Question

# Let A,B,C be finite sets. Suppose that n(A)=10,n(B)=15,n(C)=20,n(A∩B)=8 and n(B∩C)=9. Then the maximum possible value of n(A∪B∪C) is

Solution

## n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)                           −n(B∩C)−n(C∩A)+n(A∩B∩C)                        =28−[n(C∩A)−n(A∩B∩C)] We know that n(C∩A)≥n(A∩B∩C)⇒n(C∩A)−n(A∩B∩C)≥0 ∴ Maximum possible value of n(A∪B∪C) is 28Mathematics

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