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.

Answer 1

Let $P_1,P_2\mathrm{and}{P}_3$ denote the sets of persons liking products $P_1,P_2\mathrm{and}{P}_3$, 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_2$) = 14, n($P_1$$\cap$$P_3$) = 12, n($P_2\cap P_3$) = 14 and n($P_1\cap P_2\cap P_3$) = 8

Now,
Number of people who like only product $P_3$:
$$\begin{gathered} n\left(P_3\cap {P_1}^{\text{'}}\cap {P_2}^{\text{'}}\right) \\ =n\left\{P_3\cap {\left(P_1\cup P_2\right)}^{\text{'}}\right\} \\ =n\left(P_3\right)-n\left[P_3\cap \left(P_1\cup P_2\right)\right] \\ =n\left(P_3\right)-n\left[\left(P_3\cap P_1\right)\cup \left(P_3\cap P_2\right)\right] \\ =n\left(P_3\right)-\left[n\left(P_3\cap P_1\right)+n\left(P_3\cap P_2\right)-n\left(P_1\cap P_2\cap P_3\right)\right] \\ =29-\left(12+14-8\right) \\ =11 \end{gathered}$$

Therefore, the number of people who like only product $P_3$ is 11