If n(U) = 250, n(A) = 120, n(B) = 110, n(C) = 95, n(A ∩ B) = 52, n(B ∩ C) = 41, n(C ∩ A) = 37, and n(A ∩ B ∩ C) = 19, what is the number of elements not in at least one set?
Answer and explanation
Correct answer: 36
“Not in at least one set” means not belonging to the union, or the complement of A ∪ B ∪ C. Apply inclusion–exclusion: n(A ∪ B ∪ C) = 120 + 110 + 95 − 52 − 41 − 37 + 19 = 214. The pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once to correct the over-subtraction. Hence the number outside the union is n(U) − 214 = 250 − 214 = 36.
Frequently asked questions
What is the correct answer to this question?
36
Why is this the correct answer?
“Not in at least one set” means not belonging to the union, or the complement of A ∪ B ∪ C. Apply inclusion–exclusion: n(A ∪ B ∪ C) = 120 + 110 + 95 − 52 − 41 − 37 + 19 = 214. The pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once to correct the over-subtraction. Hence the number outside the union is n(U) − 214 = 250 − 214 = 36.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
