Results 1 to 4 of 4

Math Help - Max{x,y}

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Max{x,y}

    \forall x,y\in\mathbb{R}

    \text{max}\{x,y\}=\frac{1}{2}\left(x+y+|x-y|\right)

    I know

    \text{max}\{|x|-|y|,|y|-|x|\}\leq |x-y|
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member girdav's Avatar
    Joined
    Jul 2009
    From
    Rouen, France
    Posts
    675
    Thanks
    32
    What is the question ? Show the first equality ? Treat the cases x\leq y and y\leq x.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Yes.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    May 2011
    Posts
    10
    Usually one would consider non-overlapping cases, but I guess the overlapping cases here would work.

    Case 1: Suppose x \leqslant y. What does this imply? If you don't recall what this implies I'll say that addition on R satisfy commutation, monotonicity, an equality for inverses, and an equality for neutrals. In other words, for all x, y, z, c on R the following properties for + hold:

    If x \leqslant y, (x+c) \leqslant (y+c) (monotonicity)
    (x+y)=(y+x) (commutation)
    (x+-(x))=0 (inverse)
    x+0=x (neutrality)

    Since you started with x \leqslant y, do you see which property you want to use here first? Do you see what element (function of an element, depending on how you look at it) of R you want to select for "c"? If you do, I think you'll infer what I've hinted at. Then check your definition of absolute value. So in this case, abs(x-y) must equal... what? Then replace what abs(x-y) must equal in this case with abs(x-y) in the maximum formula.

    Case 2: Suppose x>y. Now I'll point out that for all x, y, c in R,
    If x>y, then (x+c)>(y+c). Now using a similar technique, along with your definition of absolute value you can show something about the absolute value in this case.

    Then you might to want to indicate the cases exhaust R and the result consequently follows.

    If the above doesn't give you enough clues, I'll happily try and spell out details here.
    Follow Math Help Forum on Facebook and Google+


/mathhelpforum @mathhelpforum