Question

If $A=\{y:y={\log }_2|x-2|,x<5,x\text{is a whole number}\},B=\{y:y=|x-2|,x\in [0,1],y\in N\},C=\{1,3,5\}$
, then $n\left(\right(A\times B)\cap (B\times C\left)\right)$ is

Answer 1

Given: $A=\{y:y={\log }_2|x-2|,x<5,x\text{is a whole number}\},B=\{y:y=|x-2|,x\in [0,1],y\in N\},C=\{1,3,5\}$

${\log }_2|x-2|$ is not defined for $x=2$
now for $x=0,1,3,4$

$|x-2|=2,1,1,2$

$\therefore A=\{0,1\}$

$y=|x-2|,x\in [0,1]\Rightarrow y\in \{1,2\}(\because y\in N)\Rightarrow B=\{1,2\}$

$C=\{1,3,5\}$

$(A\times B)\cap (B\times C)=(A\cap B)\times (B\cap C)$
$=\left\{\right(1,1\left)\right\}$