Question

Let $Z$ be the set of integers. If $A=\{x\in Z:|x-3{|}^{x^2-5x+6}=1\}$ and $B=\{x\in Z:10<3x+1<22\},$ then the number of subsets of the set $A\times B$ is

Answer 1

For $A,$
$|x-3|=1$
$\Rightarrow x-3=\pm 1$
$\Rightarrow x=2,4$
(or) $x^2-5x+6=0$
$\Rightarrow x=2,3$
But $x\ne 3$
$\therefore A=\{2,4\}$

For $B,$
$10<3x+1<22$
$\Rightarrow 3<x<7$
$B=\{4,5,6\}$
$\Rightarrow n(A\times B)=n\left(A\right)\times n\left(B\right)=2\times 3=6$
$\therefore$ Number of subsets of $A\times B$ is $2^6$