I'm given the following expression:
$I=((xw \implies (y \bar z)) \Leftrightarrow x \bar w) \overline{(xy \bar z w)}$
This expression simplifies to $x \bar w \lor x \bar y \lor xz$.
Now, I know that an expression $f$ is independent from variable $x$ if: $\frac{\partial f}{\partial x} = 0$
The text of the exercise asks: For which values of x,z does the expression I not depend on the variable y, but depend on the variable w
What I tried:
I have that $\frac{\partial I}{\partial x} = xw(1 \oplus z) = 0$.
Now, this expression is false for all $x,w$ since it is a conjunction and $(1 \oplus z)$ is always negative, right? If so, is the answer to the first part of the question that it doesn't depend on $y$ for all values $x,z$?
As for the second question: I know that an expression is independent from a variable $x_i$ if the partial derivative of that expression by $x_i$ is $=0$. However, I know no way of determining when the expression actually depends on the value. Is it when the partial derivative $=1$?