Question

There are $200$ individuals with a skin disorder, $120$ had been exposed to the chemical $C_1,50$ to chemical $C_2$, and $30$ to both the chemicals $C_1$ and $C_2$. Find the number of individuals exposed to:
$\left(i\right)$ Chemical $C_1$ but not Chemical $C_2$
$\left(ii\right)$ Chemical $C_2$ but not Chemical $C_1$
$\left(iii\right)$ Chemical $C_1$ or Chemical $C_2$

Answer 1

Given:
Number of people having skin disorder $=200$
Number of people exposed to chemical $C_1=120$
Number of people exposed to chemical $C_2=50$
Number of people exposed to both chemical $C_1\text{and}{C}_2=30$

$\left(i\right)$
Number of people exposed to chemical $C_1$ but not chemical $C_2=$ Number of people exposed to chemical $C_1-$ Number of people exposed to both chemical $C_1$ and $C_2$
$=120-30$
$=90$
Hence, the number of individuals exposed to chemical $C_1$ but not to chemical $C_2$ is $90$.

$\left(ii\right)$
Number of people exposed to chemical $C_2$ but not chemical $C_1=$ Number of people exposed to chemical $C_2-$ Number of people exposed to both chemical $C_1$ and $C_2$
$=50-30$
$=20$
Hence, the number of individuals exposed to chemical $C_2$ but not to chemical $C_1$ is $20$.

$\left(iii\right)$
Let $U$ denote the universal set consisting of individuals suffering from skin disorder, $A\text{and}B$ denote the set of individuals exposed to chemical $C_1\text{and}{C}_2$ respectively.
$\because n\left(U\right)=200,n\left(A\right)=120,n\left(B\right)=50\text{and}n(A\cap B)=30$
$\therefore n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$
$=120+50-30$
$=170-30$
$=140$
$\therefore 140$ individuals are exposed either to chemical $C_1$ or to chemical $C_2$.