Truth Table Flashcards: Connectives & Logical Equivalence
84 truth-table flashcards on logical connectives, material conditionals, nested expressions, equivalence, and counterexamples, with short explanations.
O tej talii
Practice propositional logic with 84 English truth-table flashcards. The set covers NOT (¬), AND (∧), inclusive OR (∨), exclusive OR (⊕), the material conditional (→), and the biconditional (↔). T means true and F means false. It suits learners starting formal logic or discrete mathematics.
Cards ask you to recall a connective’s truth conditions, identify a connective from its full rule, evaluate an expression from a stated assignment, and produce short output columns with an explicit row order. A few original statement-to-symbol tasks and one symbol-to-statement task connect the notation to English. Later cards practice De Morgan’s laws, double negation, conditional rewriting, biconditional and XOR expansions, negated conditionals, and the distinction between a conditional, its converse, inverse, and contrapositive.
The sequence introduces notation and simple connectives before nested expressions, logical equivalence, counterexamples, and tautology/contradiction/contingency classification. Selected equivalences have separate forward and reverse cards, placed apart. Answers start with the result and give a short reason. For a counterexample task, any assignment that makes the two outputs differ is a valid answer.
This is a focused practice set, not a full logic course or an exam syllabus. It excludes quantified statements, proof systems, circuits, and programming-language evaluation rules. It does not ask for a connective from a lone truth value, which would be ambiguous, or repeat every possible symbolic and English permutation. Material implication is used as a truth-functional rule; the cards do not treat it as a theory of causation or every everyday use of “if”.
For a short introduction, read truth values, conjunction and disjunction. For study habits, see how to use flashcards for math.
The questions, answers, examples, order, metadata, and cover were independently created with AI assistance. Logic facts were checked against the open textbook forall x: Calgary, especially its chapters on connective truth tables, complete truth tables, and semantic concepts, and by direct truth-value calculation. No textbook exercises, teaching prose, diagrams, examination questions, or competitor cards were copied. The original text and generated cover are released under CC0 1.0 to the extent applicable rights exist; reference-source text retains its own license. This is an independent resource, unaffiliated with the Open Logic Project or any school or examination provider.
Karty w tej talii
Karta 1
Pytanie
In classical propositional logic, what is a proposition?
Odpowiedź
A statement with a truth value: true or false. A question or command is not a proposition in this setting.
Karta 2
Pytanie
What does negation (¬p) do to the truth value of p?
Odpowiedź
It reverses it: true becomes false, and false becomes true.
Karta 3
Pytanie
When is the conjunction (p ∧ q) true?
Odpowiedź
Only when p and q are both true.
Karta 4
Pytanie
When is the inclusive disjunction (p ∨ q) true?
Odpowiedź
When at least one of p and q is true, including when both are true.
Karta 5
Pytanie
What does one valuation assign in a propositional truth table?
Odpowiedź
One truth value to each proposition letter. The assignment stays fixed throughout that row.
Karta 6
Pytanie
When is the material conditional (p → q) false?
Odpowiedź
Only when p is true and q is false. Here p is the antecedent and q is the consequent.
Karta 7
Pytanie
When is the biconditional (p ↔ q) true?
Odpowiedź
When p and q have the same truth value: both true or both false.
Karta 8
Pytanie
When is exclusive OR (p ⊕ q) true?
Odpowiedź
When exactly one of p and q is true. It is false when their truth values match.
Karta 9
Pytanie
What is the main connective in ((¬p) ∧ q)?
Odpowiedź
∧ (AND). It combines the whole left part, (¬p), with q.
Karta 10
Pytanie
How many rows does a complete truth table with three distinct proposition letters need?
Odpowiedź
8 rows: each of the three letters has two choices, so 2³ = 8.
Karta 11
Pytanie
If p = F, what is (¬p)?
Odpowiedź
T. Negation reverses F to T.
Karta 12
Pytanie
If p = T and q = F, what is (p ∧ q)?
