Truth Table Flashcards: Connectives & Logical Equivalence
84 truth-table flashcards on logical connectives, material conditionals, nested expressions, equivalence, and counterexamples, with short explanations.
دربارهٔ این دسته
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.
کارتهای این دسته
کارت ۱
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In classical propositional logic, what is a proposition?
پاسخ
A statement with a truth value: true or false. A question or command is not a proposition in this setting.
کارت ۲
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What does negation (¬p) do to the truth value of p?
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It reverses it: true becomes false, and false becomes true.
کارت ۳
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When is the conjunction (p ∧ q) true?
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Only when p and q are both true.
کارت ۴
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When is the inclusive disjunction (p ∨ q) true?
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When at least one of p and q is true, including when both are true.
کارت ۵
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What does one valuation assign in a propositional truth table?
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One truth value to each proposition letter. The assignment stays fixed throughout that row.
کارت ۶
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When is the material conditional (p → q) false?
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Only when p is true and q is false. Here p is the antecedent and q is the consequent.
کارت ۷
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When is the biconditional (p ↔ q) true?
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When p and q have the same truth value: both true or both false.
کارت ۸
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When is exclusive OR (p ⊕ q) true?
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When exactly one of p and q is true. It is false when their truth values match.
کارت ۹
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What is the main connective in ((¬p) ∧ q)?
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∧ (AND). It combines the whole left part, (¬p), with q.
کارت ۱۰
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How many rows does a complete truth table with three distinct proposition letters need?
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8 rows: each of the three letters has two choices, so 2³ = 8.
کارت ۱۱
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If p = F, what is (¬p)?
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T. Negation reverses F to T.
کارت ۱۲
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If p = T and q = F, what is (p ∧ q)?
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F. AND needs both inputs to be true.
کارت ۱۳
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If p = T and q = T, what is inclusive OR (p ∨ q)?
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T. Inclusive OR allows both inputs to be true.
کارت ۱۴
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If p = T and q = T, what is the material conditional (p → q)?
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T. A true antecedent with a true consequent does not make the conditional false.
کارت ۱۵
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If p = F and q = F, what is (p ↔ q)?
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T. The two truth values match, even though neither is true.
کارت ۱۶
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If p = T and q = T, what is exclusive OR (p ⊕ q)?
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F. XOR requires exactly one true input.
کارت ۱۷
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Does a true material conditional (p → q) establish that p causes q?
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No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.
کارت ۱۸
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What assignments are listed by the two-letter row order TT, TF, FT, FF?
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(p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.
کارت ۱۹
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In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?
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The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.
کارت ۲۰
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If p = F and q = T, what is the material conditional (p → q)?
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T. A material conditional with a false antecedent is true.
کارت ۲۱
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If p = T and q = F, what is (p ↔ q)?
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F. The two truth values differ.
کارت ۲۲
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If p = T and q = F, what is exclusive OR (p ⊕ q)?
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T. Exactly one input is true.
کارت ۲۳
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Which standard connective is true exactly when both inputs are true?
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Conjunction (AND), written ∧.
کارت ۲۴
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Which standard connective is false exactly when both inputs are false?
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Inclusive disjunction (OR), written ∨. The both-true case is true.
کارت ۲۵
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Which standard connective takes one input and reverses its truth value?
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Negation (NOT), written ¬.
کارت ۲۶
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If p = F and q = F, what is the material conditional (p → q)?
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T. Its only false case requires a true antecedent and a false consequent.
کارت ۲۷
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Which standard connective is true exactly when its two inputs have matching truth values?
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The biconditional (if and only if), written ↔.
کارت ۲۸
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Which standard connective is true exactly when its two inputs have different truth values?
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Exclusive OR (XOR), written ⊕.
کارت ۲۹
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What is the main connective in ((p ∨ q) → (¬r))?
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→ (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.
کارت ۳۰
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Which standard connective is false exactly when its first input is true and its second input is false?
