Tuesday, 26 April 2016

algebra precalculus - How to express max(x+y,0) in terms of max(pmx,0) and max(pmy,0)



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+|xy|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+Y0, X0 and Y0 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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