Relation R Elements Calculation (l)
Given Set $A = \{2, 3, 5, 7, 9\}$ and relation $R$ defined by $xRy \iff 2x \le 3y$. We list pairs $(x, y)$ from $A \times A$ satisfying the condition:
- For $x=2$: $2(2) \le 3y \implies 4 \le 3y$. This holds for $y \in \{2, 3, 5, 7, 9\}$. Pairs: (2,2), (2,3), (2,5), (2,7), (2,9).
- For $x=3$: $2(3) \le 3y \implies 6 \le 3y \implies 2 \le y$. This holds for $y \in \{2, 3, 5, 7, 9\}$. Pairs: (3,2), (3,3), (3,5), (3,7), (3,9).
- For $x=5$: $2(5) \le 3y \implies 10 \le 3y$. This holds for $y \in \{5, 7, 9\}$. Pairs: (5,5), (5,7), (5,9).
- For $x=7$: $2(7) \le 3y \implies 14 \le 3y$. This holds for $y \in \{5, 7, 9\}$. Pairs: (7,5), (7,7), (7,9).
- For $x=9$: $2(9) \le 3y \implies 18 \le 3y \implies 6 \le y$. This holds for $y \in \{7, 9\}$. Pairs: (9,7), (9,9).
The total number of elements in $R$ is $l = 5 + 5 + 3 + 3 + 2 = 18$.
Symmetric Relation Elements Calculation (m)
$m$ represents the minimum number of elements needed to be added to $R$ to make it symmetric. A relation is symmetric if for every pair $(x, y)$ in the relation, the pair $(y, x)$ is also in the relation. We identify pairs $(x, y) \in R$ for which $(y, x) \notin R$.
- We examine pairs $(x, y) \in R$ and check for the existence of $(y, x) \in R$:
- (2, 5) $\in R$. Check (5, 2): $2(5) = 10$ and $3(2) = 6$. Since $10 \not\le 6$, $(5, 2) \notin R$. We need to add (5, 2).
- (2, 7) $\in R$. Check (7, 2): $2(7) = 14$ and $3(2) = 6$. Since $14 \not\le 6$, $(7, 2) \notin R$. We need to add (7, 2).
- (2, 9) $\in R$. Check (9, 2): $2(9) = 18$ and $3(2) = 6$. Since $18 \not\le 6$, $(9, 2) \notin R$. We need to add (9, 2).
- (3, 5) $\in R$. Check (5, 3): $2(5) = 10$ and $3(3) = 9$. Since $10 \not\le 9$, $(5, 3) \notin R$. We need to add (5, 3).
- (3, 7) $\in R$. Check (7, 3): $2(7) = 14$ and $3(3) = 9$. Since $14 \not\le 9$, $(7, 3) \notin R$. We need to add (7, 3).
- (3, 9) $\in R$. Check (9, 3): $2(9) = 18$ and $3(3) = 9$. Since $18 \not\le 9$, $(9, 3) \notin R$. We need to add (9, 3).
- (5, 9) $\in R$. Check (9, 5): $2(9) = 18$ and $3(5) = 15$. Since $18 \not\le 15$, $(9, 5) \notin R$. We need to add (9, 5).
- All other pairs $(x, y) \in R$ have their symmetric counterpart $(y, x)$ also present in $R$. For example, $(2, 3) \in R$ and $(3, 2) \in R$. Pairs on the diagonal, like $(2, 2)$, are inherently symmetric.
- The count of the missing symmetric pairs that need to be added is $m = 7$.
Final Calculation: l + m
The question asks for the value of $l + m$.
Using the calculated values: $l = 18$ $m = 7$ $l + m = 18 + 7 = 25$.
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