Truth Table Flashcards: Connectives & Logical Equivalence
84 truth-table flashcards on logical connectives, material conditionals, nested expressions, equivalence, and counterexamples, with short explanations.
حول هذه الرزمة
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.
بطاقات هذه الرزمة
البطاقة ١
السؤال
In classical propositional logic, what is a proposition?
الإجابة
A statement with a truth value: true or false. A question or command is not a proposition in this setting.
البطاقة ٢
السؤال
What does negation (¬p) do to the truth value of p?
الإجابة
It reverses it: true becomes false, and false becomes true.
البطاقة ٣
السؤال
When is the conjunction (p ∧ q) true?
الإجابة
Only when p and q are both true.
البطاقة ٤
السؤال
When is the inclusive disjunction (p ∨ q) true?
الإجابة
When at least one of p and q is true, including when both are true.
البطاقة ٥
السؤال
What does one valuation assign in a propositional truth table?
الإجابة
One truth value to each proposition letter. The assignment stays fixed throughout that row.
البطاقة ٦
السؤال
When is the material conditional (p → q) false?
الإجابة
Only when p is true and q is false. Here p is the antecedent and q is the consequent.
البطاقة ٧
السؤال
When is the biconditional (p ↔ q) true?
الإجابة
When p and q have the same truth value: both true or both false.
البطاقة ٨
السؤال
When is exclusive OR (p ⊕ q) true?
الإجابة
When exactly one of p and q is true. It is false when their truth values match.
البطاقة ٩
السؤال
What is the main connective in ((¬p) ∧ q)?
الإجابة
∧ (AND). It combines the whole left part, (¬p), with q.
البطاقة ١٠
السؤال
How many rows does a complete truth table with three distinct proposition letters need?
الإجابة
8 rows: each of the three letters has two choices, so 2³ = 8.
البطاقة ١١
السؤال
If p = F, what is (¬p)?
الإجابة
T. Negation reverses F to T.
البطاقة ١٢
السؤال
If p = T and q = F, what is (p ∧ q)?
الإجابة
F. AND needs both inputs to be true.
البطاقة ١٣
السؤال
If p = T and q = T, what is inclusive OR (p ∨ q)?
الإجابة
T. Inclusive OR allows both inputs to be true.
البطاقة ١٤
السؤال
If p = T and q = T, what is the material conditional (p → q)?
الإجابة
T. A true antecedent with a true consequent does not make the conditional false.
البطاقة ١٥
السؤال
If p = F and q = F, what is (p ↔ q)?
الإجابة
T. The two truth values match, even though neither is true.
البطاقة ١٦
السؤال
If p = T and q = T, what is exclusive OR (p ⊕ q)?
الإجابة
F. XOR requires exactly one true input.
البطاقة ١٧
السؤال
Does a true material conditional (p → q) establish that p causes q?
الإجابة
No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.
البطاقة ١٨
السؤال
What assignments are listed by the two-letter row order TT, TF, FT, FF?
الإجابة
(p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.
البطاقة ١٩
السؤال
In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?
الإجابة
The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.
البطاقة ٢٠
السؤال
If p = F and q = T, what is the material conditional (p → q)?
الإجابة
T. A material conditional with a false antecedent is true.
البطاقة ٢١
السؤال
If p = T and q = F, what is (p ↔ q)?
الإجابة
F. The two truth values differ.
البطاقة ٢٢
السؤال
If p = T and q = F, what is exclusive OR (p ⊕ q)?
الإجابة
T. Exactly one input is true.
البطاقة ٢٣
السؤال
Which standard connective is true exactly when both inputs are true?
الإجابة
Conjunction (AND), written ∧.
البطاقة ٢٤
السؤال
Which standard connective is false exactly when both inputs are false?
الإجابة
Inclusive disjunction (OR), written ∨. The both-true case is true.
البطاقة ٢٥
السؤال
Which standard connective takes one input and reverses its truth value?
الإجابة
Negation (NOT), written ¬.
البطاقة ٢٦
السؤال
If p = F and q = F, what is the material conditional (p → q)?
الإجابة
T. Its only false case requires a true antecedent and a false consequent.
البطاقة ٢٧
السؤال
Which standard connective is true exactly when its two inputs have matching truth values?
الإجابة
The biconditional (if and only if), written ↔.
البطاقة ٢٨
السؤال
Which standard connective is true exactly when its two inputs have different truth values?
الإجابة
Exclusive OR (XOR), written ⊕.
البطاقة ٢٩
السؤال
What is the main connective in ((p ∨ q) → (¬r))?
الإجابة
→ (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.
