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