Truth Table Flashcards: Connectives & Logical Equivalence

Question 1

In classical propositional logic, what is a proposition?

Accepted Answer

A statement with a truth value: true or false. A question or command is not a proposition in this setting.

Question 2

What does negation (¬p) do to the truth value of p?

Accepted Answer

It reverses it: true becomes false, and false becomes true.

Question 3

When is the conjunction (p ∧ q) true?

Accepted Answer

Only when p and q are both true.

Question 4

When is the inclusive disjunction (p ∨ q) true?

Accepted Answer

When at least one of p and q is true, including when both are true.

Question 5

What does one valuation assign in a propositional truth table?

Accepted Answer

One truth value to each proposition letter. The assignment stays fixed throughout that row.

Question 6

When is the material conditional (p → q) false?

Accepted Answer

Only when p is true and q is false. Here p is the antecedent and q is the consequent.

Question 7

When is the biconditional (p ↔ q) true?

Accepted Answer

When p and q have the same truth value: both true or both false.

Question 8

When is exclusive OR (p ⊕ q) true?

Accepted Answer

When exactly one of p and q is true. It is false when their truth values match.

Question 9

What is the main connective in ((¬p) ∧ q)?

Accepted Answer

∧ (AND). It combines the whole left part, (¬p), with q.

Question 10

How many rows does a complete truth table with three distinct proposition letters need?

Accepted Answer

8 rows: each of the three letters has two choices, so 2³ = 8.

Question 11

If p = F, what is (¬p)?

Accepted Answer

T. Negation reverses F to T.

Question 12

If p = T and q = F, what is (p ∧ q)?

Accepted Answer

F. AND needs both inputs to be true.

Question 13

If p = T and q = T, what is inclusive OR (p ∨ q)?

Accepted Answer

T. Inclusive OR allows both inputs to be true.

Question 14

If p = T and q = T, what is the material conditional (p → q)?

Accepted Answer

T. A true antecedent with a true consequent does not make the conditional false.

Question 15

If p = F and q = F, what is (p ↔ q)?

Accepted Answer

T. The two truth values match, even though neither is true.

Question 16

If p = T and q = T, what is exclusive OR (p ⊕ q)?

Accepted Answer

F. XOR requires exactly one true input.

Question 17

Does a true material conditional (p → q) establish that p causes q?

Accepted Answer

No. Material implication is determined by truth values; it does not establish causation or capture every everyday use of “if”.

Question 18

What assignments are listed by the two-letter row order TT, TF, FT, FF?

Accepted Answer

(p, q) = (T, T), (T, F), (F, T), (F, F). Each possible assignment appears once.

Question 19

In ¬(p ∨ q), does ¬ negate just p or the whole disjunction?

Accepted Answer

The whole disjunction (p ∨ q). Evaluate that parenthesized expression before negating it.

Question 20

If p = F and q = T, what is the material conditional (p → q)?

Accepted Answer

T. A material conditional with a false antecedent is true.

Question 21

If p = T and q = F, what is (p ↔ q)?

Accepted Answer

F. The two truth values differ.

Question 22

If p = T and q = F, what is exclusive OR (p ⊕ q)?

Accepted Answer

T. Exactly one input is true.

Question 23

Which standard connective is true exactly when both inputs are true?

Accepted Answer

Conjunction (AND), written ∧.

Question 24

Which standard connective is false exactly when both inputs are false?

Accepted Answer

Inclusive disjunction (OR), written ∨. The both-true case is true.

Question 25

Which standard connective takes one input and reverses its truth value?

Accepted Answer

Negation (NOT), written ¬.

Question 26

If p = F and q = F, what is the material conditional (p → q)?

Accepted Answer

T. Its only false case requires a true antecedent and a false consequent.

Question 27

Which standard connective is true exactly when its two inputs have matching truth values?

Accepted Answer

The biconditional (if and only if), written ↔.

Question 28

Which standard connective is true exactly when its two inputs have different truth values?

Accepted Answer

Exclusive OR (XOR), written ⊕.

Question 29

What is the main connective in ((p ∨ q) → (¬r))?

Accepted Answer

→ (the material conditional). The entire disjunction is the antecedent, and (¬r) is the consequent.

Question 30

Which standard connective is false exactly when its first input is true and its second input is false?

Accepted Answer

The material conditional, written →. Input order matters.

Question 31

Give the output column for (p ∧ q), with (p, q) rows TT, TF, FT, FF.

Accepted Answer

T, F, F, F. Only the both-true row satisfies AND.

Question 32

If p = T and q = F, what is ¬(p ∧ q)?

Accepted Answer

T. First (p ∧ q) is F; negating it gives T.

Question 33

Let a mean “the archive is open” and b mean “the desk is staffed”. Symbolize “the archive is open and the desk is staffed”.

Accepted Answer

(a ∧ b). Both statements are asserted.

Question 34

Give the output column for the material conditional (p → q), with (p, q) rows TT, TF, FT, FF.

Accepted Answer

T, F, T, T. Only the true-antecedent, false-consequent row fails.

