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