البطاقة ٣٠
السؤال
Which standard connective is false exactly when its first input is true and its second input is false?
الإجابة
The material conditional, written →. Input order matters.
البطاقة ٣١
السؤال
Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.
الإجابة
T, F, F, F. Only the both-true row satisfies AND.
البطاقة ٣٢
السؤال
If p = T and q = F, what is ¬(p ∧ q)?
الإجابة
T. First (p ∧ q) is F; negating it gives T.
البطاقة ٣٣
السؤال
Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.
الإجابة
(a ∧ b). Both statements are asserted.
البطاقة ٣٤
السؤال
Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.
الإجابة
T, F, T, T. Only the true-antecedent, false-consequent row fails.
البطاقة ٣٥
السؤال
What makes a formula a tautology in classical propositional logic?
الإجابة
It is true on every possible valuation, not just the row currently being checked.
البطاقة ٣٦
السؤال
Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.
الإجابة
T, T, T, F. Only the both-false row fails inclusive OR.
البطاقة ٣٧
السؤال
When are two propositional formulas logically equivalent?
الإجابة
When their final truth values match on every valuation of their combined proposition letters.
البطاقة ٣٨
السؤال
Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.
الإجابة
T, F, F, T. The first and last rows have matching truth values.
البطاقة ٣٩
السؤال
If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?
الإجابة
F. Both p and (¬q) are F.
البطاقة ٤٠
السؤال
If p = T and q = F, what is the material conditional (p → q)?
الإجابة
F. This is its only false input combination.
البطاقة ٤١
السؤال
Give the output column for (¬p), with p rows T, F.
الإجابة
F, T. Negation reverses each row.
البطاقة ٤٢
السؤال
Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.
الإجابة
(s → a). The scan statement is the antecedent; the alert statement is the consequent.
البطاقة ٤٣
السؤال
Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.
الإجابة
F, T, T, F. Exactly one input is true in the middle two rows.
البطاقة ٤٤
السؤال
What makes a formula a contradiction in classical propositional logic?
الإجابة
It is false on every possible valuation.
البطاقة ٤٥
السؤال
If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?
الإجابة
T. The conjunction is F, but r is T, so the disjunction is T.
البطاقة ٤٦
السؤال
Simplify ¬(¬p) without changing its truth value.
الإجابة
p. Two negations restore the original truth value.
البطاقة ٤٧
السؤال
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”.
الإجابة
(d ↔ c). Both directions of the conditional are required.
البطاقة ٤٨
السؤال
What makes a propositional formula contingent?
الإجابة
It is true on at least one valuation and false on at least one other valuation.
البطاقة ٤٩
السؤال
If p = T and q = T, what is (p → (¬q)), using material implication?
الإجابة
F. Its antecedent is T and its consequent (¬q) is F.
البطاقة ٥٠
السؤال
Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.
الإجابة
((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.
البطاقة ٥١
السؤال
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”.
الإجابة
(n ∨ e). This is inclusive OR.
البطاقة ٥٢
السؤال
Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.
الإجابة
Tautology. Whether p is T or F, one disjunct is T.
البطاقة ٥٣
السؤال
If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?
الإجابة
T. The disjunction is F, so its negation is T.
البطاقة ٥٤
السؤال
Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.
الإجابة
((¬p) ∧ (¬q)). Both disjuncts must be false.
البطاقة ٥٥
السؤال
Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.
الإجابة
If the heater is on, then the window is closed. The formula itself makes no causal claim.
البطاقة ٥٦
السؤال
Classify (p ∧ (¬p)): tautology, contradiction or contingent?
الإجابة
Contradiction. The two conjuncts cannot both be true on any valuation.
البطاقة ٥٧
السؤال
Rewrite the material conditional (p → q) using only NOT and inclusive OR.
الإجابة
((¬p) ∨ q). It is false exactly when p is T and q is F.
البطاقة ٥٨
السؤال
If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?
الإجابة
T. The XOR is F and r is F, so the biconditional compares matching values.
البطاقة ٥٩
السؤال
Write an expression with exactly two NOT operators that is equivalent to p.
الإجابة
¬(¬p). Negating twice leaves every truth value unchanged.
البطاقة ٦٠
السؤال
Classify (p ∧ q): tautology, contradiction or contingent?
الإجابة
Contingent. It is T at p = T, q = T and F at p = F, q = T.
البطاقة ٦١
السؤال
What is the contrapositive of the material conditional (p → q)?
الإجابة
((¬q) → (¬p)). Swap the two sides and negate both.
البطاقة ٦٢
السؤال
Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.
الإجابة
(p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.
البطاقة ٦٣
السؤال
Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.
