

    回答

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

70. カード 70

    質問

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

    回答

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

71. カード 71

    質問

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

    回答

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

72. カード 72

    質問

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

    回答

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

73. カード 73

    質問

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

    回答

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

74. カード 74

    質問

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

    回答

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

75. カード 75

    質問

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

    回答

    (p ↔ q). This is the biconditional.

76. カード 76

    質問

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

    回答

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

77. カード 77

    質問

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

    回答

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

78. カード 78

    質問

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

    回答

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

79. カード 79

    質問

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

    回答

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

80. カード 80

    質問

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

    回答

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

81. カード 81

    質問

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

    回答

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

82. カード 82

    質問

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

    回答

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

83. カード 83

    質問

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

    回答

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

84. カード 84

    質問

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

    回答

    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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