Truth Table Flashcards: Connectives & Logical Equivalence
84 truth-table flashcards on logical connectives, material conditionals, nested expressions, equivalence, and counterexamples, with short explanations.
Про цю колоду
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.
Картки в цій колоді
Картка 1
Питання
In classical propositional logic, what is a proposition?
Відповідь
A statement with a truth value: true or false. A question or command is not a proposition in this setting.
Картка 2
Питання
What does negation (¬p) do to the truth value of p?
Відповідь
It reverses it: true becomes false, and false becomes true.
Картка 3
Питання
When is the conjunction (p ∧ q) true?
Відповідь
Only when p and q are both true.
Картка 4
Питання
When is the inclusive disjunction (p ∨ q) true?
Відповідь
When at least one of p and q is true, including when both are true.
Картка 5
Питання
What does one valuation assign in a propositional truth table?
Відповідь
One truth value to each proposition letter. The assignment stays fixed throughout that row.
Картка 6
Питання
When is the material conditional (p → q) false?
Відповідь
Only when p is true and q is false. Here p is the antecedent and q is the consequent.
Картка 7
Питання
When is the biconditional (p ↔ q) true?
Відповідь
When p and q have the same truth value: both true or both false.
Картка 8
Питання
When is exclusive OR (p ⊕ q) true?
Відповідь
When exactly one of p and q is true. It is false when their truth values match.
Картка 9
Питання
What is the main connective in ((¬p) ∧ q)?
Відповідь
∧ (AND). It combines the whole left part, (¬p), with q.
Картка 10
Питання
How many rows does a complete truth table with three distinct proposition letters need?
Відповідь
8 rows: each of the three letters has two choices, so 2³ = 8.
Картка 11
Питання
If p = F, what is (¬p)?
Відповідь
T. Negation reverses F to T.
Картка 12
Питання
If p = T and q = F, what is (p ∧ q)?
Відповідь
F. AND needs both inputs to be true.
Картка 13
Питання
If p = T and q = T, what is inclusive OR (p ∨ q)?
Відповідь
T. Inclusive OR allows both inputs to be true.
Картка 14
Питання
If p = T and q = T, what is the material conditional (p → q)?
Відповідь
T. A true antecedent with a true consequent does not make the conditional false.
Картка 15
Питання
If p = F and q = F, what is (p ↔ q)?
Відповідь
T. The two truth values match, even though neither is true.
Картка 16
Питання
If p = T and q = T, what is exclusive OR (p ⊕ q)?
Відповідь
F. XOR requires exactly one true input.
Картка 17
Питання
Does a true material conditional (p → q) establish that p causes q?
Відповідь
No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.
Картка 18
Питання
What assignments are listed by the two-letter row order TT, TF, FT, FF?
Відповідь
(p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.
Картка 19
Питання
In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?
Відповідь
The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.
Картка 20
Питання
If p = F and q = T, what is the material conditional (p → q)?
Відповідь
T. A material conditional with a false antecedent is true.
Картка 21
Питання
If p = T and q = F, what is (p ↔ q)?
Відповідь
F. The two truth values differ.
Картка 22
Питання
If p = T and q = F, what is exclusive OR (p ⊕ q)?
Відповідь
T. Exactly one input is true.
Картка 23
Питання
Which standard connective is true exactly when both inputs are true?
Відповідь
Conjunction (AND), written ∧.
Картка 24
Питання
Which standard connective is false exactly when both inputs are false?
Відповідь
Inclusive disjunction (OR), written ∨. The both-true case is true.
Картка 25
Питання
Which standard connective takes one input and reverses its truth value?
Відповідь
Negation (NOT), written ¬.
Картка 26
Питання
If p = F and q = F, what is the material conditional (p → q)?
Відповідь
T. Its only false case requires a true antecedent and a false consequent.
Картка 27
Питання
Which standard connective is true exactly when its two inputs have matching truth values?
Відповідь
The biconditional (if and only if), written ↔.
Картка 28
Питання
Which standard connective is true exactly when its two inputs have different truth values?
Відповідь
Exclusive OR (XOR), written ⊕.
Картка 29
Питання
What is the main connective in ((p ∨ q) → (¬r))?
Відповідь
→ (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.
