Prove the following
Question:

If $R=\left\{(x, y): x, y \in \mathbf{Z}, x^{2}+3 y^{2} \leq 8\right\}$ is a relation on the set of integers $\mathbf{Z}$, then the domain of $R^{-1}$ is :

1. (1) $\{-2,-1,1,2\}$

2. (2) $\{0,1\}$

3. (3) $\{-2,-1,0,1,2\}$

4. (4) $\{-1,0,1\}$

Correct Option: , 4

Solution:

Since, $R=\left\{(x, y): x, y \in \mathbf{Z}, x^{2}+3 y^{2} \leq 8\right\}$

$\therefore R=\{(1,1),(2,1),(1,-1),(0,1),(1,0)\}$

$\Rightarrow D_{R^{-1}}=\{-1,0,1\}$