>[!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
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>[!example] If $n$ is [[Even & Odd Numbers|odd]] then $n^2$ must be odd as well.
>
Direct proof:
If we assume:
$n = 2k + 1 \quad | k \in \mathbb{Z}$
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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
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