Odpowiedź
F. AND needs both inputs to be true.
Karta 13
Pytanie
If p = T and q = T, what is inclusive OR (p ∨ q)?
Odpowiedź
T. Inclusive OR allows both inputs to be true.
Karta 14
Pytanie
If p = T and q = T, what is the material conditional (p → q)?
Odpowiedź
T. A true antecedent with a true consequent does not make the conditional false.
Karta 15
Pytanie
If p = F and q = F, what is (p ↔ q)?
Odpowiedź
T. The two truth values match, even though neither is true.
Karta 16
Pytanie
If p = T and q = T, what is exclusive OR (p ⊕ q)?
Odpowiedź
F. XOR requires exactly one true input.
Karta 17
Pytanie
Does a true material conditional (p → q) establish that p causes q?
Odpowiedź
No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.
Karta 18
Pytanie
What assignments are listed by the two-letter row order TT, TF, FT, FF?
Odpowiedź
(p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.
Karta 19
Pytanie
In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?
Odpowiedź
The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.
Karta 20
Pytanie
If p = F and q = T, what is the material conditional (p → q)?
Odpowiedź
T. A material conditional with a false antecedent is true.
Karta 21
Pytanie
If p = T and q = F, what is (p ↔ q)?
Odpowiedź
F. The two truth values differ.
Karta 22
Pytanie
If p = T and q = F, what is exclusive OR (p ⊕ q)?
Odpowiedź
T. Exactly one input is true.
Karta 23
Pytanie
Which standard connective is true exactly when both inputs are true?
Odpowiedź
Conjunction (AND), written ∧.
Karta 24
Pytanie
Which standard connective is false exactly when both inputs are false?
Odpowiedź
Inclusive disjunction (OR), written ∨. The both-true case is true.
Karta 25
Pytanie
Which standard connective takes one input and reverses its truth value?
Odpowiedź
Negation (NOT), written ¬.
Karta 26
Pytanie
If p = F and q = F, what is the material conditional (p → q)?
Odpowiedź
T. Its only false case requires a true antecedent and a false consequent.
Karta 27
Pytanie
Which standard connective is true exactly when its two inputs have matching truth values?
Odpowiedź
The biconditional (if and only if), written ↔.
Karta 28
Pytanie
Which standard connective is true exactly when its two inputs have different truth values?
Odpowiedź
Exclusive OR (XOR), written ⊕.
Karta 29
Pytanie
What is the main connective in ((p ∨ q) → (¬r))?
Odpowiedź
→ (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.
Karta 30
Pytanie
Which standard connective is false exactly when its first input is true and its second input is false?
Odpowiedź
The material conditional, written →. Input order matters.
Karta 31
Pytanie
Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.
Odpowiedź
T, F, F, F. Only the both-true row satisfies AND.
Karta 32
Pytanie
If p = T and q = F, what is ¬(p ∧ q)?
Odpowiedź
T. First (p ∧ q) is F; negating it gives T.
Karta 33
Pytanie
Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.
Odpowiedź
(a ∧ b). Both statements are asserted.
Karta 34
Pytanie
Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.
Odpowiedź
T, F, T, T. Only the true-antecedent, false-consequent row fails.
Karta 35
Pytanie
What makes a formula a tautology in classical propositional logic?
Odpowiedź
It is true on every possible valuation, not just the row currently being checked.
Karta 36
Pytanie
Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.
Odpowiedź
T, T, T, F. Only the both-false row fails inclusive OR.
Karta 37
Pytanie
When are two propositional formulas logically equivalent?
Odpowiedź
When their final truth values match on every valuation of their combined proposition letters.
Karta 38
Pytanie
Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.
Odpowiedź
T, F, F, T. The first and last rows have matching truth values.
Karta 39
Pytanie
If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?
