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