Картка 30
Питання
Which standard connective is false exactly when its first input is true and its second input is false?
Відповідь
The material conditional, written →. Input order matters.
Картка 31
Питання
Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.
Відповідь
T, F, F, F. Only the both-true row satisfies AND.
Картка 32
Питання
If p = T and q = F, what is ¬(p ∧ q)?
Відповідь
T. First (p ∧ q) is F; negating it gives T.
Картка 33
Питання
Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.
Відповідь
(a ∧ b). Both statements are asserted.
Картка 34
Питання
Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.
Відповідь
T, F, T, T. Only the true-antecedent, false-consequent row fails.
Картка 35
Питання
What makes a formula a tautology in classical propositional logic?
Відповідь
It is true on every possible valuation, not just the row currently being checked.
Картка 36
Питання
Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.
Відповідь
T, T, T, F. Only the both-false row fails inclusive OR.
Картка 37
Питання
When are two propositional formulas logically equivalent?
Відповідь
When their final truth values match on every valuation of their combined proposition letters.
Картка 38
Питання
Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.
Відповідь
T, F, F, T. The first and last rows have matching truth values.
Картка 39
Питання
If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?
Відповідь
F. Both p and (¬q) are F.
Картка 40
Питання
If p = T and q = F, what is the material conditional (p → q)?
Відповідь
F. This is its only false input combination.
Картка 41
Питання
Give the output column for (¬p), with p rows T, F.
Відповідь
F, T. Negation reverses each row.
Картка 42
Питання
Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.
Відповідь
(s → a). The scan statement is the antecedent; the alert statement is the consequent.
Картка 43
Питання
Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.
Відповідь
F, T, T, F. Exactly one input is true in the middle two rows.
Картка 44
Питання
What makes a formula a contradiction in classical propositional logic?
Відповідь
It is false on every possible valuation.
Картка 45
Питання
If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?
Відповідь
T. The conjunction is F, but r is T, so the disjunction is T.
Картка 46
Питання
Simplify ¬(¬p) without changing its truth value.
Відповідь
p. Two negations restore the original truth value.
Картка 47
Питання
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”.
Відповідь
(d ↔ c). Both directions of the conditional are required.
Картка 48
Питання
What makes a propositional formula contingent?
Відповідь
It is true on at least one valuation and false on at least one other valuation.
Картка 49
Питання
If p = T and q = T, what is (p → (¬q)), using material implication?
Відповідь
F. Its antecedent is T and its consequent (¬q) is F.
Картка 50
Питання
Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.
Відповідь
((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.
Картка 51
Питання
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”.
Відповідь
(n ∨ e). This is inclusive OR.
Картка 52
Питання
Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.
Відповідь
Tautology. Whether p is T or F, one disjunct is T.
Картка 53
Питання
If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?
Відповідь
T. The disjunction is F, so its negation is T.
Картка 54
Питання
Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.
Відповідь
((¬p) ∧ (¬q)). Both disjuncts must be false.
Картка 55
Питання
Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.
Відповідь
If the heater is on, then the window is closed. The formula itself makes no causal claim.
Картка 56
Питання
Classify (p ∧ (¬p)): tautology, contradiction or contingent?
Відповідь
Contradiction. The two conjuncts cannot both be true on any valuation.
Картка 57
Питання
Rewrite the material conditional (p → q) using only NOT and inclusive OR.
Відповідь
((¬p) ∨ q). It is false exactly when p is T and q is F.
Картка 58
Питання
If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?
Відповідь
T. The XOR is F and r is F, so the biconditional compares matching values.
Картка 59
Питання
Write an expression with exactly two NOT operators that is equivalent to p.
Відповідь
¬(¬p). Negating twice leaves every truth value unchanged.
Картка 60
Питання
Classify (p ∧ q): tautology, contradiction or contingent?
Відповідь
Contingent. It is T at p = T, q = T and F at p = F, q = T.
Картка 61
Питання
What is the contrapositive of the material conditional (p → q)?
Відповідь
((¬q) → (¬p)). Swap the two sides and negate both.
Картка 62
Питання
Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.
Відповідь
(p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.
Картка 63
Питання
Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.
Відповідь
¬(p ∧ q). This is De Morgan’s law in the reverse direction.
Картка 64
Питання
Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.
