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