Question

If $A$ is the set of odd prime numbers and $B=\{x\in Z:-6<\frac{2x-5}{3}\le 7\text{and}-4\le \frac{x-7}{2}<4\},$ then the value of $n(A\cap B)$ is

Answer 1

$A=\{3,5,7,11,13,17,19,...\}$

$B=\{x\in Z:-6<\frac{2x-5}{3}\le 7\text{and}-4\le \frac{x-7}{2}<4\}$
$\Rightarrow -6<\frac{2x-5}{3}\le 7$
$\Rightarrow -18<2x-5\le 21$
$\Rightarrow -13<2x\le 26$
$\Rightarrow \frac{-13}{2}<x\le 13...\left(1\right)$

and $-4\le \frac{x-7}{2}<4$
$\Rightarrow -8\le x-7<8$
$\Rightarrow -1\le x<15...\left(2\right)$
From $\left(1\right)$ and $\left(2\right),$
$-1\le x\le 13$
$B=\{-1,0,1,...,12,13\}$

$A\cap B=\{3,5,7,11,13\}$
$\therefore n(A\cap B)=5$