In a survey, n(U) = 240, n(A) = 112, n(B) = 104, n(C) = 93, n(A ∩ B) = 46, n(B ∩ C) = 38, n(C ∩ A) = 35, and n(A ∩ B ∩ C) = 17. How many elements are in none of the sets?
Advertisement
Answer and explanation
Correct answer: 33
Use the inclusion–exclusion formula for three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substitution gives 112 + 104 + 93 − 46 − 38 − 35 + 17 = 207. The elements in none of the sets are outside the union, so their number is n(U) − n(A ∪ B ∪ C) = 240 − 207 = 33.
Frequently asked questions
What is the correct answer to this question?
33
Why is this the correct answer?
Use the inclusion–exclusion formula for three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substitution gives 112 + 104 + 93 − 46 − 38 − 35 + 17 = 207. The elements in none of the sets are outside the union, so their number is n(U) − n(A ∪ B ∪ C) = 240 − 207 = 33.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
Advertisement