Let $S$ denote all integers. Define a relation $R$ on $S$ as $aRb$ if $ab\ge 0$ where $a,b\in S$. Then $R$ is:

1. Reflexive: $aRa\Rightarrow a\cdot a=a^2$. Since $a^2\ge 0$ for all integers, $R$ is reflexive.2. Symmetric: If $aRb\Rightarrow ab\ge 0$. Since $ab=ba$, then $ba\ge 0\Rightarrow bRa$. $R$ is symmetric.3. Transitive: If $aRb$ and $bRc\Rightarrow ab\ge 0$ and $bc\ge 0$.Counterexample: Let $a=-1$, $b=0$, $c=1$.$a\cdot b=(-1)(0)=0\ge 0$ (True)$b\cdot c=(0)(1)=0\ge 0$ (True)But $a\cdot c=(-1)(1)=-1<0$ (False).So, $R$ is not transitive.Conclusion: Reflexive, symmetric but not transitive.