# In a survey it was found that 21 persons liked product

Question:

In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1 ; 14 persons liked products P2 and P3 and 8 liked all the three products. Find how many liked product P3 only.

Solution:

Let $P_{1}, P_{2}$ and $P_{3}$ denote the sets of persons liking products $P_{1}, P_{2}$ and $P_{3}$, respectively. Also, let $U$ be the universal set.

Thus, we have:

$n\left(P_{1}\right)=21, n\left(P_{2}\right)=26$ and $n\left(P_{3}\right)=29$

And,

$n\left(P_{1} \cap P_{2}\right)=14, n\left(P_{1} \cap P_{3}\right)=12, n\left(P_{2} \cap P_{3}\right)=14$ and $n\left(P_{1} \cap P_{2} \cap P_{3}\right)=8$

Now,

Number of people who like only product $P_{3}$ :

$n\left(P_{3} \cap P_{1}^{\prime} \cap P_{2}^{\prime}\right)$

$=n\left\{P_{3} \cap\left(P_{1} \cup P_{2}\right)^{\prime}\right\}$

$=n\left(P_{3}\right)-n\left[P_{3} \cap\left(P_{1} \cup P_{2}\right)\right]$

$=n\left(P_{3}\right)-n\left[\left(P_{3} \cap P_{1}\right) \cup\left(P_{3} \cap P_{2}\right)\right]$

$=n\left(P_{3}\right)-\left[n\left(P_{3} \cap P_{1}\right)+n\left(P_{3} \cap P_{2}\right)-n\left(P_{1} \cap P_{2} \cap P_{3}\right)\right]$

$=29-(12+14-8)$

= 11

Therefore, the number of people who like only product $P_{3}$ is 11