What is the truth value of the statement: 'If the moon is made of cheese, then 2 + 2 = 5'?
ATrue, because both parts are false and they cancel out
BFalse, because the conclusion is false
CTrue, because the hypothesis is false, making the implication vacuously true
DUndefined, because neither part is a meaningful mathematical claim
A conditional P → Q is false ONLY when P is true and Q is false. Here P ('the moon is made of cheese') is false, so the implication is vacuously true — it made a promise only about what happens when P holds, and P never holds. The truth of Q is irrelevant.
Question 2 Multiple Choice
Two students debate the meaning of 'P OR Q' in mathematics. Student A says it means exactly one of P or Q is true (but not both). Student B says it means at least one of P or Q is true (including the case where both are true). Which student is correct?
AStudent A — mathematical OR is exclusive, like everyday 'either/or'
BStudent B — mathematical OR is inclusive, true whenever at least one component is true
CBoth are correct — context determines which interpretation applies
DNeither — OR is only defined when P and Q have opposite truth values
Mathematical disjunction (∨) is inclusive OR: P ∨ Q is true whenever at least one of P, Q is true, including the case where both are true. This differs from everyday English 'either/or,' which often implies exclusivity. Student A's version is called XOR (exclusive or) and is a different connective.
Question 3 True / False
The sentence 'What is the square root of 9?' is a false statement because the answer is 3, not implied.
TTrue
FFalse
Answer: False
This sentence is not a statement at all — it is a question. Statements are declarative sentences that have a definite truth value (true or false). Questions, commands, and exclamations are not statements and cannot be called true or false. Logic only operates on statements.
Question 4 True / False
The conditional statement P → Q is logically equivalent to ¬P ∨ Q.
TTrue
FFalse
Answer: True
This equivalence is fundamental: 'If P then Q' is false only when P is true and Q is false — the same conditions under which ¬P ∨ Q is false (¬P is false and Q is false). Checking all four truth-value combinations confirms they always agree. This equivalence is used to transform implications into disjunctions, which is often easier to reason about.
Question 5 Short Answer
Explain why the statement 'If it is raining, then the ground is wet' can be TRUE on a sunny day when it is not raining and the ground is also dry.
Think about your answer, then reveal below.
Model answer: A conditional P → Q is vacuously true whenever P is false, regardless of Q's truth value. The statement makes a promise only about what happens when P (it is raining) holds — if it never rains, the promise is never violated.
Vacuous truth is one of the most counterintuitive features of material implication. The conditional is a guarantee: whenever rain occurs, wetness follows. On a dry, sunny day, the rain condition never activates, so there is no opportunity to violate the guarantee. This is not a trick — it reflects the logical structure of 'if-then' claims, which assert nothing about what happens when the hypothesis is false.