The statement 'If it rains, the ground gets wet' (P → Q) is true. Which of the following must also be true?
AIf the ground gets wet, it rained (Q → P)
BIf it doesn't rain, the ground doesn't get wet (¬P → ¬Q)
CIf the ground is not wet, it did not rain (¬Q → ¬P)
DP and Q have the same truth value in every scenario
The contrapositive (¬Q → ¬P) is logically equivalent to the original implication (P → Q) — they have identical truth tables. The converse (Q → P) and inverse (¬P → ¬Q) are NOT equivalent to the original; they are equivalent to each other, but both can be false when the original is true. This is one of the most important equivalences in logic and the foundation of proof by contrapositive.
Question 2 Multiple Choice
A student checks one row of a truth table and finds that P is true and Q is true. She concludes P ≡ Q. Which error has she made?
AShe should have checked whether P and Q are both false instead
BLogical equivalence requires identical truth values in every possible row, not just one
CShe needs to use De Morgan's laws, not truth tables, to verify equivalence
DThere is no error — if both are true simultaneously, they are equivalent
Logical equivalence means P and Q must agree (both true or both false) in EVERY possible assignment of truth values to their atomic components. Finding one row where they agree proves nothing — you must check all rows. The most common misconception is confusing 'both happen to be true' with 'they always agree.' For example, 'it is raining' and 'the sky is blue' might both be true right now, but they are clearly not logically equivalent.
Question 3 True / False
De Morgan's law ¬(P ∧ Q) ≡ (¬P ∨ ¬Q) means that 'It is not the case that both P and Q are true' is logically equivalent to 'At least one of P or Q is false.'
TTrue
FFalse
Answer: True
This is exactly what De Morgan's law says. ¬(P ∧ Q) asserts the conjunction fails — P and Q are not both true. That is precisely the condition ¬P ∨ ¬Q: either P is false, or Q is false (or both). The truth tables of both sides match in every row. De Morgan's laws let you push negations inward through conjunctions and disjunctions, which is essential for rewriting logical statements in useful forms.
Question 4 True / False
If two statements P and Q are logically equivalent, then P and Q should be tautologies (true in most circumstances).
TTrue
FFalse
Answer: False
Logical equivalence only requires that P and Q agree with each other in every row — both true or both false in every assignment. They can both be false sometimes and true other times, as long as they always match. For example, 'P' and '¬¬P' are logically equivalent (double negation), but P itself is not a tautology — it can be true or false. Tautologies (like P ∨ ¬P) are true in every row, but equivalence is a relationship between two statements, not a property of one.
Question 5 Short Answer
Why can't you establish logical equivalence by checking just one or two scenarios, and what must you check instead?
Think about your answer, then reveal below.
Model answer: Logical equivalence requires that two statements produce the same truth value in every possible assignment of truth values to their atomic components. A single scenario where they agree shows only that they can match, not that they always match. You must verify every row of the joint truth table — all 2ⁿ combinations of truth values for n atomic variables — or use known equivalences as rewriting rules to transform one statement into the other.
This is the core of what makes equivalence a strong claim. Two statements might agree in most scenarios while differing in one edge case. That one disagreement is enough to destroy equivalence. Truth tables make this systematic: you must account for all inputs. Once you have a verified toolkit of equivalences (De Morgan, contrapositive, double negation, distributive laws), you can use them as guaranteed-correct rewriting rules without re-checking truth tables each time.