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

Answer: (D)

10

Answer 2

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.