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