<p>Let A = {1, 2, 3, 4, 5}. R: xRy ⇔4x ≤5y. Let m = |R| and n = minimum elements to add
for symmetry. Find m + n:</p>
Step-by-Step Solution
Key Concept: Build the full 5 \times 5 grid checking 4x \leq5y (i.e. y \geq4x/5). For each (x, y) \in R, check if (y, x) \in R
(i.e. 4y \leq5x). Every asymmetric pair contributes 1 to n.
<p><strong>Step 1</strong>: Build R — check 4x \leq5y:</p><br>x\y<br>1<br>2<br>3<br>4<br>5<br>1<br>✓<br>✓<br>✓<br>✓<br>✓<br>2<br>✗<br>✓<br>✓<br>✓<br>✓<br>3<br>✗<br>✗<br>✓<br>✓<br>✓<br>4<br>✗<br>✗<br>✗<br>✓<br>✓<br>5<br>✗<br>✗<br>✗<br>✓<br>✓<br>Note: x = 5, y = 4: 4(5) = 20 \leq5(4) = 20 ✓(borderline, included). m = 5 + 4 + 3 + 2 + 2 = 16.<p><strong>Step 2</strong>: Find pairs needing reverse added: check (y, x) \in R iff 4y \leq5x. The only off-diagonal symmetric pair is</p><br>(4, 5) ↔(5, 4): 4(5) = 20 = 5(4), both already in R. All other above-diagonal pairs need their reverse added:<br>(2, 1), (3, 1), (4, 1), (5, 1), (3, 2), (4, 2), (5, 2), (4, 3), (5, 3) — total n = 9.<p><strong>Step 3</strong>: m + n = 16 + 9 = 25.</p>
Correct Answer: 3