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