# Coprimality Relation is Symmetric

## Theorem

Consider the coprimality relation on the set of integers:

$\forall x, y \in \Z: x \perp y \iff \gcd \set {x, y} = 1$

where $\gcd \set {x, y}$ denotes the greatest common divisor of $x$ and $y$.

Then:

$\perp$ is symmetric.

## Proof

 $\displaystyle x$ $\perp$ $\displaystyle y$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \gcd \set {x, y}$ $=$ $\displaystyle 1$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \gcd \set {y, x}$ $=$ $\displaystyle 1$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $\perp$ $\displaystyle x$ $\quad$ $\quad$

Hence the result by definition of symmetric relation.

$\blacksquare$