Even Integer Plus 5 is Odd

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.

$\blacksquare$

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

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.