>[!summary] A direct proof is a type of proof in which you assume the hypothesis from the proposition to be true, then use logic to prove the conclusion to be true. >[!info]+ Read Time ⏱ **2 mis** # Definition A direct proof is a type of proof that specifically applies to [[Conditional Statements|conditional statements]]. This type of proof follows this step structure: 1. Assume that the hypothesis from the [[Propositions|proposition]] is true; 2. Then, using definitions, proven results or facts to justify why the conclusion must be true ## Examples >[!example] If $a$ and $b$ are [[Even & Odd Numbers|even]] [[Integers|integers]], then a + b must be equal. > Direct Proof: If we assume: $a = 2m$, $b = 2n$ where $m,n \in \mathbb{Z}$ > Then: $(a+b) = 2m + 2n = 2(m+n)$ Since m and n are [[Integers|integers]] the result is divisible by 2, hence even --- >[!example] If $n$ is [[Even & Odd Numbers|odd]] then $n^2$ must be odd as well. > Direct proof: If we assume: $n = 2k + 1 \quad | k \in \mathbb{Z}$ > Then: >$\begin{array}{c} n^2 = (2k + 1)^2 = 2(2k^2 + 2k) + 1 \\ \\ \text{Let $m = 2k^2 + 2k$} \\ \\ 2(2k^2 + 2k) + 1 = 2m + 1 \end{array}$ > $2m + 1$ is in the form of an odd number, and thus is odd --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidian-MathMatter) to support the project and help others discover it! ---