For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2}$ . Then $\equiv$ is an equivalence relation on $\mathbb{R}$ , there are infinitely many equivalence classes, one of them consists of one element and the rest consist of two elements.
Solution
True. To show that \equiv is reflexive we need to show that \forall x\in\mathbb{R} :x=x. Let x\in\mathbb{R} , x\equiv\mbox{\ensuremath{x}} if x^{2}=x^{2}, which is obvious.
[x] ={y\in\mathbb{R} |x\equiv y} =[0]={y\in\mathbb{R}|0\equiv y}={0} . Hence y^{2} =0^{2} =0 which implies that y=0 .