Question

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

• A. $\{-1,0,1\}$
• B. $\{-2,-1,1,2\}$
• C. $\{0,1\}$
• D. $\{-2,-1,0,1,2\}$

Answer 1

The correct option is A $\{-1,0,1\}$

$3y^2\le 8-x^2$

$R=\left\{\begin{array}{c}(0,1),(0,-1),(-1,0),(1,1),(1,-1),\\ (-1,1),(1,-1)(-1,-1),(2,0),(-2,0),\\ (-2,0),(2,1),(2,-1),(-2,1),(-2,-1)\end{array}\right\}$

$\Rightarrow R:\{-2,-1,0,1,2\}\to \{-1,0,-1\}$

Hence ${\text{R}}^{-1}:\{-1,0,1\}\to \{-2,-1,0,1,2\}$