For $A$:
$P(E_1\cup E_2)=P(E_1)+P(E_2)-P(E_1\cap E_2)$
$=\frac{1}{2}+\frac{1}{3}-\frac{1}{5}$
$=\frac{19}{30}$ $\neq \frac{2}{3}$ $\therefore$ $A$ is not True
For$B$:
If $E_1$ and $E_2$ are independent then
$P(E_1\cap E_2)=P(E_1)P(E_2)$
$=\frac{1}{2}\times\frac{1}{3}$
$=\frac{1}{6}$ $\neq \frac{1}{5}$ $\therefore$ $B$ is not True
For $D$:
$P\left ( \frac{E_1}{E_2} \right )=\frac{P(E_1\cap E_2)}{P(E_2)}$
$=\frac{\frac{1}{5}}{\frac{1}{3}}=\frac{3}{5} \neq \frac{4}{5}$ $\therefore$ $D$ is not True
So, the answer is $C$, $E_1$ and $E_2$ are not independent