I have a problem with this task. If anyone had a similar problem it would help me.
The task is:
In the set $S\in[-\pi,\pi]$ a binary relation is defined ρ with $xρy⟺sgn(\sin(x-\pi))=sgn(\sin(y))$. Examine whether the relation is an equivalence relation.
I tried this:
Reflexivity: $(\forall x \in S) x\rho x$
$sgn(\sin(x-x))=sgn(\sin(x))\\sgn(0)=sgn(sin(x))\\sgn(\sin(x))=0\\\sin(x)=0\\x=arcsin(0)=0\in [-\pi,\pi]$
The relation is reflexive
Symmetry:
$(\forall x,y \in S) x\rho x \iff y\rho x\\sgn(\sin(x-\pi))=sgn(\sin(y))\iff sgn(\sin(y)))=sgn(\sin(x-\pi))$
The relation is symmetric.
And now I don't know how to prove transitivity ?
Thanks in advance !