How to express $\max(x+y,0)$ in terms of $\max(\pm x, 0)$ and $\max(\pm y, 0)$

Question

Suppose we define $X^+, X^-$ as $\max(X, 0)$ and $\max(-X, 0)$ respectively. Then, given $Z = X + Y$, I've been trying to figure out how to express $Z^+$ and $Z^-$ in terms of $X^\pm$ and $Y^\pm$, which is supposedly possible.

I know that $\max(x, y) = \frac{x+y+|x-y|}2$, and so $Z^+ = X^+ + Y^+ + \frac{|X+Y|-|X|-|Y|}{2}$, but I'm unsure what to do with this remaining term, I can't seem to figure out how to express it in terms of the other quantities. I have considered breaking he domain $X, Y$ up into regions where $X+Y\ge 0$, $X\ge 0$ and $Y\ge 0$ and flip-flopping the signs, but this seemed like too many cases to be the true solution.

How exactly do you do this? I can't seem to see it.

Answer 1

If we substitute $X=t-1$ and $Y=t+1$, then the maps $t\mapsto X^\pm$, $t\mapsto Y^\pm$ have their kinks at $t=\pm1$. You want to construct from these a function with a kink at $t=0$. This is not possible with elemetary functions alone

Answer 2

You cannot express $(X+Y)^+$ alone in terms of $X^\pm$ and $Y^\pm$, and likewise $(X+Y)^-$, but you can express the two of them together: $$ (X+Y)^+ - (X+Y)^- = X + Y = (X^+ - X^-) + (Y^+ - Y^-). $$ (Part (b) of that exercise in Rosenthal's book kind of gives it away.)