Let A = \2, 3, 5, 7, 9\$A = \{2, 3, 5, 7, 9\}$. Let R$R$ be the relation on A$A$ defined by xRy$xRy$ if and only if 2x leq 3y$2x \leq 3y$. Let ell$\ell$ be the number of elements in R$R$, and m$m$ be the minimum number of elements required to be added in R$R$ to make it a symmetric relation. Then ell + m$\ell + m$ is equal to:

### Related Formula textA relation R text is symmetric if (x,y) in R implies (y,x) in R.$$\text{A relation } R \text{ is symmetric if } (x,y) \in R \implies (y,x) \in R.$$ ### Core Logic Check condition y geq frac2x3$y \geq \frac{2x}{3}$ for each x in A = \2, 3, 5, 7, 9\$x \in A = \{2, 3, 5, 7, 9\}$ to generate ordered pairs (x,y)$(x,y)$. Count the pairs to get ell$\ell$. Then find asymmetric pairs to get m$m$. ### Step 1: Enumerate elements of R x = 2 implies y geq 4/3 = 1.33 implies y in \2, 3, 5, 7, 9\$x = 2 \implies y \geq 4/3 = 1.33 \implies y \in \{2, 3, 5, 7, 9\}$ (5 elements) x = 3 implies y geq 6/3 = 2 implies y in \2, 3, 5, 7, 9\$x = 3 \implies y \geq 6/3 = 2 \implies y \in \{2, 3, 5, 7, 9\}$ (5 elements) x = 5 implies y geq 10/3 = 3.33 implies y in \5, 7, 9\$x = 5 \implies y \geq 10/3 = 3.33 \implies y \in \{5, 7, 9\}$ (3 elements) x = 7 implies y geq 14/3 = 4.66 implies y in \5, 7, 9\$x = 7 \implies y \geq 14/3 = 4.66 \implies y \in \{5, 7, 9\}$ (3 elements) x = 9 implies y geq 18/3 = 6 implies y in \7, 9\$x = 9 \implies y \geq 18/3 = 6 \implies y \in \{7, 9\}$ (2 elements) Total elements in R$R$ is ell = 5 + 5 + 3 + 3 + 2 = 18$\ell = 5 + 5 + 3 + 3 + 2 = 18$. ### Step 2: Determine Missing Symmetric Pairs We need to check which reverse pairs (y,x)$(y,x)$ are missing. (2, 5) in R$(2, 5) \in R$, but (5, 2) notin R$(5, 2) \notin R$. (2, 7) in R$(2, 7) \in R$, but (7, 2) notin R$(7, 2) \notin R$. (2, 9) in R$(2, 9) \in R$, but (9, 2) notin R$(9, 2) \notin R$. (3, 5) in R$(3, 5) \in R$, but (5, 3) notin R$(5, 3) \notin R$. (3, 7) in R$(3, 7) \in R$, but (7, 3) notin R$(7, 3) \notin R$. (3, 9) in R$(3, 9) \in R$, but (9, 3) notin R$(9, 3) \notin R$. (5, 9) in R$(5, 9) \in R$, but (9, 5) notin R$(9, 5) \notin R$. These are m = 7$m = 7$ pairs that need to be added. Thus, ell + m = 18 + 7 = 25$\ell + m = 18 + 7 = 25$. ### Pattern Recognition When counting pairs for small sets, list them manually by rows. To make symmetric, any non-diagonal pair (x,y)$(x,y)$ that is present while (y,x)$(y,x)$ is absent counts as 1$1$ missing element. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Sets and Relations