Let us consider a variable X with values $\{-1,1\}$, and another one Y with values $\{-2,-1,1,2\}$.
X and Y have the following joint probability function $P(x,y)$:
From this table, the covariance is zero.
Does this mean that there are no linear relationships between X and Y (because covariance function should measures linear relationships)?
Then, is it possible to calculate the non linear relation between X and Y or, at least, prove that their relationship is non linear?
In this case, $y$ takes the values $\pm\left(\frac{3}{2}x-\frac{1}{2}\right)$ with equal probabilities. Since $y$ take both positive and negative values with equal probabilities, we can't say that $y$ increases or decreases with $x$. However, it is not possible to deduce such relationships easily, for a general case.