What are the relation properties of = {(, ) | gcd(, ) = 1}

Question

I am guessing that it is reflexive since it relates to itself $gcd(x,y)=gcd(x,y)$
But I can't think of a way to picture the relation to find if it's transitive and symmetric.

Answer 1

I think you've mixed up "reflexive" which asserts every element is related to itself $xRx$ (and is not true here) with "symmetric" which says all relations are two-way, $xRy \iff yRx$ (which is true).

Because the relation is symmetric but not reflexive, we immediately know that it is not transitive, since $xRy$ and $yRx$ do not lead to $xRx$.

As for picturing the relation, in plain words it says $x$ and $y$ are coprime, or have no common factors greater than $1$. Since $x$ definitely has a common factor with itself, that eliminates reflexivity easily, and the rest follows as above.