Even Integer Plus 5 is Odd

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Theorem

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

Then $x + 5$ is odd.


Direct Proof

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

Then $x + 5$ is odd.


Let $x$ be an even integer.

Then by definition:

$x = 2 n$

for some integer $n$.

Let $y = 2 n + 5$.

Then:

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

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

$\blacksquare$


Indirect Proof

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

Then $x + 5$ is odd.


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:

\(\displaystyle x\) \(=\) \(\displaystyle 2 n - 5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 n - 6} + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \paren {n - 3} + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2 r + 1\) 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.


$\blacksquare$


Proof by Contradiction

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

Then $x + 5$ is odd.


Let $x$ be an even integer.

Then by definition:

$x = 2 n$

for some integer $n$.


Aiming for a contradiction, suppose $y = x + 5 = 2 m$ for some integer $m$.

Then:

\(\displaystyle x\) \(=\) \(\displaystyle 2 m - 5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 m - 6} + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \paren {m - 3} + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2 r + 1\) where $r = m - 3 \in \Z$

Hence $x$ is odd.

But this contradicts our premise that $x$ is even.

Hence the result by Proof by Contradiction.

$\blacksquare$


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.