Question

If $A=\{x:x^2-5x+6=0\}$ and $B=\{y:y\in Z,3<|y-2|\le 5\},$ then the number of relations from $A$ to $B$ is

Answer 1

$A=\{x:x^2-5x+6=0\}$
$x^2-5x+6=0$
$\Rightarrow (x-2)(x-3)=0$
$\Rightarrow x=2,3$
$\Rightarrow A=\{2,3\}$

$B=\{y:y\in Z,3<|y-2|\le 5\}$
$3<|y-2|\le 5$
Case $:1$ When $y\ge 2$
$3<y-2\le 5$
$\Rightarrow 5<y\le 7$

Case $:2$ When $y<2$
$3<2-y\le 5$
$\Rightarrow 1<-y\le 3$
$\Rightarrow -3\le y<-1$

Since $y\in Z,$
$y=-3,-2,6,7$
$\Rightarrow B=\{-3,-2,6,7\}$

Number of relations from $A$ to $B$ is $2^{n\left(A\right)\cdot n\left(B\right)}=2^{2\cdot 4}=256$