Odpowiedź
F. Both p and (¬q) are F.
Karta 40
Pytanie
If p = T and q = F, what is the material conditional (p → q)?
Odpowiedź
F. This is its only false input combination.
Karta 41
Pytanie
Give the output column for (¬p), with p rows T, F.
Odpowiedź
F, T. Negation reverses each row.
Karta 42
Pytanie
Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.
Odpowiedź
(s → a). The scan statement is the antecedent; the alert statement is the consequent.
Karta 43
Pytanie
Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.
Odpowiedź
F, T, T, F. Exactly one input is true in the middle two rows.
Karta 44
Pytanie
What makes a formula a contradiction in classical propositional logic?
Odpowiedź
It is false on every possible valuation.
Karta 45
Pytanie
If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?
Odpowiedź
T. The conjunction is F, but r is T, so the disjunction is T.
Karta 46
Pytanie
Simplify ¬(¬p) without changing its truth value.
Odpowiedź
p. Two negations restore the original truth value.
Karta 47
Pytanie
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”.
Odpowiedź
(d ↔ c). Both directions of the conditional are required.
Karta 48
Pytanie
What makes a propositional formula contingent?
Odpowiedź
It is true on at least one valuation and false on at least one other valuation.
Karta 49
Pytanie
If p = T and q = T, what is (p → (¬q)), using material implication?
Odpowiedź
F. Its antecedent is T and its consequent (¬q) is F.
Karta 50
Pytanie
Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.
Odpowiedź
((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.
Karta 51
Pytanie
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”.
Odpowiedź
(n ∨ e). This is inclusive OR.
Karta 52
Pytanie
Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.
Odpowiedź
Tautology. Whether p is T or F, one disjunct is T.
Karta 53
Pytanie
If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?
Odpowiedź
T. The disjunction is F, so its negation is T.
Karta 54
Pytanie
Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.
Odpowiedź
((¬p) ∧ (¬q)). Both disjuncts must be false.
Karta 55
Pytanie
Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.
Odpowiedź
If the heater is on, then the window is closed. The formula itself makes no causal claim.
Karta 56
Pytanie
Classify (p ∧ (¬p)): tautology, contradiction or contingent?
Odpowiedź
Contradiction. The two conjuncts cannot both be true on any valuation.
Karta 57
Pytanie
Rewrite the material conditional (p → q) using only NOT and inclusive OR.
Odpowiedź
((¬p) ∨ q). It is false exactly when p is T and q is F.
Karta 58
Pytanie
If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?
Odpowiedź
T. The XOR is F and r is F, so the biconditional compares matching values.
Karta 59
Pytanie
Write an expression with exactly two NOT operators that is equivalent to p.
Odpowiedź
¬(¬p). Negating twice leaves every truth value unchanged.
Karta 60
Pytanie
Classify (p ∧ q): tautology, contradiction or contingent?
Odpowiedź
Contingent. It is T at p = T, q = T and F at p = F, q = T.
Karta 61
Pytanie
What is the contrapositive of the material conditional (p → q)?
Odpowiedź
((¬q) → (¬p)). Swap the two sides and negate both.
Karta 62
Pytanie
Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.
Odpowiedź
(p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.
Karta 63
Pytanie
Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.
Odpowiedź
¬(p ∧ q). This is De Morgan’s law in the reverse direction.
Karta 64
Pytanie
Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.
Odpowiedź
T, T, F, T. The only false row has p = F and q = T.
Karta 65
Pytanie
What is the converse of (p → q)?
Odpowiedź
(q → p). Swap the antecedent and consequent without negating either.
Karta 66
Pytanie
Rewrite (p ↔ q) as an AND of two material conditionals.
Odpowiedź
((p → q) ∧ (q → p)). Both directions must hold.
Karta 67
Pytanie
Rewrite ((¬p) ∧ (¬q)) as one negation of an inclusive disjunction.
Odpowiedź
¬(p ∨ q). This is De Morgan’s law in the reverse direction.
