In a survey it was found that 21 persons liked product *P*_{1}, 26 liked product *P*_{2} and 29 liked product *P*_{3}. If 14 persons liked products *P*_{1} and *P*_{2}; 12 persons liked product *P*_{3} and *P*_{1} ; 14 persons liked products *P*_{2} and *P*_{3} and 8 liked all the three products. Find how many liked product *P*_{3} only.

#### Solution

Let \[P_1 , P_2 \text{ and } P_3\] denote the sets of persons liking products\[P_1 , P_2 \text{ and } P_3\] respectively.

Also, let U be the universal set.

Thus, we have:

*n*( \[P_1\]= 21, *n*(\[P_2\]= 26 and *n*(\[P_3\] 29

And,*n*(\[P_1\]\[\cap\]\[P_3\]= 12, *n*(\[P_2 \cap P_3\]*n*(\[P_1 \cap P_2 \cap P_3\]= 8

Now,

Number of people who like only product \[P_3\]

= `n (P_3∩ P_1′∩ P_2′)`

= `n {P_3∩ (P_1∪ P_2)′}`

\[ = n \left( P_3 \right) - n\left[ P_3 \cap \left( P_1 \cup P_2 \right) \right]\]

\[ = n\left( P_3 \right) - n\left[ \left( P_3 \cap P_1 \right) \cup \left( P_3 \cap P_2 \right) \right]\]

\[ = n\left( P_3 \right) - \left[ n\left( P_3 \cap P_1 \right) + n\left( P_3 \cap P_2 \right) - n\left( P_1 \cap P_2 \cap P_3 \right) \right]\]

\[ = 29 - \left( 12 + 14 - 8 \right)\]

\[ = 11\]

Therefore, the number of people who like only product \[P_3\]is 11