My question is: For the relation $p$ described below, determine if $p$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $p$ is an equivalence relation, describe the equivalence classes.

$p$ = $\mathbb{R}$, and $a\mathrel{p} b$ if and only if $\sin a$ = $\sin b$

Okay so now we know that $p$ is symmetric, reflexive and transitive, we know that it is an equivalence relation. Although how would I describe the equivalence classes?



reflexive, $a p a$ since $\sin a=\sin a$.

symmetric, If $a p b$, then $\sin a=\sin b$, and hence $\sin b=\sin a$, which shows that $b p a$ .

transitive, If $apb$ and $bpc$, then $\sin a=\sin b$ and $\sin b=\sin c$, and hence $\sin a=\sin c$.

So $apc$.

$[a]=\{b\in \Bbb R: \sin a=\sin b\}$.

  • $\begingroup$ OP: and the equivalence classes are? $\endgroup$ Oct 21 '14 at 1:57
  • $\begingroup$ Thank you for your answer Paul, I have a whole bunch of these questions but now I realise they are quite simple. Also Sammy Black is correct.. How would I go about describing the equivalence classes? $\endgroup$ Oct 21 '14 at 2:09
  • $\begingroup$ Is it the case of since sin a = sin b \iff sin a - sin b = 0 $\endgroup$ Oct 21 '14 at 2:32
  • $\begingroup$ It is: $[a]=\{b\in \Bbb R: \sin a=\sin b\}$. $\endgroup$
    – Paul
    Oct 21 '14 at 3:15

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