Question

Let the relations \( R_1 \) and \( R_2 \) on the set
\( X = \{ 1, 2, 3, \dots, 20 \} \) be given by
\( R_1 = \{ (x, y) : 2x - 3y = 2 \} \) and
\( R_2 = \{ (x, y) : -5x + 4y = 0 \} \).
If \( M \) and \( N \) be the minimum number of elements required to be added in \( R_1 \) and \( R_2 \), respectively, in order to make the relations symmetric, then \( M + N \) equals:

• A. 8
• B. 16
• C. 12
• D. 10

Answer 1

The Correct Option is D

From the set \( X = \{1, 2, 3, \ldots, 20\} \):

For \( R_1 = \{(4,2), (7,4), (10,6), (13,8), (16,10), (19,12)\} \), 6 elements need to be added to make it symmetric.

For \( R_2 = \{(4,5), (8,10), (12,15), (16,20)\} \), 4 elements need to be added.

Thus: \( x = 1, 2, 3, \ldots, 20 \)

\( R_1 = (x, y) : 2x - 3y = 2 \)

\( R_2 = (x, y) : -5x + 4y = 0 \)

\( R_1 = \{(4, 2), (7, 4), (10, 6), (13, 8), (16, 10), (19, 12)\} \)

\( R_2 = \{(4, 5), (8, 10), (12, 15), (16, 20)\} \)

In \( R_1 \), 6 elements needed.

In \( R_2 \), 4 elements needed.

So, total \( 6 + 4 = 10 \) elements.