Max Operation Representation on Real Numbers

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Theorem

Let $x, y \in \R$.


Then:

$\max \set{x, y} = \dfrac 1 2 \paren {x + y + \size {x - y} }$

where $\max$ denotes the max operation.


Proof

From the Trichotomy Law for Real Numbers exactly one of the following holds:

$x < y$ and so $\max \set {x, y} = y$
$x = y$ and so $\max \set {x, y} = x = y$
$y < x$ and so $\max \set {x, y} = x$


By the definition of the absolute value function for each case respectively we have:

$\size {x - y} = y - x$
$\size {x - y} = 0$
$\size {x - y} = x - y$


Thus the equation holds by $+$ being commutative and associative as for each case:

$\dfrac 1 2 \paren {x + y + y - x} = y$
$\dfrac 1 2 \paren {x + y + 0} = x = y$
$\dfrac 1 2 \paren {x + y + x - y} = x$

$\blacksquare$


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