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प्रश्न
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.
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उत्तर
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
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