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