0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ It does fail transitivity, but it even fails reflexivity.... $\endgroup$ – Greg Martin Oct 14 '14 at 21:22
  • 1
    $\begingroup$ 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
1
$\begingroup$

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?)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.