Question 35

What makes a formula a tautology in classical propositional logic?

Accepted Answer

It is true on every possible valuation, not just the row currently being checked.

Question 36

Give the output column for inclusive OR (p ∨ q), with (p, q) rows TT, TF, FT, FF.

Accepted Answer

T, T, T, F. Only the both-false row fails inclusive OR.

Question 37

When are two propositional formulas logically equivalent?

Accepted Answer

When their final truth values match on every valuation of their combined proposition letters.

Question 38

Give the output column for (p ↔ q), with (p, q) rows TT, TF, FT, FF.

Accepted Answer

T, F, F, T. The first and last rows have matching truth values.

Question 39

If p = F and q = T, what is (p ∨ (¬q)), using inclusive OR?

Accepted Answer

F. Both p and (¬q) are F.

Question 40

If p = T and q = F, what is the material conditional (p → q)?

Accepted Answer

F. This is its only false input combination.

Question 41

Give the output column for (¬p), with p rows T, F.

Accepted Answer

F, T. Negation reverses each row.

Question 42

Let s mean “the scan succeeds” and a mean “the alert appears”. Symbolize “if the scan succeeds, the alert appears”, using material implication.

Accepted Answer

(s → a). The scan statement is the antecedent; the alert statement is the consequent.

Question 43

Give the output column for exclusive OR (p ⊕ q), with (p, q) rows TT, TF, FT, FF.

Accepted Answer

F, T, T, F. Exactly one input is true in the middle two rows.

Question 44

What makes a formula a contradiction in classical propositional logic?

Accepted Answer

It is false on every possible valuation.

Question 45

If p = T, q = F and r = T, what is ((p ∧ q) ∨ r), using inclusive OR?

Accepted Answer

T. The conjunction is F, but r is T, so the disjunction is T.

Question 46

Simplify ¬(¬p) without changing its truth value.

Accepted Answer

p. Two negations restore the original truth value.

Question 47

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”.

Accepted Answer

(d ↔ c). Both directions of the conditional are required.

Question 48

What makes a propositional formula contingent?

Accepted Answer

It is true on at least one valuation and false on at least one other valuation.

Question 49

If p = T and q = T, what is (p → (¬q)), using material implication?

Accepted Answer

F. Its antecedent is T and its consequent (¬q) is F.

Question 50

Use De Morgan’s law to rewrite ¬(p ∧ q) with negations only on letters.

Accepted Answer

((¬p) ∨ (¬q)). At least one conjunct must be false; ∨ is inclusive OR.

Question 51

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”.

Accepted Answer

(n ∨ e). This is inclusive OR.

Question 52

Classify (p ∨ (¬p)): tautology, contradiction or contingent? Use classical logic and inclusive OR.

Accepted Answer

Tautology. Whether p is T or F, one disjunct is T.

Question 53

If p = F and q = F, what is ¬(p ∨ q), using inclusive OR?

Accepted Answer

T. The disjunction is F, so its negation is T.

Question 54

Use De Morgan’s law to rewrite ¬(p ∨ q) with negations only on letters. Use inclusive OR.

Accepted Answer

((¬p) ∧ (¬q)). Both disjuncts must be false.

Question 55

Let w mean “the window is closed” and h mean “the heater is on”. Read (h → w) as an English material conditional.

Accepted Answer

If the heater is on, then the window is closed. The formula itself makes no causal claim.

Question 56

Classify (p ∧ (¬p)): tautology, contradiction or contingent?

Accepted Answer

Contradiction. The two conjuncts cannot both be true on any valuation.

Question 57

Rewrite the material conditional (p → q) using only NOT and inclusive OR.

Accepted Answer

((¬p) ∨ q). It is false exactly when p is T and q is F.

Question 58

If p = T, q = T and r = F, what is ((p ⊕ q) ↔ r), where ⊕ is exclusive OR?

Accepted Answer

T. The XOR is F and r is F, so the biconditional compares matching values.

Question 59

Write an expression with exactly two NOT operators that is equivalent to p.

Accepted Answer

¬(¬p). Negating twice leaves every truth value unchanged.

Question 60

Classify (p ∧ q): tautology, contradiction or contingent?

Accepted Answer

Contingent. It is T at p = T, q = T and F at p = F, q = T.

Question 61

What is the contrapositive of the material conditional (p → q)?

Accepted Answer

((¬q) → (¬p)). Swap the two sides and negate both.

Question 62

Rewrite ¬(p → q) using AND and NOT, with → meaning material implication.

Accepted Answer

(p ∧ (¬q)). A material conditional fails exactly when its antecedent is true and its consequent is false.

Question 63

Rewrite ((¬p) ∨ (¬q)) as one negation of a conjunction, using inclusive OR.

Accepted Answer

¬(p ∧ q). This is De Morgan’s law in the reverse direction.

Question 64

Give the output column for (p ∨ (¬q)), with (p, q) rows TT, TF, FT, FF and inclusive OR.

Accepted Answer

T, T, F, T. The only false row has p = F and q = T.

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