>[!summary] Proof by contraposition is assuming the negation of our conclusion, and proving our hypothesis is false. >[!info]+ Read Time ⏱ **2 mins** # Definition Proof by contraposition is the opposite of a [[Direct Proof|direct proof]] in that, instead of proving that our conclusion is true by assuming our hypothesis is true, we assume our conclusion is false and then prove our hypothesis is false ([[Conditional Statements|conditional statements]]). In other words, we follow these steps: 1. Assume the negation of our conclusion (B) 2. Using logic and definitions, we prove that our hypothesis (A) is false Or mathematically $\neg B \to \neg A$ ## Examples >[!example] Claim: If $n^2$ is [[Even & Odd Numbers|even]] then $n$ is even. > Proof by contraposition: If $n^2$ is [[Even & Odd Numbers|odd]] then $n$ is odd > Let $n = 2k +1$, where k is an integer, so $n$ is odd by definition > Then: >$ n^2 = (2k +1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) +1 >$ If we let $2k^2 + 2k = m$ where m is also an integer, then by simplification: >$ 2(2k^2 +2k) +1 = 2m +1 >$ This by definition, an odd number. > We proved $n^2$ if odd than $n$ is odd, so then we proved $n^2$ if $n$ is even. --- >[!example] Claim: If $ab$ is odd then $a$ and $b$ are odd. > Proof by contraposition: If either $a$ or $b$ are even then $ab$ is even > If we let $a$ be even $a = 2k$ where k is an integer > Then: $ab = 2(kb)$ >Which is always even (also true if we let $b$ be even) > Thus proving our claim --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. ---