Question

In a pollution study of $1500$ Indian rivers the following data were reported. $520$ were polluted by sulphur compounds, $335$ polluted by phosphate, $425$ were polluted by crude oil, $100$ were polluted by both crude oil and sulphur compounds, $180$ were polluted by both sulphur compounds and phosphates, $150$ were polluted by both phosphates and crude oil and $28$ were polluted by sulphur compounds, phosphates, and crude oil. How many of the rivers were polluted by at least one of the three impurities ? How many of the rivers were polluted by exactly one of the three impurities ?

Answer 1

Ans $878,S$ only $268$, $P$ only $33$, C only $203$.
Here $n\left(U\right)=1500$, $n\left(S\right)=520$, $n\left(P\right)=335$,
$n\left(C\right)=425$, $n(C\cap S)=100$, $n(S\cap P)=180$,
$n(P\cap C)=150$
and $n(S\cap P\cap C)=28$,
Where $S,P$ and $C$ denote respectively the st of river polluted by sulphur compounds, phos-phates and crude oil.
Now we draw a Venn-diagram respectively the set $U$ of
$1500$ Indian rivers by a rectangle and its three subsets $S.P$ and $C$ by closed curves inside $U$. We first write $28$ in the region
$S\cap P\cap C$
Since $n(C\cap S)=100$, out of which $28$ have $100-28$, That is $72$ in the remaining par of $C\cap S$. Similarly we write $152$ and $122$ in the remaining parts of $S\cap P$ and $P\cap C$ respectively. Finally we put $268$ in ${}^{\prime }S$ Only',$33$ in $P^{\prime }$ Only' and $203$ in $C^{\prime }$ Only'.
now the number of river polluted by at least one of the three impurities is given by
$n(S\cup P\cup C)=28+72+152+122+268+33+203=878$
or $S_1-S_2+S_3=(520+335+425)-(100+180+15)+28=1280-430+28=1308-430=878$
Only Sulphur
$=520-(152+72-28)=520--252=268$
Number of rivers polluted by exactly sulphur compounds$=268$
Number of river polluted by exactly phosphates $=33$ and the Numb, of rivers polluted by exactly crude oil $=203$.