\text{Let }A=\{2,3,4,5,9\}\text{and relation }R=\left\{\right(x,y):2x\le 3y,x,y\in A\}.\text{If}m\text{is the number of elements in }R\text{ and}\mathrm{ n}\text{ is the number of elements to be added in}\mathrm{R }\text{to make it symmetric,}

$$\begin{gathered} \text{then }m+n\text{ is equal to} \\ \end{gathered}$$

` tags. Let's write out the response carefully. 25

The correct option is 25.

Given the set:

$A=\{2,3,4,5,9\}$

And the relation defined on set $A$:

$R=\left\{\right(x,y):2x\le 3y,x,y\in A\}$

Step 1: Find the number of elements in $R$ ($m$)

We test the condition $2x\le 3y$ for each element $x\in A$:

1. For $x=2$: $2\left(2\right)=4\le 3y\Rightarrow y\ge \frac{4}{3}\approx 1.33$
Possible values for $y\in A$ are $2,3,4,5,9$ (5 pairs):
$(2,2),(2,3),(2,4),(2,5),(2,9)$

2. For $x=3$: $2\left(3\right)=6\le 3y\Rightarrow y\ge 2$
Possible values for $y\in A$ are $2,3,4,5,9$ (5 pairs):
$(3,2),(3,3),(3,4),(3,5),(3,9)$

3. For $x=4$: $2\left(4\right)=8\le 3y\Rightarrow y\ge \frac{8}{3}\approx 2.67$
Possible values for $y\in A$ are $3,4,5,9$ (4 pairs):
$(4,3),(4,4),(4,5),(4,9)$

4. For $x=5$: $2\left(5\right)=10\le 3y\Rightarrow y\ge \frac{10}{3}\approx 3.33$
Possible values for $y\in A$ are $4,5,9$ (3 pairs):
$(5,4),(5,5),(5,9)$

5. For $x=9$: $2\left(9\right)=18\le 3y\Rightarrow y\ge 6$
Possible values for $y\in A$ is $9$ (1 pair):
$(9,9)$

Adding all the valid pairs gives the total number of elements in $R$:

$m=5+5+4+3+1=18$

Step 2: Find the number of elements to be added to make $R$ symmetric ($n$)

A relation $R$ is symmetric if $(x,y)\in R\Rightarrow (y,x)\in R$.

Let's check the symmetric requirement for non-diagonal elements in $R$:

- $(2,3)\in R$ and $(3,2)\in R$ (Already symmetric)
- $(3,4)\in R$ and $(4,3)\in R$ (Already symmetric)
- $(4,5)\in R$ and $(5,4)\in R$ (Already symmetric)

The remaining elements in $R$ do not have their corresponding symmetric pairs:

- $(2,4)\in R$, so we need to add $(4,2)$
- $(2,5)\in R$, so we need to add $(5,2)$
- $(2,9)\in R$, so we need to add $(9,2)$
- $(3,5)\in R$, so we need to add $(5,3)$
- $(3,9)\in R$, so we need to add $(9,3)$
- $(4,9)\in R$, so we need to add $(9,4)$
- $(5,9)\in R$, so we need to add $(9,5)$

Thus, the number of elements that must be added to make $R$ symmetric is:

$n=7$

Step 3: Calculate $m+n$

$m+n=18+7=25$