Key Concepts and Formulas

1. Relation: A relation $R$ on a set $A$ is a subset of $A \times A$. If $(x, y) \in R$, we denote it as $x \mathrm{R} y$.
2. Reflexive Relation: A relation $R$ on $A$ is reflexive if for all $x \in A$, $(x, x) \in R$.
3. Symmetric Relation: A relation $R$ on $A$ is symmetric if for all $x, y \in A$, whenever $(x, y) \in R$, then $(y, x) \in R$.

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Step-by-Step Solution

Step 1: Determine the relation $R$ and its number of elements, $l$.

The set is $A = \{-2, -1, 0, 1, 2, 3\}$. The relation $R$ is defined by $x \mathrm{R} y$ if and only if $y = \max\{x, 1\}$. We need to find all ordered pairs $(x, y)$ where $x \in A$ and $y = \max\{x, 1\}$.

• For $x = -2$, $y = \max\{-2, 1\} = 1$. So, $(-2, 1) \in R$.
• For $x = -1$, $y = \max\{-1, 1\} = 1$. So, $(-1, 1) \in R$.
• For $x = 0$, $y = \max\{0, 1\} = 1$. So, $(0, 1) \in R$.
• For $x = 1$, $y = \max\{1, 1\} = 1$. So, $(1, 1) \in R$.
• For $x = 2$, $y = \max\{2, 1\} = 2$. So, $(2, 2) \in R$.
• For $x = 3$, $y = \max\{3, 1\} = 3$. So, $(3, 3) \in R$.

Therefore, the relation $R$ is $R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$. The number of elements in $R$ is $l = |R| = 6$.

Step 2: Determine the minimum number of elements to add to make $R$ reflexive, $m$.

A relation $R$ on $A$ is reflexive if $(x, x) \in R$ for all $x \in A$. The elements of $A$ are $\{-2, -1, 0, 1, 2, 3\}$. The required reflexive pairs are: $(-2, -2), (-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$.

We check which of these are already in $R$:

• $(-2, -2) \notin R$
• $(-1, -1) \notin R$
• $(0, 0) \notin R$
• $(1, 1) \in R$
• $(2, 2) \in R$
• $(3, 3) \in R$

The elements that need to be added to make $R$ reflexive are $\{(-2, -2), (-1, -1), (0, 0)\}$. The minimum number of elements to add for reflexivity is $m = 3$.

Step 3: Determine the minimum number of elements to add to make $R$ symmetric, $n$.

A relation $R$ is symmetric if for every $(x, y) \in R$, $(y, x)$ is also in $R$. We examine each pair in $R = \{(-2, 1), (-1, 1), (0, 1), (1, 1), (2, 2), (3, 3)\}$ for its symmetric counterpart.

• For $(-2, 1) \in R$, we need $(1, -2) \in R$. Since $(1, -2) \notin R$, we must add it.
• For $(-1, 1) \in R$, we need $(1, -1) \in R$. Since $(1, -1) \notin R$, we must add it.
• For $(0, 1) \in R$, we need $(1, 0) \in R$. Since $(1, 0) \notin R$, we must add it.
• For $(1, 1) \in R$, we need $(1, 1) \in R$. This is already satisfied.
• For $(2, 2) \in R$, we need $(2, 2) \in R$. This is already satisfied.
• For $(3, 3) \in R$, we need $(3, 3) \in R$. This is already satisfied.

The unique elements that need to be added to make $R$ symmetric are $\{(1, -2), (1, -1), (1, 0)\}$. The minimum number of elements to add for symmetry is $n = 3$.

Step 4: Calculate $l+m+n$.

We have $l = 6$, $m = 3$, and $n = 3$. The sum is $l + m + n = 6 + 3 + 3 = 12$.

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Common Mistakes & Tips

• When checking for reflexivity, ensure you consider all elements of set $A$.
• For symmetry, if $(x, y) \in R$ and $x=y$, the pair $(x,x)$ is already symmetric with itself, so no additional element is needed for this specific pair.
• When counting elements to add for symmetry, only count unique pairs that are required to be added.

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Summary

We first determined the elements of the given relation $R$ by applying the rule $y = \max\{x, 1\}$ for each element $x$ in set $A$, finding that $l=6$. Then, we identified the missing pairs $(x,x)$ for reflexivity, which amounted to $m=3$. Finally, we found the missing symmetric pairs $(y,x)$ for each $(x,y) \in R$, determining that $n=3$. The sum $l+m+n$ was calculated as $6+3+3=12$.

The final answer is $\boxed{12}$.