# direct proof

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## What do we mean by a direct proof?

$\text{In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given}$ $\text{statement by a straightforward combination of established facts, usually axioms, existing lemmas}$ $\text{and theorems, without making any further assumptions.}$

Directly prove that if n is an odd integer then $n^2$ is also an odd integer.

Let $p$ be the statement that $n$ is an odd integer and $q$ be the statement that $n^2$ is an odd integer. Assume that $n$ is an odd integer, then by definition $n = 2k + 1$ for some integer $k$. We will now use this to show that $n^2$ is also an odd integer.

$n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1$

Hence we have shown that $n^2$ has the form of an integer since $2k^2+2k$ is an integer.Our

direct proofends here. Statement $p$ holds true.