Let R_1 and R_2 be two relations defined on R by a,R_1,b ab0 and a,R_2,b a b.

Question

Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by $a\,R_1\,b \iff ab\geq0$ and $a\,R_2\,b \iff a\geq b$. Then:

Accepted Answer

$R_1$: Reflexive ($x\cdot x=x^2\geq0$). Symmetric ($xy\geq0\iff yx\geq0$). Not transitive: $x=2,y=0,z=-2$ gives $xy=0\geq0$, $yz=0\geq0$, but $xz=-4<0$. So $R_1$ is not an equivalence relation.

$R_2$: $a\geq b$ does not imply $b\geq a$ --- not symmetric. So $R_2$ is not an equivalence relation.

Neither is an equivalence relation.

Solutions · 1