Suppose we have the following question:
Consider rolling a fair six-sided die, so that $S = \{1, 2, 3, 4, 5, 6\}$.
Let $X(s) = s$, and $Y(s) = s^3 + 2$. Let $Z = XY$. Compute $Z(s)$ for all s ∈ S.
My textbook says the following:
Basically what this means is that $Z(2)=X(2)Y(2)=(2)(2^3+2)=20$
However, why can't we think of it like this:
Since we have $Z=XY$, if $Z=2$, that leads to two scenarios, that is, $X=1,Y=2$ or $X=2, Y=1$
So, $Z(z=2)=P(x=2,y=1)+P(x=1,y=2)$
What's wrong with my thinking here? Why would the above not work?