الإجابة
¬(p ∧ q). This is De Morgan’s law in the reverse direction.
البطاقة ٦٤
السؤال
Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.
الإجابة
T, T, F, T. The only false row has p = F and q = T.
البطاقة ٦٥
السؤال
What is the converse of (p → q)?
الإجابة
(q → p). Swap the antecedent and consequent without negating either.
البطاقة ٦٦
السؤال
Rewrite (p ↔ q) as an AND of two material conditionals.
الإجابة
((p → q) ∧ (q → p)). Both directions must hold.
البطاقة ٦٧
السؤال
Rewrite ((¬p) ∧ (¬q)) as one negation of an inclusive disjunction.
الإجابة
¬(p ∨ q). This is De Morgan’s law in the reverse direction.
البطاقة ٦٨
السؤال
Give one valuation showing that (p ∨ q) and (p ∧ q) are not equivalent. Use inclusive OR.
الإجابة
p = T, q = F gives T for the OR and F for the AND. The swapped assignment also works.
البطاقة ٦٩
السؤال
What is the inverse of (p → q)?
الإجابة
((¬p) → (¬q)). Negate both sides without swapping them.
البطاقة ٧٠
السؤال
Rewrite ((¬p) ∨ q) as a single material conditional, using inclusive OR.
الإجابة
(p → q). Both expressions fail exactly when p is T and q is F.
البطاقة ٧١
السؤال
Rewrite exclusive OR (p ⊕ q) using AND, inclusive OR and NOT.
الإجابة
((p ∨ q) ∧ ¬(p ∧ q)). Require at least one true input and rule out both being true.
البطاقة ٧٢
السؤال
A formula is true for p = T and q = F. Is that enough to call it a tautology?
الإجابة
No. A tautology must be true on every valuation. One true row establishes only that it can be true.
البطاقة ٧٣
السؤال
Is a material conditional (p → q) logically equivalent to its contrapositive ((¬q) → (¬p))?
الإجابة
Yes. Both are false exactly when p is T and q is F.
البطاقة ٧٤
السؤال
Give one valuation showing that (p → q) and its converse (q → p) are not equivalent. Use material implication.
الإجابة
p = T, q = F. Then (p → q) is F and (q → p) is T.
البطاقة ٧٥
السؤال
Rewrite ((p → q) ∧ (q → p)) using one connective, with both arrows meaning material implication.
الإجابة
(p ↔ q). This is the biconditional.
البطاقة ٧٦
السؤال
What does one valuation with different outputs prove about two formulas?
الإجابة
They are not logically equivalent. Equivalence requires agreement on every valuation.
البطاقة ٧٧
السؤال
Give one valuation showing that (p → q) and its inverse ((¬p) → (¬q)) are not equivalent. Use material implication.
الإجابة
p = T, q = F. The original is F, while the inverse has a false antecedent and is T.
البطاقة ٧٨
السؤال
Give one valuation showing that ¬(p ∧ q) and ((¬p) ∧ (¬q)) are not equivalent.
الإجابة
p = T, q = F. The negated conjunction is T; the conjunction of negations is F.
البطاقة ٧٩
السؤال
Which connective does ((p ∨ q) ∧ ¬(p ∧ q)) express? Here ∨ is inclusive OR.
الإجابة
Exclusive OR: (p ⊕ q). Exactly one input must be true.
البطاقة ٨٠
السؤال
If (p ↔ q) is true on one row, does that show that the formulas p and q are logically equivalent?
الإجابة
No. They match on that row only. Logical equivalence requires the biconditional to be true on every valuation.
البطاقة ٨١
السؤال
Are the converse (q → p) and inverse ((¬p) → (¬q)) of (p → q) equivalent to each other? Use material implication.
الإجابة
Yes. They are contrapositives of each other, and both are false exactly when q is T and p is F.
البطاقة ٨٢
السؤال
At p = T, q = F and r = F, compare ((p ∨ q) ∧ r) with (p ∨ (q ∧ r)). Use inclusive OR.
الإجابة
The first is F; the second is T. Parentheses change which operations combine first.
البطاقة ٨٣
السؤال
Rewrite (p ∧ (¬q)) as the negation of one material conditional.
الإجابة
¬(p → q). The conjunction describes exactly the conditional’s false case.
البطاقة ٨٤
السؤال
Among AND (∧), inclusive OR (∨), XOR (⊕), material conditional (→) and biconditional (↔), which has outputs T, F, F, T for (p, q) rows TT, TF, FT, FF?
الإجابة
Biconditional (↔). It is true on the two rows where the inputs match.
٨٤ بطاقة
Truth Table Flashcards: Connectives & Logical Equivalence
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