Відповідь
T, T, F, T. The only false row has p = F and q = T.
Картка 65
Питання
What is the converse of (p → q)?
Відповідь
(q → p). Swap the antecedent and consequent without negating either.
Картка 66
Питання
Rewrite (p ↔ q) as an AND of two material conditionals.
Відповідь
((p → q) ∧ (q → p)). Both directions must hold.
Картка 67
Питання
Rewrite ((¬p) ∧ (¬q)) as one negation of an inclusive disjunction.
Відповідь
¬(p ∨ q). This is De Morgan’s law in the reverse direction.
Картка 68
Питання
Give one valuation showing that (p ∨ q) and (p ∧ q) are not equivalent. Use inclusive OR.
Відповідь
p = T, q = F gives T for the OR and F for the AND. The swapped assignment also works.
Картка 69
Питання
What is the inverse of (p → q)?
Відповідь
((¬p) → (¬q)). Negate both sides without swapping them.
Картка 70
Питання
Rewrite ((¬p) ∨ q) as a single material conditional, using inclusive OR.
Відповідь
(p → q). Both expressions fail exactly when p is T and q is F.
Картка 71
Питання
Rewrite exclusive OR (p ⊕ q) using AND, inclusive OR and NOT.
Відповідь
((p ∨ q) ∧ ¬(p ∧ q)). Require at least one true input and rule out both being true.
Картка 72
Питання
A formula is true for p = T and q = F. Is that enough to call it a tautology?
Відповідь
No. A tautology must be true on every valuation. One true row establishes only that it can be true.
Картка 73
Питання
Is a material conditional (p → q) logically equivalent to its contrapositive ((¬q) → (¬p))?
Відповідь
Yes. Both are false exactly when p is T and q is F.
Картка 74
Питання
Give one valuation showing that (p → q) and its converse (q → p) are not equivalent. Use material implication.
Відповідь
p = T, q = F. Then (p → q) is F and (q → p) is T.
Картка 75
Питання
Rewrite ((p → q) ∧ (q → p)) using one connective, with both arrows meaning material implication.
Відповідь
(p ↔ q). This is the biconditional.
Картка 76
Питання
What does one valuation with different outputs prove about two formulas?
Відповідь
They are not logically equivalent. Equivalence requires agreement on every valuation.
Картка 77
Питання
Give one valuation showing that (p → q) and its inverse ((¬p) → (¬q)) are not equivalent. Use material implication.
Відповідь
p = T, q = F. The original is F, while the inverse has a false antecedent and is T.
Картка 78
Питання
Give one valuation showing that ¬(p ∧ q) and ((¬p) ∧ (¬q)) are not equivalent.
Відповідь
p = T, q = F. The negated conjunction is T; the conjunction of negations is F.
Картка 79
Питання
Which connective does ((p ∨ q) ∧ ¬(p ∧ q)) express? Here ∨ is inclusive OR.
Відповідь
Exclusive OR: (p ⊕ q). Exactly one input must be true.
Картка 80
Питання
If (p ↔ q) is true on one row, does that show that the formulas p and q are logically equivalent?
Відповідь
No. They match on that row only. Logical equivalence requires the biconditional to be true on every valuation.
Картка 81
Питання
Are the converse (q → p) and inverse ((¬p) → (¬q)) of (p → q) equivalent to each other? Use material implication.
Відповідь
Yes. They are contrapositives of each other, and both are false exactly when q is T and p is F.
Картка 82
Питання
At p = T, q = F and r = F, compare ((p ∨ q) ∧ r) with (p ∨ (q ∧ r)). Use inclusive OR.
Відповідь
The first is F; the second is T. Parentheses change which operations combine first.
Картка 83
Питання
Rewrite (p ∧ (¬q)) as the negation of one material conditional.
Відповідь
¬(p → q). The conjunction describes exactly the conditional’s false case.
Картка 84
Питання
Among AND (∧), inclusive OR (∨), XOR (⊕), material conditional (→) and biconditional (↔), which has outputs T, F, F, T for (p, q) rows TT, TF, FT, FF?
Відповідь
Biconditional (↔). It is true on the two rows where the inputs match.
84 картки
Truth Table Flashcards: Connectives & Logical Equivalence
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