Let the relations and on the set be given by and . If M and N be the minimum number of elements required to be added in and , respectively, in order to make the relations symmetric, then M+N equals

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Solution

Q.
Let the relations $R_{1}$ and $R_{2}$ on the set $X = {1, 2, 3, . . ., 20}$ be given by $R_{1} = {(x, y) : 2 x - 3 y = 2}$ and $R_{2} = {(x, y) : - 5 x + 4 y = 0}$ . 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 [2024]

1 10

2 8

3 16

4 12

Ans.
(1)

   $R_{1} = {(x, y) : 2 x - 3 y = 2}$

   $R_{1} = {(4, 2), (7, 4), (10, 6), (13, 8), (16, 10), (19, 12)}$

So, 6 elements are needed to make $R_{1}$ symmetric

$\Rightarrow M = 6$

$R_{2} = {(x, y) : - 5 x + 4 y = 0}$

$R_{2} = {(4, 5), (8, 10), (12, 15), (16, 20)}$

So, 4 elements are needed to make $R_{2}$ symmetric

   $\Rightarrow N = 4$

   $∴ M + N = 6 + 4 = 10$

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