Coprimality Relation is Symmetric

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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.

$\blacksquare$