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