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The material conditional, written →. Input order matters.
کارت ۳۱
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Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.
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T, F, F, F. Only the both-true row satisfies AND.
کارت ۳۲
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If p = T and q = F, what is ¬(p ∧ q)?
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T. First (p ∧ q) is F; negating it gives T.
کارت ۳۳
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Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.
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(a ∧ b). Both statements are asserted.
کارت ۳۴
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Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.
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T, F, T, T. Only the true-antecedent, false-consequent row fails.
کارت ۳۵
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What makes a formula a tautology in classical propositional logic?
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It is true on every possible valuation, not just the row currently being checked.
کارت ۳۶
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Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.
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T, T, T, F. Only the both-false row fails inclusive OR.
کارت ۳۷
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When are two propositional formulas logically equivalent?
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When their final truth values match on every valuation of their combined proposition letters.
کارت ۳۸
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Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.
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T, F, F, T. The first and last rows have matching truth values.
کارت ۳۹
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If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?
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F. Both p and (¬q) are F.
کارت ۴۰
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If p = T and q = F, what is the material conditional (p → q)?
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F. This is its only false input combination.
کارت ۴۱
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Give the output column for (¬p), with p rows T, F.
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F, T. Negation reverses each row.
کارت ۴۲
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Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.
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(s → a). The scan statement is the antecedent; the alert statement is the consequent.
کارت ۴۳
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Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.
پاسخ
F, T, T, F. Exactly one input is true in the middle two rows.
کارت ۴۴
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What makes a formula a contradiction in classical propositional logic?
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It is false on every possible valuation.
کارت ۴۵
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If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?
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T. The conjunction is F, but r is T, so the disjunction is T.
کارت ۴۶
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Simplify ¬(¬p) without changing its truth value.
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p. Two negations restore the original truth value.
کارت ۴۷
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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”.
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(d ↔ c). Both directions of the conditional are required.
کارت ۴۸
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What makes a propositional formula contingent?
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It is true on at least one valuation and false on at least one other valuation.
کارت ۴۹
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If p = T and q = T, what is (p → (¬q)), using material implication?
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F. Its antecedent is T and its consequent (¬q) is F.
کارت ۵۰
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Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.
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((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.
کارت ۵۱
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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”.
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(n ∨ e). This is inclusive OR.
کارت ۵۲
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Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.
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Tautology. Whether p is T or F, one disjunct is T.
کارت ۵۳
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If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?
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T. The disjunction is F, so its negation is T.
کارت ۵۴
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Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.
پاسخ
((¬p) ∧ (¬q)). Both disjuncts must be false.
کارت ۵۵
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Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.
پاسخ
If the heater is on, then the window is closed. The formula itself makes no causal claim.
کارت ۵۶
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Classify (p ∧ (¬p)): tautology, contradiction or contingent?
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Contradiction. The two conjuncts cannot both be true on any valuation.
کارت ۵۷
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Rewrite the material conditional (p → q) using only NOT and inclusive OR.
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((¬p) ∨ q). It is false exactly when p is T and q is F.
کارت ۵۸
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If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?
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T. The XOR is F and r is F, so the biconditional compares matching values.
کارت ۵۹
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Write an expression with exactly two NOT operators that is equivalent to p.
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¬(¬p). Negating twice leaves every truth value unchanged.
کارت ۶۰
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Classify (p ∧ q): tautology, contradiction or contingent?
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Contingent. It is T at p = T, q = T and F at p = F, q = T.
کارت ۶۱
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What is the contrapositive of the material conditional (p → q)?
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((¬q) → (¬p)). Swap the two sides and negate both.
کارت ۶۲
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Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.
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(p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.
کارت ۶۳
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Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.
پاسخ
¬(p ∧ q). This is De Morgan’s law in the reverse direction.
کارت ۶۴
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Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.
پاسخ
T, T, F, T. The only false row has p = F and q = T.
کارت ۶۵
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What is the converse of (p → q)?
