Given P → Q ('If it rains, the ground gets wet'), which of the following is logically equivalent to the original conditional?
AQ → P (If the ground is wet, it rained)
B¬P → ¬Q (If it didn't rain, the ground isn't wet)
C¬Q → ¬P (If the ground isn't wet, it didn't rain)
D¬P → Q (If it didn't rain, the ground is still wet)
The contrapositive ¬Q → ¬P is the only form logically equivalent to P → Q — they have identical truth values under every assignment. Option A is the converse and option B is the inverse; these are equivalent to each other but not to the original. The contrapositive merely reverses and negates both sides simultaneously, which preserves the logical content exactly.
Question 2 Multiple Choice
A mathematician wants to prove 'If n² is even, then n is even' but finds it easier to work with the negation of the conclusion. Which substitution is logically valid?
AProve 'If n is even, then n² is even' — the converse, which is equivalent
BProve 'If n is odd, then n² is odd' — the contrapositive, which is equivalent
CProve 'If n² is odd, then n is even' — by negating only the hypothesis
DEither the converse or the contrapositive — both are equivalent to the original
The contrapositive ('If n is odd, then n² is odd') is logically equivalent to the original, so proving it proves the original. This is a direct application of proof by contrapositive. The converse ('If n is even, then n² is even') is actually true here, but that's a coincidence of this particular statement — it is not equivalent in general. Option D is wrong because the converse is not equivalent to the original.
Question 3 True / False
The converse and contrapositive of a conditional statement are logically equivalent to each other.
TTrue
FFalse
Answer: False
The contrapositive (¬Q → ¬P) is logically equivalent to the original (P → Q). The converse (Q → P) is logically equivalent to the inverse (¬P → ¬Q). Converse and contrapositive are NOT equivalent to each other — they are equivalent to different things. Confusing these is one of the most common sources of invalid proofs.
Question 4 True / False
If P → Q is false, then its contrapositive ¬Q → ¬P must also be false.
TTrue
FFalse
Answer: True
The contrapositive is logically equivalent to the original — they have the same truth value in every possible scenario. If P → Q is false (P is true but Q is false), then substituting into the contrapositive: Q is false so ¬Q is true, and P is true so ¬P is false — making ¬Q → ¬P also false. They are not just 'related'; they are the same logical statement written differently.
Question 5 Short Answer
Why is substituting the contrapositive valid in a proof, while substituting the converse is not?
Think about your answer, then reveal below.
Model answer: The contrapositive (¬Q → ¬P) is logically equivalent to the original (P → Q) — they are true under exactly the same conditions. Proving one proves the other. The converse (Q → P) makes a different claim about the world; it may be true when the original is false, or false when the original is true. Substituting the converse would prove a different statement, not the one you set out to prove.
Logical equivalence is the key concept: two statements are equivalent if they always have the same truth value, regardless of what P and Q mean. Truth tables confirm that P → Q and ¬Q → ¬P share a truth column; P → Q and Q → P do not. This equivalence is what licenses the substitution in proof by contrapositive — you are not making an assumption, you are exploiting an identity.