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 proof ends here. Statement \(p\) holds true.