Let $$A = \{-2, -1, 0, 1, 2, 3\}$$. Let R be a relation on A defined by $$xRy$$ if and only if $$y = \max\{x, 1\}$$. Let $$\ell$$ 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 relations, respectively. Then $$\ell + m + n$$ is equal to

The given set is $$A=\{-2,-1,0,1,2,3\}$$, so $$|A|=6$$.

Relation $$R$$ is defined by  $$xRy \Longleftrightarrow y=\max\{x,1\}$$.

Step 1 - List the ordered pairs in $$R$$
 • If $$x\lt 1$$ (i.e. $$x=-2,-1,0$$), then $$\max\{x,1\}=1$$, hence $$y=1$$.
 • If $$x\ge 1$$ (i.e. $$x=1,2,3$$), then $$\max\{x,1\}=x$$, hence $$y=x$$.

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

The number of elements in $$R$$ is
$$\ell = 6.$$

Step 2 - Make $$R$$ reflexive
A relation is reflexive if $$(a,a)\in R$$ for every $$a\in A$$.

Pairs already present: $$(1,1),(2,2),(3,3).$$
Missing reflexive pairs: $$(-2,-2),(-1,-1),(0,0).$$

Hence the minimum number of pairs to be added is
$$m = 3.$$

Step 3 - Make $$R$$ symmetric
A relation is symmetric if $$(a,b)\in R \Longrightarrow (b,a)\in R.$$

Check each pair in $$R$$:
 • $$(1,1),(2,2),(3,3)$$ are self-symmetric.
 • $$(-2,1)$$ needs $$(1,-2).$$
 • $$(-1,1)$$ needs $$(1,-1).$$
 • $(0,1)$$ needs $$(1,0).$$

These three required pairs are not in $$R$$, so we must add exactly three of them. Thus
$$n = 3.$$

Step 4 - Compute $$\ell + m + n$$
$$\ell + m + n = 6 + 3 + 3 = 12.$$

Therefore, $$\ell + m + n = 12$$, which corresponds to Option A.