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