Which of the following correctly negates the compound statement 'n is even AND n is positive'?
An is not even AND n is not positive
Bn is not even OR n is not positive
Cn is odd AND n is negative
Dn is even OR n is positive
By De Morgan's law, the negation of (P ∧ Q) is (¬P ∨ ¬Q) — the AND flips to OR, and each component is negated. Option 0 mistakenly keeps AND, which is ¬P ∧ ¬Q, a stronger claim that requires both to be false. The correct negation only requires that at least one condition fail — hence OR.
Question 2 Multiple Choice
Statement P is true and statement Q is also true. What is the truth value of P ∨ Q?
AFalse — both being true makes 'or' ambiguous
BTrue — mathematical OR is inclusive, so it is true whenever at least one component is true, including when both are true
CUndefined — the truth value depends on context
DFalse — 'or' in logic means exactly one is true (exclusive or)
Mathematical OR (∨) is inclusive: P ∨ Q is true whenever at least one of P or Q is true — including when both are true. This differs from the exclusive 'or' common in everyday English ('cake or pie, not both'). In logic and mathematics, exclusive-or is a separate connective (XOR). Unless XOR is explicitly stated, assume inclusive OR.
Question 3 True / False
In formal logic, 'P or Q' is true primarily when exactly one of P or Q is true, not both.
TTrue
FFalse
Answer: False
Mathematical OR (disjunction, ∨) is inclusive: P ∨ Q is true whenever at least one of P, Q is true — this includes the case where both are true. The statement describes exclusive-or (XOR), which is a different connective. A common source of confusion is that everyday English 'or' often implies exclusivity, but formal logic defaults to inclusive OR.
Question 4 True / False
The statement 'x > 3 AND x < 7' is logically equivalent to the negation of '(x ≤ 3 OR x ≥ 7)'.
TTrue
FFalse
Answer: True
By De Morgan's law, ¬(A ∨ B) = ¬A ∧ ¬B. Here ¬(x ≤ 3 ∨ x ≥ 7) = (¬(x ≤ 3)) ∧ (¬(x ≥ 7)) = (x > 3) ∧ (x < 7). The two expressions describe identical sets of values. This illustrates how negation distributes through OR by flipping it to AND — a non-obvious but reliable transformation.
Question 5 Short Answer
Why does mathematical OR differ from the common English use of 'or,' and why does this distinction matter for constructing logical proofs?
Think about your answer, then reveal below.
Model answer: Mathematical OR (∨) is inclusive: P ∨ Q is true even when both P and Q are true. Everyday English 'or' often implies exclusivity (one but not both). The distinction matters in proofs because when you assume 'P or Q' as a hypothesis, you must handle the case where both hold — you cannot automatically rule it out. Proof by cases on a disjunction requires a case for 'both P and Q,' and omitting it produces an invalid proof.
Many proof errors come from treating OR as exclusive when it isn't. For example, proving a statement for 'n is even or n is prime' requires covering n = 2, which is both. Inclusive OR also interacts with negation via De Morgan's laws: ¬(P ∨ Q) = ¬P ∧ ¬Q, not ¬P ∨ ¬Q.