Question

Let $A = \{-3, -1, 0, 2, 4\}$. Let $R$ be a relation on $A$ defined by $R = \{(x, y) : y = \min\{x, 0\}, x, y \in A\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added to $R$ to make it reflexive and symmetric, respectively. Then, $l + m + n$ is equal to

Answer 1

9

Answer 2

Here's how to solve the problem:

1. Determine the elements of R: Given $A = \{-3, -1, 0, 2, 4\}$ and $R = \{(x, y) : y = \min\{x, 0\}, x, y \in A\}$, we find the elements of $R$ by evaluating $y = \min\{x, 0\}$ for each $x \in A$:

   • For $x = -3$: $y = \min\{-3, 0\} = -3$. So, $(-3, -3) \in R$.
   • For $x = -1$: $y = \min\{-1, 0\} = -1$. So, $(-1, -1) \in R$.
   • For $x = 0$: $y = \min\{0, 0\} = 0$. So, $(0, 0) \in R$.
   • For $x = 2$: $y = \min\{2, 0\} = 0$. So, $(2, 0) \in R$.
   • For $x = 4$: $y = \min\{4, 0\} = 0$. So, $(4, 0) \in R$.

   Thus, $R = \{(-3, -3), (-1, -1), (0, 0), (2, 0), (4, 0)\}$ and $l = |R| = 5$.

2. Determine the minimum number of elements to add to R to make it reflexive (m): A relation $R$ on a set $A$ is reflexive if for every element $a \in A$, the pair $(a, a)$ is in $R$. For the set $A = \{-3, -1, 0, 2, 4\}$, the pairs required for reflexivity are $\{(-3, -3), (-1, -1), (0, 0), (2, 2), (4, 4)\}$. We need to add $(2, 2)$ and $(4, 4)$ to $R$ to make it reflexive. Therefore, $m = 2$.

3. Determine the minimum number of elements to add to R to make it symmetric (n): A relation $R$ on a set $A$ is symmetric if for every pair $(x, y) \in R$, the pair $(y, x)$ is also in $R$.

   • $(-3, -3)$, $(-1, -1)$, and $(0, 0)$ are already symmetric.
   • For $(2, 0) \in R$, we need $(0, 2)$.
   • For $(4, 0) \in R$, we need $(0, 4)$.

   Thus, we need to add $(0, 2)$ and $(0, 4)$ to make $R$ symmetric. Therefore, $n = 2$.

4. Calculate l + m + n: $l + m + n = 5 + 2 + 2 = 9$.

Therefore, the final answer is 9.