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