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