Question

Let $A=\{x:x\text{is a prime factor of}30\}$ and $B=\{y:y\in N,-3\le y+3<8\}$. If $R$ is a relation from $A$ to $B$, then $R^{-1}$ can be

• A. $\left\{\right(2,2),(3,3\left)\right\}$
• B. $\left\{\right(3,2),(4,2),(4,3\left)\right\}$
• C. $\left\{\right(2,1),(3,1),(5,2\left)\right\}$
• D. $\left\{\right(1,5),(3,3),(4,2\left)\right\}$

Answer 1

Answer: (A), (B), (D)

$\left\{\right(1,5),(3,3),(4,2\left)\right\}$

$A=\{x:x\text{is a prime factor of}30\}$
Prime factors of $30$ are $2,3,5$
$A=\{2,3,5\}$
$B=\{y:y\in N,-3\le y+3<8\}$
$-3\le y+3<8$
$\Rightarrow -6\le y<5$
Since $y\in N$
$B=\{1,2,3,4\}$

$B\times A=\left\{\right(1,2),(1,3),(1,5),(2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(4,2),(4,3),(4,5\left)\right\}$
Here, $R$ is a relation from $A$ to $B$, then
$R^{-1}$ is a relation from $B$ to $A$

Now, $\left\{\right(2,2),(3,3\left)\right\}\subset B\times A$
$\left\{\right(3,2),(4,2),(4,3\left)\right\}\subset B\times A$
​​​​​​​​​​​​​​$\left\{\right(2,1),(3,1),(5,2\left)\right\}\subset ̸B\times A$
$\left\{\right(1,5),(3,3),(4,2\left)\right\}\subset B\times A$