The Flipper

"If P then Q" looks like a one-way street. Hypothesis on the left, conclusion on the right, an arrow in between. But hidden inside every implication is its contrapositive — the same statement rotated and flipped, with both parts negated and their roles swapped. The astonishing fact is that these two forms are logically equivalent: they carry identical information. Proving one proves the other.

The flipper below makes this visible. Drag the rotation slider and watch P \Rightarrow Q pivot 180°. As the rotation completes, the card lands as \lnot Q \Rightarrow \lnot P and turns green — confirming that the two forms agree on every possible truth assignment.

Rotation:

The flipper. At $t = 0$ the card shows the original $P \Rightarrow Q$. As you drag, it rotates through an intermediate phase (neither form is fully readable mid-rotation) and lands at $\lnot Q \Rightarrow \lnot P$. Both endpoints display the truth-value row $T, F, T, T$ — the two statements agree on every combination of $P$ and $Q$.

Why the rotation captures the equivalence

The rotation is not just aesthetic. Two things happen during the flip:

Positional swap. The hypothesis and conclusion trade places. P moves from left to right; Q moves from right to left.
Sign flip. Each proposition gets a negation. P becomes \lnot P; Q becomes \lnot Q.

Together these produce \lnot Q \Rightarrow \lnot P. Either operation on its own would give a different statement: swap alone produces the converse Q \Rightarrow P, and negate alone produces the inverse \lnot P \Rightarrow \lnot Q. Only swap and negate together gives the contrapositive.

Why both operations are needed for equivalence: a truth table has four rows, one for each combination of P and Q. The original P \Rightarrow Q produces the pattern T, F, T, T across these rows. Swapping P and Q (the converse) produces T, T, F, T — different. Negating both (the inverse) also produces T, T, F, T — same as the converse, different from the original. But the combination of swap-and-negate returns you to T, F, T, T, matching the original. The two individual operations cancel out in truth-table terms, and only their combination preserves the pattern.

A quick check on a concrete statement

Take the true claim: "If n^2 is odd, then n is odd." Set P = "n^2 is odd" and Q = "n is odd."

Original (P \Rightarrow Q): If n^2 is odd, then n is odd. True.
Contrapositive (\lnot Q \Rightarrow \lnot P): If n is not odd (i.e., even), then n^2 is not odd (i.e., even). Also true — because n = 2k gives n^2 = 4k^2, which is even.

Both statements are true, for the same underlying reason. Proving either one in any valid way proves the other. This is exactly what proof by contrapositive exploits: if you find the forward implication hard, try the backward negated one. The two are interchangeable.

The two patterns they should not be confused with

The flipper shows one equivalence. Two close relatives are not equivalent to the original:

Converse Q \Rightarrow P: "If n is odd, then n^2 is odd." This one happens to be true here, but for a general P and Q the converse is a separate claim. The converse of "if it is raining, the ground is wet" is "if the ground is wet, it is raining" — obviously different.
Inverse \lnot P \Rightarrow \lnot Q: "If n^2 is not odd, then n is not odd." Equivalent to the converse, not the original.

Two of the four forms match, two do not. The flipper shows the matching pair. Do not let the rotation trick you into thinking converse and inverse are the same game — they are different theorems and need their own proofs.

The one-line takeaway

The contrapositive is the original rotated 180° and re-painted: same content, different wrapping. Proving \lnot Q \Rightarrow \lnot P is proving P \Rightarrow Q. The flipper is the animation of that fact.