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:
- $\dfrac 1 2 \paren {x + y + y - x} = y$
- $\dfrac 1 2 \paren {x + y + 0} = x = y$
- $\dfrac 1 2 \paren {x + y + x - y} = x$
$\blacksquare$
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 18 \ \text {(a)}$