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