User:Ascii/Coprime Relation for Integers is Non-Reflexive

Theorem

The relation "is coprime to" on the integers is non-reflexive.

That is, it is neither reflexive:

$\exists n \in \Z: \neg n \perp n$

nor antireflexive:

$\exists n \in \Z: n \perp n$

where $\perp$ denotes "is coprime to".

Proof

Coprime Relation for Integers is Not Reflexive

Consider $2$.

From Integer Divides Itself, $2$ is a divisor of $2$.

From Absolute Value of Integer is not less than Divisors, $2$ is the greatest divisor of $2$.

So, the greatest common divisor of $2$ and itself is $2$: $\gcd \set {2, 2} = 2$

Hence $\neg \, 2 \perp 2$ and $\perp$ is not reflexive.

$\Box$

Coprime Relation for Integers is Not Antireflexive

Consider $1$.

From Integer Divides Itself, $1$ is a divisor of $1$.

From Absolute Value of Integer is not less than Divisors, $1$ is the greatest divisor of $1$.

So, the greatest common divisor of $1$ and itself is $1$: $\gcd \set {1, 1} = 1$

Hence $1 \perp 1$ and $\perp$ is not antireflexive.

$\blacksquare$