Proving n² odd implies n odd

Intro to Proofs · Axiom Academy

EXAMPLE Proving n^2 odd implies n odd Master the contrapositive proof technique with a classic number-theory theorem Prove that for every integer n , if n^2 is odd, then n is odd. We'll do it the slick way — by proving the contrapositive . Excellent work — you just ran a complete contrapositive proof. Here's what made it tick: When a direct proof is awkward: proving head-on means reasoning backwards from n^2 . The contrapositive sidesteps that. Logical equivalence: a statement and its contrapositive are equivalent — proving one proves the other. The easier road: "if n is even, then n^2 is even" is a clean forward computation: . Definitions do the work: we used n = 2k (even) and "a multiple of 2 is even" to drive the algebra to its conclusion. Proof skeleton: assume the hypothesis → apply definitions → manipulate algebraically → land the conclusion → invoke the contrapositive. The contrapositive is one of the most reliable moves in a prover's toolkit. Whenever the direct direction looks tangled, check whether is easier — in number theory and analysis, it often is.

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