<p>On X = {1, . . . , 20}: R1 = {(x, y) : 2x −3y = 2} and R2 = {(x, y) : −5x + 4y = 0}. Let
M, N be minimum elements to add for symmetry. M + N =?</p>
Step-by-Step Solution
Key Concept: List all pairs in each relation within X. For each pair, check if its reverse is also in the relation.
Add all missing reverses.
<p><strong>Step 1</strong>: R1: 2x -3y = 2 \Rightarrow y = 2(x-1)</p><br>3<br>. Need 3 | (x -1), so x \in {1, 4, 7, 10, 13, 16, 19}. But x = 1 gives y = 0 /\in X.<br>Valid pairs: (4, 2), (7, 4), (10, 6), (13, 8), (16, 10), (19, 12). 6 pairs.<br>14<br><br>JEE Main 2019–2024 | Relations<br>Complete Solutions Booklet<p><strong>Step 2</strong>: Check reverses: (y, x) would need 2y -3x = 2. For (2, 4): 2(2) -3(4) = -8 ̸= 2. All fail. All 6 reverses</p><br>must be added. M = 6.<p><strong>Step 3</strong>: R2: -5x + 4y = 0 \Rightarrow y = 5x</p><br>4 . Need 4 | x, so x \in {4, 8, 12, 16, 20}. But x = 20 gives y = 25 > 20 /\in X.<br>Valid pairs: (4, 5), (8, 10), (12, 15), (16, 20). 4 pairs.<p><strong>Step 4</strong>: Check reverses: (y, x) would need -5y + 4x = 0. For (5, 4): -25 + 16 = -9 ̸= 0. All fail. All 4 reverses</p><br>must be added. N = 4.<p><strong>Step 5</strong>: M + N = 6 + 4 = 10.</p>
Correct Answer: 4