Contrapositive vs Contradiction

Intro to Proofs · Axiom Academy

LESSON Contrapositive vs Contradiction Two indirect proofs that look alike — and the one structural difference that tells you which to reach for. To prove "If P , then Q " , you instead prove its contrapositive : "If not Q , then not P ." Watch the implication below flip — the arrow reverses and both ends get negated. The two statements are logically equivalent , so proving the flipped one proves the original. To prove a statement S , assume it is false ( ), then follow valid steps until you hit an impossibility — a fact and its own negation at once. Below, the assumption travels down the chain and slams into the wall where R meets . Since leads to nonsense, S must be true. 3. Same Theorem, Two Structures Let's prove one theorem both ways and watch the shape of each argument. Notice that contrapositive is a single straight road to a true statement, while contradiction loops back to collide with its own starting assumption. 4. Which Method Should You Use? Here's the decision in one picture: ask whether the claim is a clean "If P then Q " . If yes, the token rolls left to contrapositive ; if it's an existence claim or already negative, it rolls right to contradiction . "If P then Q " with easy negations → contrapositive "There exists no x such that…" → contradiction " is irrational" → contradiction "If n^2 is even then n is even" → contrapositive (cleaner) Multiple conditions must combine → contradiction

This is the written version of the interactive lesson above. See the full Intro to Proofs course.