Reversing the Flow

Intro to Proofs · Axiom Academy

Sometimes proving you can't get somewhere is easier going backwards. Meet the contrapositive — and watch an implication turn around. You're on a river with a current. Some destinations are easy to reach, and some are flat-out impossible — but proving impossibility by trying every way to steer is hopeless: there are infinitely many strategies. The trick is to stop pushing forward and reverse the flow : ask not "can I get there?" but "if I were there, where must I have come from?" Watch one implication turn into another. The forward claim start at A → reach B flips: the arrow reverses, each end negates, and it settles on its contrapositive not‑B → not‑A . Same truth, read the other way. A statement and its contrapositive are two ways of saying the very same thing — the arrow just points the other way. The river: when forward fails, reverse Your boat rows at 3 units/sec ; the current pushes right . Drag the current strength. While it's weaker than your boat you can crawl upstream to the target. Push it past 3 and the current wins — you always drift right, so the target is unreachable. The teal band is the contrapositive answer: the only places you could have started and still reached the target. Watch it slide off your actual start. Forward: "start at A → reach B." Reversed: "reached B → must have started in the band." Your start sits outside the band ⟹ B is impossible. Why reversing is allowed: same truth, every time

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