Let $A=\{-2,-1{,}0{,}1{,}2{,}3\}$. Define a relation $R$ on $A$ by $xRy$ if and only if $y=\max(x,1)$. If $l$ is the number of elements in $R$, and $m$ and $n$ are the minimum numbers of ordered pairs that must be added to $R$ to make it reflexive and symmetric respectively, then $l+m+n$ is equal to:

Step 1: Determine the relation R and its size l

The relation $R$ is defined for elements $x, y$ in set $A$ such that $y$ is the maximum of $x$ and $1$. We need to find all ordered pairs $(x,y)$ that satisfy this condition for each $x \in A$. We will then count the number of elements in $R$, which is $l$.

Step 2: Calculate the elements of R

For each $x \in A$: - If $x = -2$, $\max(-2{,}1) = 1$, so $(-2{,}1) \in R$. - If $x = -1$, $\max(-1{,}1) = 1$, so $(-1{,}1) \in R$. - If $x = 0$, $\max(0{,}1) = 1$, so $(0{,}1) \in R$. - If $x = 1$, $\max(1{,}1) = 1$, so $(1{,}1) \in R$. - If $x = 2$, $\max(2{,}1) = 2$, so $(2{,}2) \in R$. - If $x = 3$, $\max(3{,}1) = 3$, so $(3{,}3) \in R$. Thus, $R = \{(-2{,}1), (-1{,}1), (0{,}1), (1{,}1), (2{,}2), (3{,}3)\}$. The number of elements in $R$ is $l = 6$.

Step 3: Determine m for reflexivity

For a relation to be reflexive, every element $x \in A$ must have $(x,x) \in R$. The set $A = \{-2,-1{,}0{,}1{,}2{,}3\}$. The reflexive pairs are $(-2,-2), (-1,-1), (0{,}0), (1{,}1), (2{,}2), (3{,}3)$. From $R$, we have $(1{,}1), (2{,}2), (3{,}3)$. The missing reflexive pairs are $(-2,-2), (-1,-1), (0{,}0)$. Therefore, $m = 3$ pairs must be added to make $R$ reflexive.

Step 4: Determine n for symmetry

For a relation to be symmetric, if $(x,y) \in R$, then $(y,x)$ must also be in $R$. We check each pair in $R$: - $(-2{,}1) \in R$, but $(1,-2) \notin R$. So, $(1,-2)$ must be added. - $(-1{,}1) \in R$, but $(1,-1) \notin R$. So, $(1,-1)$ must be added. - $(0{,}1) \in R$, but $(1{,}0) \notin R$. So, $(1{,}0)$ must be added. - $(1{,}1) \in R$, and $(1{,}1) \in R$. No addition needed. - $(2{,}2) \in R$, and $(2{,}2) \in R$. No addition needed. - $(3{,}3) \in R$, and $(3{,}3) \in R$. No addition needed. Therefore, $n = 3$ pairs must be added to make $R$ symmetric.

Step 5: Calculate l + m + n

We have found $l=6$, $m=3$, and $n=3$. Now we sum these values to get the final answer.

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