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(q → p). Swap the antecedent and consequent without negating either.
کارت ۶۶
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Rewrite (p ↔ q) as an AND of two material conditionals.
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((p → q) ∧ (q → p)). Both directions must hold.
کارت ۶۷
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Rewrite ((¬p) ∧ (¬q)) as one negation of an inclusive disjunction.
پاسخ
¬(p ∨ q). This is De Morgan’s law in the reverse direction.
کارت ۶۸
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Give one valuation showing that (p ∨ q) and (p ∧ q) are not equivalent. Use inclusive OR.
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p = T, q = F gives T for the OR and F for the AND. The swapped assignment also works.
کارت ۶۹
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What is the inverse of (p → q)?
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((¬p) → (¬q)). Negate both sides without swapping them.
کارت ۷۰
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Rewrite ((¬p) ∨ q) as a single material conditional, using inclusive OR.
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(p → q). Both expressions fail exactly when p is T and q is F.
کارت ۷۱
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Rewrite exclusive OR (p ⊕ q) using AND, inclusive OR and NOT.
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((p ∨ q) ∧ ¬(p ∧ q)). Require at least one true input and rule out both being true.
کارت ۷۲
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A formula is true for p = T and q = F. Is that enough to call it a tautology?
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No. A tautology must be true on every valuation. One true row establishes only that it can be true.
کارت ۷۳
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Is a material conditional (p → q) logically equivalent to its contrapositive ((¬q) → (¬p))?
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Yes. Both are false exactly when p is T and q is F.
کارت ۷۴
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Give one valuation showing that (p → q) and its converse (q → p) are not equivalent. Use material implication.
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p = T, q = F. Then (p → q) is F and (q → p) is T.
کارت ۷۵
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Rewrite ((p → q) ∧ (q → p)) using one connective, with both arrows meaning material implication.
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(p ↔ q). This is the biconditional.
کارت ۷۶
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What does one valuation with different outputs prove about two formulas?
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They are not logically equivalent. Equivalence requires agreement on every valuation.
کارت ۷۷
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Give one valuation showing that (p → q) and its inverse ((¬p) → (¬q)) are not equivalent. Use material implication.
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p = T, q = F. The original is F, while the inverse has a false antecedent and is T.
کارت ۷۸
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Give one valuation showing that ¬(p ∧ q) and ((¬p) ∧ (¬q)) are not equivalent.
پاسخ
p = T, q = F. The negated conjunction is T; the conjunction of negations is F.
کارت ۷۹
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Which connective does ((p ∨ q) ∧ ¬(p ∧ q)) express? Here ∨ is inclusive OR.
پاسخ
Exclusive OR: (p ⊕ q). Exactly one input must be true.
کارت ۸۰
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If (p ↔ q) is true on one row, does that show that the formulas p and q are logically equivalent?
پاسخ
No. They match on that row only. Logical equivalence requires the biconditional to be true on every valuation.
کارت ۸۱
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Are the converse (q → p) and inverse ((¬p) → (¬q)) of (p → q) equivalent to each other? Use material implication.
پاسخ
Yes. They are contrapositives of each other, and both are false exactly when q is T and p is F.
کارت ۸۲
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At p = T, q = F and r = F, compare ((p ∨ q) ∧ r) with (p ∨ (q ∧ r)). Use inclusive OR.
پاسخ
The first is F; the second is T. Parentheses change which operations combine first.
کارت ۸۳
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Rewrite (p ∧ (¬q)) as the negation of one material conditional.
پاسخ
¬(p → q). The conjunction describes exactly the conditional’s false case.
کارت ۸۴
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Among AND (∧), inclusive OR (∨), XOR (⊕), material conditional (→) and biconditional (↔), which has outputs T, F, F, T for (p, q) rows TT, TF, FT, FF?
پاسخ
Biconditional (↔). It is true on the two rows where the inputs match.
۸۴ کارت
Truth Table Flashcards: Connectives & Logical Equivalence
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