Consider $\langle\Bbb{Z}_6, +_6\rangle$. Let $a\sim b$ if and only if $\{a,b\}$ generates $\langle\Bbb{Z}_6, +_6\rangle$. $a,b \in \Bbb{Z}_6$. Is $\sim$ an equivalence relation?
I know an equivalence relation must have the properties of being reflexive, symmetric, and transitive. I believe the relation described above fails transitivity. Any thoughts would be appreciated.
It is not an equivalence relation. In fact, it is not (even) reflexive, since $\{2,2\}$ does not generate $\mathbb{Z}/6$. It also fails one of the other two properties of an equivalence relation (which one?)