>[!summary] Proof by cases applies conditional statements where you create cases in the hypothesis that return to the same conclusion. >[!info]+ Read Time ⏱ **2 mins** # Definition Proof by cases is very similar to [[Direct Proof|direct proof]] in that it applies [[Conditional Statements|conditional statements]], but the step structure differs slightly: 1. Assume the hypothesis from the proposition is true (**Hypothesis has cases, e.g. Hypothesis is an integer and irrational number**) 2. Then, using definitions, proven results or facts to justify why the conclusion must be true The highlighted part of the steps is the difference between proof by cases and [[Direct Proof|direct proof]] ## Examples >[!example] Absolute value function for all real numbers x For all real numbers x, > $ |x| = \begin{cases} x, & \text{if } x \ge 0 \\ -x, & \text{if } x < 0 \end{cases} >$ **Proof by cases:** > Case 1: $x \geq 0$ Then the $|x| = x$ by definition > Case 2: $x \leq 0$ Then the $|x| = -x$ by definition, since x is negative and taking the negative of a negative returns positive --- >[!example] For any [[Integers|integer]] $n, \space n^2$ is even if and only if $n$ is [[Even & Odd Numbers|even]] > **Proof by cases:** > Case 1: $n$ is even If n is even, assume $n = 2k$ where k is an integer > Then $n^2 = 4k^2$, which will always be even > Case 2: $n$ is [[Even & Odd Numbers|odd]] If n is odd assume $n = 2k + 1$ where k is an integer > Then >$ \begin{array}{c} n^2 = (2k+1)^2 = 4k^2 + 4k+1 = 2(2k^2 + 2k ) +1 \\ \text{Assume $2k^2 + 2k = m$ where $m$ is also an integer by defintion} \\\\ 2(2k^2 + 2k ) +1 = 2m +1 \end{array} >$ It is in the form of an odd number, so $n^2$ returns an odd number, proving the statement. --- <!-- Light Mode Newsletter Embed --> <div class="mm-form-light"> <iframe src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17" id="inline-Y0kpQVpjJQuxEfX59m17" title="Join Math & Matter Newsletter (Light)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Dark Mode Newsletter Embed --> <div class="mm-form-dark"> <iframe src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1" id="inline-lbeDLm24VjuaFxhjccA1" title="Join Math & Matter Newsletter (Dark)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Provider script (only once) --> <script src="https://updates.cyberleadhub.com/js/form_embed.js"></script>