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