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