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