Key Concepts and Formulas

• Relation on a Set: A relation $R$ on a set $A$ is a subset of the Cartesian product $A \times A$. An element $(x,y) \in R$ means $x$ is related to $y$, denoted by $xRy$.
• Symmetric Relation: A relation $R$ on a set $A$ is symmetric if for every $(x,y) \in R$, it is also true that $(y,x) \in R$.
• Minimum Elements for Symmetry: To make a relation $R$ symmetric, we need to add all pairs $(y,x)$ such that $(x,y) \in R$ but $(y,x) \notin R$. The number of such pairs is the minimum number of elements to be added.

Step-by-Step Solution

Step 1: Determine the elements of relation R and calculate m.

We are given the set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ defined by $xRy$ if and only if $4x \leq 5y$. We need to find all ordered pairs $(x,y)$ where $x, y \in A$ that satisfy this inequality.

• For $x=1$: $4(1) \leq 5y \Rightarrow 4 \leq 5y \Rightarrow y \geq \frac{4}{5}$. Since $y \in \{1,2,3,4,5\}$, all values of $y$ are valid. Pairs: $(1,1), (1,2), (1,3), (1,4), (1,5)$. (5 pairs)
• For $x=2$: $4(2) \leq 5y \Rightarrow 8 \leq 5y \Rightarrow y \geq \frac{8}{5} = 1.6$. Since $y \in \{1,2,3,4,5\}$, valid $y$ are $\{2,3,4,5\}$. Pairs: $(2,2), (2,3), (2,4), (2,5)$. (4 pairs)
• For $x=3$: $4(3) \leq 5y \Rightarrow 12 \leq 5y \Rightarrow y \geq \frac{12}{5} = 2.4$. Since $y \in \{1,2,3,4,5\}$, valid $y$ are $\{3,4,5\}$. Pairs: $(3,3), (3,4), (3,5)$. (3 pairs)
• For $x=4$: $4(4) \leq 5y \Rightarrow 16 \leq 5y \Rightarrow y \geq \frac{16}{5} = 3.2$. Since $y \in \{1,2,3,4,5\}$, valid $y$ are $\{4,5\}$. Pairs: $(4,4), (4,5)$. (2 pairs)
• For $x=5$: $4(5) \leq 5y \Rightarrow 20 \leq 5y \Rightarrow y \geq \frac{20}{5} = 4$. Since $y \in \{1,2,3,4,5\}$, valid $y$ are $\{4,5\}$. Pairs: $(5,4), (5,5)$. (2 pairs)

The relation $R$ is the union of all these pairs: $R = \{(1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,4), (5,5)\}$ The number of elements in $R$ is $m = 5 + 4 + 3 + 2 + 2 = 16$.

Step 2: Identify elements required for symmetry and calculate n.

To make $R$ symmetric, for every pair $(x,y) \in R$, the pair $(y,x)$ must also be in $R$. We need to find pairs $(x,y) \in R$ such that $(y,x) \notin R$. The number of such unique pairs $(y,x)$ will be $n$.

Let's examine each pair in $R$:

• $(1,1) \in R$. $(1,1) \in R$. (Symmetric)

• $(1,2) \in R$. Check $(2,1)$: $4(2) = 8$, $5(1) = 5$. $8 \not\leq 5$, so $(2,1) \notin R$. Add $(2,1)$.

• $(1,3) \in R$. Check $(3,1)$: $4(3) = 12$, $5(1) = 5$. $12 \not\leq 5$, so $(3,1) \notin R$. Add $(3,1)$.

• $(1,4) \in R$. Check $(4,1)$: $4(4) = 16$, $5(1) = 5$. $16 \not\leq 5$, so $(4,1) \notin R$. Add $(4,1)$.

• $(1,5) \in R$. Check $(5,1)$: $4(5) = 20$, $5(1) = 5$. $20 \not\leq 5$, so $(5,1) \notin R$. Add $(5,1)$.

• $(2,2) \in R$. $(2,2) \in R$. (Symmetric)

• $(2,3) \in R$. Check $(3,2)$: $4(3) = 12$, $5(2) = 10$. $12 \not\leq 10$, so $(3,2) \notin R$. Add $(3,2)$.

• $(2,4) \in R$. Check $(4,2)$: $4(4) = 16$, $5(2) = 10$. $16 \not\leq 10$, so $(4,2) \notin R$. Add $(4,2)$.

• $(2,5) \in R$. Check $(5,2)$: $4(5) = 20$, $5(2) = 10$. $20 \not\leq 10$, so $(5,2) \notin R$. Add $(5,2)$.

• $(3,3) \in R$. $(3,3) \in R$. (Symmetric)

• $(3,4) \in R$. Check $(4,3)$: $4(4) = 16$, $5(3) = 15$. $16 \not\leq 15$, so $(4,3) \notin R$. Add $(4,3)$.

• $(3,5) \in R$. Check $(5,3)$: $4(5) = 20$, $5(3) = 15$. $20 \not\leq 15$, so $(5,3) \notin R$. Add $(5,3)$.

• $(4,4) \in R$. $(4,4) \in R$. (Symmetric)

• $(4,5) \in R$. Check $(5,4)$: $4(5) = 20$, $5(4) = 20$. $20 \leq 20$, so $(5,4) \in R$. (Symmetric)

• $(5,4) \in R$. Check $(4,5)$: $4(4) = 16$, $5(5) = 25$. $16 \leq 25$, so $(4,5) \in R$. (Symmetric)

• $(5,5) \in R$. $(5,5) \in R$. (Symmetric)

The pairs $(x,y)$ for which $(y,x) \notin R$ are: $(2,1), (3,1), (4,1), (5,1)$ $(3,2), (4,2), (5,2)$ $(4,3), (5,3)$

The set of elements to be added for symmetry is $\{(2,1), (3,1), (4,1), (5,1), (3,2), (4,2), (5,2), (4,3), (5,3)\}$. The number of elements to be added is $n = 4 + 3 + 2 = 9$.

Step 3: Calculate m + n.

We found $m = 16$ and $n = 9$. Therefore, $m + n = 16 + 9 = 25$.

Common Mistakes & Tips

• When checking for symmetry, ensure you are checking if the reverse pair $(y,x)$ is already present in the original relation $R$. If $(y,x)$ is already in $R$, it does not contribute to $n$.
• Pairs of the form $(x,x)$ are always symmetric. If $(x,x) \in R$, then $(x,x) \in R$, so these pairs never contribute to $n$.
• Be careful with the inequality signs and ensure all possible values from set $A$ are considered for each variable.

Summary

We first determined all pairs $(x,y)$ in $A \times A$ satisfying $4x \leq 5y$ to find the number of elements $m$ in relation $R$. This yielded $m=16$. Subsequently, we identified pairs $(x,y) \in R$ for which the reverse pair $(y,x)$ was not present in $R$. The count of these missing reverse pairs gave us the minimum number of elements $n$ required to make $R$ symmetric, resulting in $n=9$. Finally, we calculated $m+n = 16+9=25$.

The final answer is $\boxed{25}$.