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

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?

## 1 Answer

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 $apc$.

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

• OP: and the equivalence classes are? – Sammy Black Oct 21 '14 at 1:57
• 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? – I hate math Oct 21 '14 at 2:09
• Is it the case of since sin a = sin b \iff sin a - sin b = 0 – I hate math Oct 21 '14 at 2:32
• It is: $[a]=\{b\in \Bbb R: \sin a=\sin b\}$. – Paul Oct 21 '14 at 3:15