Question

Let $Z$ be the set of integers. If
$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

• A. $2^{10}$
• B. $2^{12}$
• C. $2^{15}$
• D. $2^{18}$

Answer 1

The correct option is C $2^{15}$

$2^{(x+2)(x^2-5x+6)}=1=2^0$
$\therefore x=-2,2\text{or}3$
$A=\{-2,2,3\}$

and $B=\{x\in Z:-3<2x-1<9\}$
$\therefore -3<2x-1<9$
$\Rightarrow -1<x<5$
$\Rightarrow B=\{0,1,2,3,4\}$

As $n(A\times B)=n\left(A\right)\times n\left(B\right)=3\times 5=15$
$\therefore$ Number of subsets of $A\times B$ is $2^{15}$