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