The Contrapositive Method

Intro to Proofs · Axiom Academy

LESSON The Contrapositive Method A powerful proof technique: instead of proving p → q directly, prove ¬q → ¬p. They are logically equivalent. 1. What Is the Contrapositive? The contrapositive flips the arrow and negates both parts of an implication. Watch p → q reverse into ¬q → ¬p: the arrow turns around and each statement is replaced by its negation. Remarkably, the result is always logically equivalent to the original. An implication p → q is false in exactly one situation : when p is true but q is false. In every other case it is true. The animation builds the truth table and highlights that single false row — and shows that ¬q → ¬p fails on the same row, so the two columns are identical top to bottom. The one false row is p = T, q = F. A true hypothesis cannot lead to a false conclusion. On that same row ¬q = T and ¬p = F — a true hypothesis with a false conclusion. Same failure, same row. Matching in all four rows means the two statements have the same truth value, always. Proving one is, logically, proving the other — pick whichever is easier. Reach for the contrapositive when the negation of the conclusion (¬q) hands you something concrete to grab, while the hypothesis (p) is vague. The animation contrasts the two starting points: a direct proof that stalls on abstract information, versus a contrapositive that starts from ¬q and walks straight to the goal. Starting from p gives vague or abstract information that is hard to manipulate.

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