Question 63

Consider the given expression:

By the definition of the absolute value function $|z|$:

• Case 1: $x \ge y$.
  Then $|x - y| = x - y$.
  Substituting into the expression:
  $$E = \frac{(x + y) - (x - y)}{2} = \frac{x + y - x + y}{2} = \frac{2y}{2} = y$$
  Since $x \ge y$, the minimum of $x$ and $y$ is $y$. Thus, $E = \min(x, y)$.

• Case 2: $x < y$.
  Then $|x - y| = -(x - y) = y - x$.
  Substituting into the expression:
  $$E = \frac{(x + y) - (y - x)}{2} = \frac{x + y - y + x}{2} = \frac{2x}{2} = x$$
  Since $x < y$, the minimum of $x$ and $y$ is $x$. Thus, $E = \min(x, y)$.

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

(Note: the corresponding formula with addition, $\frac{(x + y) + |x - y|}{2}$, equals $\max(x, y)$).