Solutions

Geometrically, the formulae make perfect sense. Of course, intuition is no substitute for a formal proof. Let us start by proving the max function:

• If $\bm{x \leq y}$

$$\begin{aligned} x \leq y & \implies y - x \geq 0 \\ & \implies \max{(x, y)} = \frac{x + y + (y - x)}{2} \\ & \implies \max({x, y}) = \frac{2y}{2} = y \end{aligned}$$

• If $\bm{y \leq x}$

$$\begin{aligned} y \leq x & \implies y - x \leq 0 \\ & \implies \max{(x, y)} = \frac{x + y - (y - x)}{2} \\ & \implies \max{(x, y)} = \frac{2x}{2} = x \end{aligned}$$

To prove the min function:

• If $\bm{x \leq y}$

$$\begin{aligned} x \leq y & \implies y - x \geq 0 \\ & \implies \min{(x, y)} = \frac{x + y - (y - x)}{2} \\ & \implies \min({x, y}) = \frac{2x}{2} = x \end{aligned}$$

• If $\bm{y \leq x}$

$$\begin{aligned} y \leq x & \implies y - x \leq 0 \\ & \implies \min{(x, y)} = \frac{x + y - (- (y - x))}{2} \\ & \implies \min{(x, y)} = \frac{x + y + (y - x)}{2} \\ & \implies \min{(x, y)} = \frac{2y}{2} = y \end{aligned}$$