Key Concepts and Formulas

1. Relation: A relation $R$ on a set $A$ is a subset of the Cartesian product $A \times A$. An ordered pair $(x,y) \in R$ signifies that $x$ is related to $y$, denoted as $xRy$.
2. Reflexive Relation: A relation $R$ on a set $A$ is reflexive if for every element $a \in A$, the ordered pair $(a,a)$ is in $R$.
3. Symmetric Relation: A relation $R$ on a set $A$ is symmetric if for every ordered pair $(x,y) \in R$, the ordered pair $(y,x)$ is also in $R$.

Step-by-Step Solution

1. Determining the Elements of Relation R and Calculating $l$

The given set is $A=\{-3,-2,-1,0,1,2,3\}$. The relation $R$ on $A$ is defined by $x \mathrm{R} y$ if and only if $2x-y \in \{0,1\}$. This condition implies that either $2x-y = 0$ or $2x-y = 1$. We can rewrite these as $y = 2x$ or $y = 2x-1$. We need to find all pairs $(x,y)$ such that $x \in A$, $y \in A$, and $y$ satisfies one of these conditions. We examine each element of $A$ for $x$.

• For $x=-3$:
  • $y = 2(-3) = -6$. Since $-6 \notin A$, this pair is not in $R$.
  • $y = 2(-3)-1 = -7$. Since $-7 \notin A$, this pair is not in $R$.
• For $x=-2$:
  • $y = 2(-2) = -4$. Since $-4 \notin A$, this pair is not in $R$.
  • $y = 2(-2)-1 = -5$. Since $-5 \notin A$, this pair is not in $R$.
• For $x=-1$:
  • $y = 2(-1) = -2$. Since $-2 \in A$, the pair $(-1,-2)$ is in $R$.
  • $y = 2(-1)-1 = -3$. Since $-3 \in A$, the pair $(-1,-3)$ is in $R$.
• For $x=0$:
  • $y = 2(0) = 0$. Since $0 \in A$, the pair $(0,0)$ is in $R$.
  • $y = 2(0)-1 = -1$. Since $-1 \in A$, the pair $(0,-1)$ is in $R$.
• For $x=1$:
  • $y = 2(1) = 2$. Since $2 \in A$, the pair $(1,2)$ is in $R$.
  • $y = 2(1)-1 = 1$. Since $1 \in A$, the pair $(1,1)$ is in $R$.
• For $x=2$:
  • $y = 2(2) = 4$. Since $4 \notin A$, this pair is not in $R$.
  • $y = 2(2)-1 = 3$. Since $3 \in A$, the pair $(2,3)$ is in $R$.
• For $x=3$:
  • $y = 2(3) = 6$. Since $6 \notin A$, this pair is not in $R$.
  • $y = 2(3)-1 = 5$. Since $5 \notin A$, this pair is not in $R$.

The set of ordered pairs in relation $R$ is: $R = \{(-1,-2), (-1,-3), (0,0), (0,-1), (1,2), (1,1), (2,3)\}$ The number of elements in $R$ is $l$. Counting these pairs, we get: $l = 7$

2. Calculating $m$ for Reflexivity

For a relation $R$ on set $A$ to be reflexive, every element $a \in A$ must be related to itself, i.e., $(a,a) \in R$. The set $A$ has 7 elements: $\{-3,-2,-1,0,1,2,3\}$. We need to check if the following pairs are present in $R$: $(-3,-3), (-2,-2), (-1,-1), (0,0), (1,1), (2,2), (3,3)$.

From our calculation of $R$:

• $(0,0) \in R$.
• $(1,1) \in R$.

The following pairs are missing from $R$ for it to be reflexive: $(-3,-3), (-2,-2), (-1,-1), (2,2), (3,3)$.

To make $R$ reflexive, we need to add these 5 missing pairs. Therefore, the minimum number of elements to be added to make $R$ reflexive is $m=5$.

3. Calculating $n$ for Symmetry

For a relation $R$ to be symmetric, if $(x,y) \in R$, then $(y,x)$ must also be in $R$. We examine each pair in $R$ and check if its reverse is also in $R$.

• $(-1,-2) \in R$. We check if $(-2,-1) \in R$. For $(-2,-1)$, $x=-2, y=-1$. $2x-y = 2(-2)-(-1) = -4+1 = -3$. Since $-3 \notin \{0,1\}$, $(-2,-1) \notin R$. So, to make $R$ symmetric, we must add $(-2,-1)$.
• $(-1,-3) \in R$. We check if $(-3,-1) \in R$. For $(-3,-1)$, $x=-3, y=-1$. $2x-y = 2(-3)-(-1) = -6+1 = -5$. Since $-5 \notin \{0,1\}$, $(-3,-1) \notin R$. So, to make $R$ symmetric, we must add $(-3,-1)$.
• $(0,0) \in R$. The reverse is $(0,0)$, which is already in $R$.
• $(0,-1) \in R$. We check if $(-1,0) \in R$. For $(-1,0)$, $x=-1, y=0$. $2x-y = 2(-1)-0 = -2$. Since $-2 \notin \{0,1\}$, $(-1,0) \notin R$. So, to make $R$ symmetric, we must add $(-1,0)$.
• $(1,2) \in R$. We check if $(2,1) \in R$. For $(2,1)$, $x=2, y=1$. $2x-y = 2(2)-1 = 4-1 = 3$. Since $3 \notin \{0,1\}$, $(2,1) \notin R$. So, to make $R$ symmetric, we must add $(2,1)$.
• $(1,1) \in R$. The reverse is $(1,1)$, which is already in $R$.
• $(2,3) \in R$. We check if $(3,2) \in R$. For $(3,2)$, $x=3, y=2$. $2x-y = 2(3)-2 = 6-2 = 4$. Since $4 \notin \{0,1\}$, $(3,2) \notin R$. So, to make $R$ symmetric, we must add $(3,2)$.

The pairs that need to be added to make $R$ symmetric are: $(-2,-1), (-3,-1), (-1,0), (2,1), (3,2)$. The number of elements to be added to make $R$ symmetric is $n=5$.

4. Calculating $l+m+n$

We have found: $l = 7$ (number of elements in $R$) $m = 5$ (minimum elements to make $R$ reflexive) $n = 5$ (minimum elements to make $R$ symmetric)

The required sum is $l+m+n = 7 + 5 + 5 = 17$.

Common Mistakes & Tips

• Set Membership: Always ensure that both elements of an ordered pair $(x,y)$ belong to the given set $A$.
• Reflexivity Check: For reflexivity, remember to check every element in set $A$ for the pair $(a,a)$.
• Symmetry Check: When checking for symmetry, systematically verify if the reverse of each existing pair in $R$ is also present. If a pair $(x,y)$ is in $R$ but $(y,x)$ is not, then $(y,x)$ must be added.

Summary

We first determined the elements of the given relation $R$ by applying the condition $2x-y \in \{0,1\}$ for all pairs $(x,y)$ where $x, y \in A$. This gave us $l=7$. For reflexivity, we identified the missing pairs $(a,a)$ for all $a \in A$, leading to $m=5$. For symmetry, we checked each pair $(x,y) \in R$ and found its corresponding $(y,x)$ was missing, requiring $n=5$ additions. The final sum $l+m+n$ was calculated as $7+5+5=17$.

The final answer is \boxed{17}.