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