Karta 68
Pytanie
Give one valuation showing that (p ∨ q) and (p ∧ q) are not equivalent. Use inclusive OR.
Odpowiedź
p = T, q = F gives T for the OR and F for the AND. The swapped assignment also works.
Karta 69
Pytanie
What is the inverse of (p → q)?
Odpowiedź
((¬p) → (¬q)). Negate both sides without swapping them.
Karta 70
Pytanie
Rewrite ((¬p) ∨ q) as a single material conditional, using inclusive OR.
Odpowiedź
(p → q). Both expressions fail exactly when p is T and q is F.
Karta 71
Pytanie
Rewrite exclusive OR (p ⊕ q) using AND, inclusive OR and NOT.
Odpowiedź
((p ∨ q) ∧ ¬(p ∧ q)). Require at least one true input and rule out both being true.
Karta 72
Pytanie
A formula is true for p = T and q = F. Is that enough to call it a tautology?
Odpowiedź
No. A tautology must be true on every valuation. One true row establishes only that it can be true.
Karta 73
Pytanie
Is a material conditional (p → q) logically equivalent to its contrapositive ((¬q) → (¬p))?
Odpowiedź
Yes. Both are false exactly when p is T and q is F.
Karta 74
Pytanie
Give one valuation showing that (p → q) and its converse (q → p) are not equivalent. Use material implication.
Odpowiedź
p = T, q = F. Then (p → q) is F and (q → p) is T.
Karta 75
Pytanie
Rewrite ((p → q) ∧ (q → p)) using one connective, with both arrows meaning material implication.
Odpowiedź
(p ↔ q). This is the biconditional.
Karta 76
Pytanie
What does one valuation with different outputs prove about two formulas?
Odpowiedź
They are not logically equivalent. Equivalence requires agreement on every valuation.
Karta 77
Pytanie
Give one valuation showing that (p → q) and its inverse ((¬p) → (¬q)) are not equivalent. Use material implication.
Odpowiedź
p = T, q = F. The original is F, while the inverse has a false antecedent and is T.
Karta 78
Pytanie
Give one valuation showing that ¬(p ∧ q) and ((¬p) ∧ (¬q)) are not equivalent.
Odpowiedź
p = T, q = F. The negated conjunction is T; the conjunction of negations is F.
Karta 79
Pytanie
Which connective does ((p ∨ q) ∧ ¬(p ∧ q)) express? Here ∨ is inclusive OR.
Odpowiedź
Exclusive OR: (p ⊕ q). Exactly one input must be true.
Karta 80
Pytanie
If (p ↔ q) is true on one row, does that show that the formulas p and q are logically equivalent?
Odpowiedź
No. They match on that row only. Logical equivalence requires the biconditional to be true on every valuation.
Karta 81
Pytanie
Are the converse (q → p) and inverse ((¬p) → (¬q)) of (p → q) equivalent to each other? Use material implication.
Odpowiedź
Yes. They are contrapositives of each other, and both are false exactly when q is T and p is F.
Karta 82
Pytanie
At p = T, q = F and r = F, compare ((p ∨ q) ∧ r) with (p ∨ (q ∧ r)). Use inclusive OR.
Odpowiedź
The first is F; the second is T. Parentheses change which operations combine first.
Karta 83
Pytanie
Rewrite (p ∧ (¬q)) as the negation of one material conditional.
Odpowiedź
¬(p → q). The conjunction describes exactly the conditional’s false case.
Karta 84
Pytanie
Among AND (∧), inclusive OR (∨), XOR (⊕), material conditional (→) and biconditional (↔), which has outputs T, F, F, T for (p, q) rows TT, TF, FT, FF?
Odpowiedź
Biconditional (↔). It is true on the two rows where the inputs match.
84 karty
Truth Table Flashcards: Connectives & Logical Equivalence
Otworzy się Nibomo, żeby od razu zacząć naukę.