Binary relation basic question

Question

We have a relation r between two non-zero integers, can the relation i r j $\leftrightarrow $ pgcd(i,j) =1 be an equivalence relation?

My guess is no, because if i=j pgcd is not 1, so reflexivity is not fulfilled here. Is this a correct approach?

Answer 1

I'm assuming you are asking whether $i \, R \, j \Leftrightarrow\mathrm{gcd}(i,j)=1$ defines an equivalence relation on the set of integers. The answer is no. The relation $R$ is not reflexive, as you noticed already. It is not transitive either: $\mathrm{gcd}(i,j) = \mathrm{gcd}(j,k) = 1$ does not imply $\mathrm{gcd}(i,k)=1$. Take $i,j,k = 3,5,6$ as a counterexample.