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
\(\ds x\) | \(\perp\) | \(\ds y\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \gcd \set {x, y}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \gcd \set {y, x}\) | \(=\) | \(\ds 1\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(\perp\) | \(\ds x\) |
Hence the result by definition of symmetric relation.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $4 \ \text{(a)}$