The expression $\frac{(x+y)-|x-y|}{2}$ is equal to

Consider the two cases based on the values of $x$ and $y$:

1. Case 1: $x \ge y$

   • Then $|x - y| = x - y$.

   • Substituting into the expression:

     $$\frac{(x + y) - (x - y)}{2} = \frac{2y}{2} = y = \min(x, y)$$

2. Case 2: $x < y$

   • Then $|x - y| = -(x - y) = y - x$.

   • Substituting into the expression:

     $$\frac{(x + y) - (y - x)}{2} = \frac{2x}{2} = x = \min(x, y)$$

In both cases, the expression evaluates to the minimum of $x$ and $y$.

*(Note: Conversely, $\frac{(x+y)+|x-y|}{2} = \max(x, y)$).*