Even Integer Plus 5 is Odd/Direct Proof

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $x \in \Z$ be an even integer.

Then $x + 5$ is odd.


Direct Proof

Let $x$ be an even integer.

Then by definition:

$x = 2 n$

for some integer $n$.

Let $y = 2 n + 5$.

Then:

\(\ds y\) \(=\) \(\ds 2 n + 5\)
\(\ds \) \(=\) \(\ds 2 n + 2 \times 2 + 1\)
\(\ds \) \(=\) \(\ds 2 \paren {n + 2} + 1\)
\(\ds \) \(=\) \(\ds 2 r + 1\) where $r = n + 2 \in \Z$

Hence $y = 2 n + 5$ is an odd integer by definition.

$\blacksquare$


Historical Note

There is nothing profound about this result.

Gary Chartrand used it as a simple demonstration of the construction of various kinds of proof in his Introductory Graph Theory of $1977$.

It is questionable whether the indirect proof and the Proof by Contradiction actually constitute different proofs of this result, but both are included on $\mathsf{Pr} \infty \mathsf{fWiki}$ anyway, in case they are found to be instructional.


He sets a similar theorem as an exercise:

Prove the implication "If $x$ is an odd integer, then $y = x - 3$ is an even integer" using the three proof techniques: ...

but it has been considered not sufficiently different from this one to be actually included on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a separate result to be proved.


For similar reasons, several other of the trivial exercises in applied logic that he sets have also been omitted from this site.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Even_Integer_Plus_5_is_Odd/Direct_Proof&oldid=390384"