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.

  • $\begingroup$ It does fail transitivity, but it even fails reflexivity.... $\endgroup$ – Greg Martin Oct 14 '14 at 21:22
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    $\begingroup$ TeX remarks: use \langle and \rangle ($\langle$ and $\rangle$) instead of less-than and greater-than ($<$ and $>$) for better spacing of these delimiters; and use \in $\in$ instead of \epsilon ($\epsilon$) for the set-membership relation. $\endgroup$ – Greg Martin Oct 14 '14 at 21:22
  • $\begingroup$ I changed $<\bullet,\bullet>$ to $\langle\bullet,\bullet\rangle$, $a$~$b$ to $a\sim b$, $a,b\,\epsilon\,\bullet$ to $a,b\in\bullet$, and {$a,b$} to $\{a,b\}$. All of that is standard usage. ${}\qquad{}$ $\endgroup$ – Michael Hardy Oct 14 '14 at 21:32

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?)


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