Min Operation Representation on Real Numbers

Theorem
Let $x, y \in \R$.

Then:


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

where $\min$ denotes the min operation.

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


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

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:


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

Also see

 * Max Operation Representation on Real Numbers