Question
Let $$A,\,B,\,C$$ be finite sets. Suppose that $$n\left( A \right) = 10,\,n\left( B \right) = 15,\,n\left( C \right) = 20,\,n\left( {A \cap B} \right) = 8$$ and $$n\left( {B \cap C} \right) = 9.$$ Then the possible value of $$n\left( {A \cup B \cup C} \right)$$ is :
A.
26
B.
27
C.
28
D.
any of the three values 26, 27, 28 is possible
Answer :
any of the three values 26, 27, 28 is possible
Solution :
$$\eqalign{
& {\text{We have}} \cr
& n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right) + n\left( {A \cap B \cap C} \right) \cr
& = 10 + 15 + 20 - 8 - 9 - n\left( {C \cap A} \right) + n\left( {A \cap B \cap C} \right) \cr
& = 28 - \left\{ {n\left( {C \cap A} \right) - n\left( {A \cap B \cap C} \right)} \right\}.....({\text{i}}) \cr
& {\text{Since }}\,n\left( {C \cap A} \right) \geqslant n\left( {A \cap B \cap C} \right) \cr
& {\text{We have }}n\left( {C \cap A} \right) - n\left( {A \cap B \cap C} \right) \geqslant 0.....({\text{ii}}) \cr
& {\text{From (i) and (ii) : }}n\left( {A \cup B \cup C} \right) \leqslant 28.....({\text{iii}}) \cr
& {\text{Now, }}n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) \cr
& = 10 + 15 - 8 = 17 \cr
& {\text{and }}n\left( {B \cup C} \right) = n\left( B \right) + n\left( C \right) - n\left( {B \cap C} \right) \cr
& = 15 + 20 - 9 = 26 \cr
& {\text{Since, }}n\left( {A \cup B \cup C} \right) \geqslant n\left( {A \cup C} \right)\,{\text{and }}n\left( {A \cup B \cup C} \right) \geqslant n\left( {B \cup C} \right){\text{,}} \cr
& {\text{we have }}n\left( {A \cup B \cup C} \right) \geqslant 17{\text{ and }}n\left( {A \cup B \cup C} \right) \geqslant 26 \cr
& {\text{Hence }}n\left( {A \cup B \cup C} \right) \geqslant 26.....({\text{iv}}) \cr
& {\text{From (iii) and (iv) we obtain}} \cr
& {\text{26}} \leqslant n\left( {A \cup B \cup C} \right) \leqslant 28 \cr
& {\text{Also }}n\left( {A \cup B \cup C} \right){\text{ is a positive integer}} \cr
& \therefore \,n\left( {A \cup B \cup C} \right) = 26{\text{ or }}27{\text{ or }}28 \cr} $$