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

Answer: The expression always equals min(x, y), the smaller of the two numbers.

Step 1 — remove the modulus by case analysis:

• Case 1 — x ≥ y: here y is the smaller number, and |x−y| = x−y. So the numerator is (x+y) − (x−y) = 2y, and the expression = 2y/2 = y.

• Case 2 — x < y: here x is the smaller number, and |x−y| = y−x. So the numerator is (x+y) − (y−x) = 2x, and the expression = 2x/2 = x.

Step 2 — why this is the minimum (not just "x or y"):

Look at WHICH variable comes out in each case. In Case 1 we assumed x ≥ y, so the smaller one is y — and the answer is y. In Case 2 we assumed x < y, so the smaller one is x — and the answer is x. So in every case the result is exactly the smaller of the two values. "The smaller of the two, whichever it happens to be" is precisely the definition of the minimum, written min(x, y). It is never the maximum, because the maximum is the larger value, which is the one that got cancelled out.

Quick check: x = 5, y = 3 → (5+3 − |5−3|)/2 = (8 − 2)/2 = 3 = min(5, 3). x = 2, y = 4 → (2+4 − |2−4|)/2 = (6 − 2)/2 = 2 = min(2, 4). The output is always the smaller value.

Therefore the correct option is "the minimum of x and y". (Note: the companion identity ((x+y) + |x−y|)/2 gives the maximum.)