>[!summary] A direct proof is a type of proof in which you assume the hypothesis from the proposition to be true, then using logic proving the conclusion to be true. >[!info]+ Read Time ⏱ **1 min** # Definition A direct proof is a type of proof that specifically applies [[Conditional Statements|conditional statements]]. This type of proof follow this step structure: 1. Assume that hypothesis from [[Propositions|proposition]] is true; 2. Then using definitions, proven results or facts to justify why the conclusion must be true Proofs in physics often uses this proof, and is likely the most common you will see on this site. ## Examples >[!example] If $a$ and $b$ are [[Even & Odd Numbers|even]] [[Logic/Integers.md|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 the 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! ---