Let $$A = \{-3, -2, -1, 0, 1, 2, 3\}$$ and $$R$$ be a relation on $$A$$ defined by $$xRy$$ if and only if $$2x - y \in \{0, 1\}$$. Let $$l$$ be the number of elements in $$R$$. Let $$m$$ and $$n$$ be the minimum number of elements required to be added in $$R$$ to make it reflexive and symmetric, respectively. Then $$l + m + n$$ is equal to :

The set is $$A=\{-3,-2,-1,0,1,2,3\}$$, so $$|A|=7$$.
The relation $$R$$ is defined by $$xRy \iff 2x-y\in\{0,1\}$$, i.e. $$y=2x$$ or $$y=2x-1$$.

Case 1: $$y=2x$$

Check each $$x\in A$$:

$$\begin{array}{c|c} x & 2x \\ \hline -3 & -6\;(\notin A)\\ -2 & -4\;(\notin A)\\ -1 & -2\;(\in A)\\ 0 & 0\;(\in A)\\ 1 & 2\;(\in A)\\ 2 & 4\;(\notin A)\\ 3 & 6\;(\notin A) \end{array}$$

Pairs obtained: $$(-1,-2),\,(0,0),\,(1,2).$$

Case 2: $$y=2x-1$$

Again test every $$x$$:

$$\begin{array}{c|c} x & 2x-1 \\ \hline -3 & -7\;(\notin A)\\ -2 & -5\;(\notin A)\\ -1 & -3\;(\in A)\\ 0 & -1\;(\in A)\\ 1 & 1\;(\in A)\\ 2 & 3\;(\in A)\\ 3 & 5\;(\notin A) \end{array}$$

Pairs obtained: $$(-1,-3),\,(0,-1),\,(1,1),\,(2,3).$$

Combining both cases, $$R$$ contains

$$\{(-1,-2),\,(-1,-3),\,(0,0),\,(0,-1),\,(1,2),\,(1,1),\,(2,3)\}.$$

Number of elements in $$R$$: $$l=7.$$

Making $$R$$ reflexive

A relation is reflexive when every $$a\in A$$ satisfies $$(a,a)\in R$$.
Present self-pairs: $$(0,0),\,(1,1).$$
Missing self-pairs: $$(-3,-3),\,(-2,-2),\,(-1,-1),\,(2,2),\,(3,3).$$
Minimum additions needed: $$m=5.$$

Making $$R$$ symmetric

For symmetry, whenever $$(a,b)\in R$$ we also need $$(b,a).$$
List missing converse pairs:

$$(-1,-2)\;\Rightarrow\;(-2,-1)$$
$$(-1,-3)\;\Rightarrow\;(-3,-1)$$
$$(0,-1)\;\Rightarrow\;(-1,0)$$
$$(1,2)\;\Rightarrow\;(2,1)$$
$$(2,3)\;\Rightarrow\;(3,2)$$

All five are distinct and not already in $$R$$, so
minimum additions required: $$n=5.$$

Finally,
$$l+m+n = 7+5+5 = 17.$$

Hence $$l + m + n = 17$$  ⇒  Option B.