User:Ascii/Coprime Relation for Integers is Not Antireflexive
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Theorem
The relation "is coprime to" on the integers is not antireflexive.
That is:
- $\exists n \in \Z: n \perp n$
where $\perp$ denotes "is coprime to".
Proof
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.
$\blacksquare$