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