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± and Y±, which is supposedly possible.
I know that max(x,y)=x+y+|x−y|2, and so Z+=X++Y++|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≥0, X≥0 and Y≥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
You cannot express (X+Y)+ alone in terms of X± and Y±, 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.)
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