In a pollution study of 1500 Indian rivers the following data were reported. 520 were pulleted by sulphur compounds, 335 were polluted by phosphates, 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 atleast one of the three impurities?
How many of the rivers were polluted by exactly one of the three impurities?

Explanation

Let S, P and C denote the sets of rivers polluted by sulphur compounds, by phosphates and by crude oil respectively, and let a, b, c, d, e, f, g denote the elements (impurities) contained in the bounded region as shown in the diagram.

Then, `a+d+e+g=520`
`c+d+f+g=425`
`b+e+f+g=335impliesd+g=100`
`e+g=180impliesf+g=150`
g = 28
After solving, we get
g = 28, f = 122, e = 152, b = 33, d = 72, c = 203 and a = 268 The number of rivers were polluted by atleast one of the three impurities
`=(a+b+c+d+e+f+g)=878`
and the number of rivers were polluted by exactly one of the three impurities,
a + b + c = 268 + 33 + 203 = 504