Max Operation Representation on Real Numbers

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:


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

Also see

 * Min Operation Representation on Real Numbers