Truth Table Flashcards: Connectives & Logical Equivalence

Cards in this deck

1. Card 1

   Question

   In classical propositional logic, what is a proposition?

   Answer

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

2. Card 2

   Question

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

   Answer

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

3. Card 3

   Question

   When is the conjunction (p ∧ q) true?

   Answer

   Only when p and q are both true.

4. Card 4

   Question

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

   Answer

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

5. Card 5

   Question

   What does one valuation assign in a propositional truth table?

   Answer

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

6. Card 6

   Question

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

   Answer

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

7. Card 7

   Question

   When is the biconditional (p ↔ q) true?

   Answer

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

8. Card 8

   Question

   When is exclusive OR (p ⊕ q) true?

   Answer

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

9. Card 9

   Question

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

   Answer

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

10. Card 10

    Question

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

    Answer

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

11. Card 11

    Question

    If p = F, what is (¬p)?

    Answer

    T. Negation reverses F to T.

12. Card 12

    Question

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

    Answer

    F. AND needs both inputs to be true.

13. Card 13

    Question

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

    Answer

    T. Inclusive OR allows both inputs to be true.

14. Card 14

    Question

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

    Answer

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

15. Card 15

    Question

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

    Answer

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

16. Card 16

    Question

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

    Answer

    F. XOR requires exactly one true input.

17. Card 17

    Question

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

    Answer

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

18. Card 18

    Question

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

    Answer

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

19. Card 19

    Question

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

    Answer

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

20. Card 20

    Question

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

    Answer

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

21. Card 21

    Question

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

    Answer

    F. The two truth values differ.

22. Card 22

    Question

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

    Answer

    T. Exactly one input is true.

23. Card 23

    Question

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

    Answer

    Conjunction (AND), written ∧.

24. Card 24

    Question

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

    Answer

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

25. Card 25

    Question

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

    Answer

    Negation (NOT), written ¬.

26. Card 26

    Question

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

    Answer

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

27. Card 27

    Question

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

    Answer

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

28. Card 28

    Question

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

    Answer

    Exclusive OR (XOR), written ⊕.

29. Card 29

    Question

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

    Answer

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

30. Card 30

    Question

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

    Answer

    The material conditional, written →. Input order matters.

31. Card 31

    Question

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

    Answer

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

32. Card 32

    Question

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

    Answer

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

33. Card 33

    Question

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

    Answer

    (a ∧ b). Both statements are asserted.

34. Card 34

    Question

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

    Answer

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

35. Card 35

    Question

    What makes a formula a tautology in classical propositional logic?

    Answer

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

36. Card 36

    Question

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

    Answer

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

37. Card 37

    Question

    When are two propositional formulas logically equivalent?

    Answer

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

38. Card 38

    Question

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

    Answer

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

39. Card 39

    Question

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

    Answer

    F. Both p and (¬q) are F.

40. Card 40

    Question

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

    Answer

    F. This is its only false input combination.

41. Card 41

    Question

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

    Answer

    F, T. Negation reverses each row.

42. Card 42

    Question

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

    Answer

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

43. Card 43

    Question

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

    Answer

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

44. Card 44

    Question

    What makes a formula a contradiction in classical propositional logic?

    Answer

    It is false on every possible valuation.

45. Card 45

    Question

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

    Answer

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

46. Card 46

    Question

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

    Answer

    p. Two negations restore the original truth value.

47. Card 47

    Question

    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”.

    Answer

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

48. Card 48

    Question

    What makes a propositional formula contingent?

    Answer

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

49. Card 49

    Question

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

    Answer

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

50. Card 50

    Question

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

    Answer

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

51. Card 51

    Question

    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”.

    Answer

    (n ∨ e). This is inclusive OR.

52. Card 52

    Question

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

    Answer

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

53. Card 53

    Question

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

    Answer

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

54. Card 54

    Question

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

    Answer

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

55. Card 55

    Question

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

    Answer

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

56. Card 56

    Question

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

    Answer

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

57. Card 57

    Question

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

    Answer

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

58. Card 58

    Question

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

    Answer

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

59. Card 59

    Question

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

    Answer

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

60. Card 60

    Question

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

    Answer

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

61. Card 61

    Question

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

    Answer

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

62. Card 62

    Question

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

    Answer

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

63. Card 63

    Question

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

    Answer

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

64. Card 64

    Question

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

    Answer

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

65. Card 65

    Question

    What is the converse of (p → q)?

    Answer

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

66. Card 66

    Question

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

    Answer

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

67. Card 67

    Question

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

    Answer

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

68. Card 68

    Question

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

    Answer

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

69. Card 69

    Question

    What is the inverse of (p → q)?

    Answer

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

70. Card 70

    Question

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

    Answer

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

71. Card 71

    Question

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

    Answer

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

72. Card 72

    Question

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

    Answer

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

73. Card 73

    Question

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

    Answer

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

74. Card 74

    Question

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

    Answer

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

75. Card 75

    Question

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

    Answer

    (p ↔ q). This is the biconditional.

76. Card 76

    Question

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

    Answer

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

77. Card 77

    Question

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

    Answer

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

78. Card 78

    Question

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

    Answer

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

79. Card 79

    Question

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

    Answer

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

80. Card 80

    Question

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

    Answer

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

81. Card 81

    Question

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

    Answer

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

82. Card 82

    Question

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

    Answer

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

83. Card 83

    Question

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

    Answer

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

84. Card 84

    Question

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

    Answer

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

84 cards

Truth Table Flashcards: Connectives & Logical Equivalence

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