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Problem involving three sets


Q. In a city, three daily newspapers A B and C are published. 42% read newspaper A, 51% read nespaper B, 68% read newspaper C, 30% read newspaper A and B, 28% read neapaper B and C, 36% read A and C, 8% read none of them. Find the percent of people who read all the three papers. Represent the information in a venn diagram.

Solution:
Let U represent the set of people under survey.

Here,   n(A)=42%         n(B)=51%                             n(C)=68%         n(A⋂B)=30%                       n(B⋂C)=28%     n(A⋂C)=36%                     n(A∪B∪C)'=8%

Now,   n(A∪B∪C)= n(U) - n(A∪B∪C)'                                        =100% - 8%                                                    =92%
Again, 
n(A∪B∪C)= n(A) +n(B) + n(C)- n(B⋂C) -                   n (A⋂ B)- n (A⋂C) + n(A⋂B⋂C)
or, 92%= 42% +51% +68% -28% -30%                       -36% + n(A⋂B⋂C)                        or, 92%= 161% - 94 + n(A⋂B⋂C)              or, 92% = 67% + n(A⋂B⋂C)                        or, 92% - 67%=  n(A⋂B⋂C)                          or, 25% =  n(A⋂B⋂C)

Therefore, 25% of the people read all of the three newspapers.

Venn-diagram: