Even Integer Plus 5 is Odd/Indirect Proof
Theorem
Let $x \in \Z$ be an even integer.
Then $x + 5$ is odd.
Proof
Let $x$ be an even integer.
Let $y = 2 n + 5$.
Assume $y = x + 5$ is not an odd integer.
Then:
- $y = x + 5 = 2 n$
where $n \in \Z$.
Then:
\(\ds x\) | \(=\) | \(\ds 2 n - 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 n - 6} + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \paren {n - 3} + 1\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 r + 1\) | where $r = n - 3 \in \Z$ |
Hence $x$ is odd.
That is, it is false that $x$ is even.
It follows by the Rule of Transposition that if $x$ is even, then $y$ is odd.
$\blacksquare$
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
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Example $\text A.3$