Which Is the Contrapositive? A Ten-Question Quiz With Quantifier Traps

Identifying the contrapositive of a clean textbook implication is easy: swap and negate. Identifying the contrapositive of a statement with quantifiers ("for all", "there exists") or compound conditions ("a is odd and b is even") is where almost everyone slips. The negation rules apply, but they apply in a specific order, and one wrong step lands you on the converse or the inverse instead.

This quiz gives you ten statements, each with three candidate rewrites. Drag the slider to the statement number; the diagram highlights the correct contrapositive and exposes the two tempting wrong answers.

The quiz

Question:

Ten original implications, one at a time. The green panel shows the correct contrapositive (use Reveal to hide and test yourself). The two grey panels show the common traps: the converse (swap only) and the inverse (negate only). The trap-watch note explains the specific slip each question rewards.

How to run the playbook on any statement

Every one of the ten items follows the same three-step playbook, applied carefully.

Identify the hypothesis P and conclusion Q. Separate them cleanly. If the statement begins with "for all", isolate the inner implication.
Negate both. Compute \lnot P and \lnot Q. Apply De Morgan when either side is a compound (AND becomes OR; OR becomes AND). Apply the quantifier flip when either side contains "for all" or "there exists" (∀ becomes ∃¬; ∃ becomes ∀¬).
Swap and rewrite. The contrapositive is \lnot Q \Rightarrow \lnot P.

Why the order matters: if you negate without swapping, you get the inverse — a different statement. If you swap without negating, you get the converse — also a different statement. Only the combination of both operations returns you to an equivalent form. See The Flipper — Watch P ⇒ Q Rotate 180° Into ¬Q ⇒ ¬P for the rotation-based visual of why both steps are required.

The four trap families

Reading through the ten questions, you see the same four trap families show up again and again:

Plain-swap trap. Hypothesis and conclusion trade places, but no negations are added. Produces the converse. Question 2 (x = 3 \Rightarrow x^2 = 9) is the textbook version; x = -3 is the counter-example that exposes it.
Plain-negate trap. Both sides get a "not" but stay in position. Produces the inverse. Seen in questions 5 and 10.
De Morgan trap. A compound "and" or "or" in the hypothesis or conclusion requires you to flip the connective when negating. Students often forget: the negation of "a = 0 or b = 0" is "a \neq 0 and b \neq 0," not "a \neq 0 or b \neq 0." Questions 6 and 9 live here.
Quantifier trap. The implication lives under a universal quantifier ("for all x..."). The quantifier itself does not flip in the contrapositive — it stays as "for all x" and only the inner implication flips. If you try to negate the quantifier, you have produced the negation of the whole statement, which is neither the contrapositive nor the converse. See Negation Lab — ∀ Becomes ∃¬, and ∃ Becomes ∀¬ for the quantifier-flip rules in isolation.

Worked pass through Q6

Original: "If ab = 0, then a = 0 or b = 0."

P: ab = 0.
Q: a = 0 or b = 0.
\lnot Q: not "a = 0 or b = 0" — which by De Morgan is "a \neq 0 and b \neq 0."
\lnot P: ab \neq 0.
Contrapositive: "If a \neq 0 and b \neq 0, then ab \neq 0."

The De Morgan step is the point of failure. Students often write the contrapositive as "If a \neq 0 or b \neq 0, then ab \neq 0" — keeping the "or" from the original Q. That statement is strictly weaker and, in fact, false (take a = 1, b = 0: then "a \neq 0 or b \neq 0" is true, but ab = 0, so the claim fails). De Morgan turned the "or" into an "and" during negation, and missing that step gives the wrong answer.

When the quiz is easy vs. when it is hard

The questions with the cleanest one-word hypotheses ("even", "odd", "differentiable") are the easiest — just negate the word and swap. The questions with compound conditions (AND, OR) and quantifiers are the hard ones, because the negation interacts with the logical structure. A good rule: the harder the statement, the more important it is to write the negation step explicitly on paper before you claim the contrapositive.

Every time you do a proof by contrapositive in an actual theorem, this quiz is the first half of the problem — you are reading a statement and computing its contrapositive before you prove anything. Getting that computation wrong means your proof is about a different theorem. Make sure you are proving the right statement, and the hardest part is often already behind you.