Set Theory

Question

Suppose A, B, and C are subsets of a universal set U. Also suppose that n(U) = 150 n(A) = n(B) = 2n(C) = 50, $A\cap B\cap C = ∅$  and $n(A\cap B) = 2n(A\cap C) = 2n(B\cap C) = 10$. How many elements are in at least two of the sets A, B, and C?               

Answer 1

The question is simple with lots of unnecessary information and checks if you understand the given information and the meaning of the question.

To find the number of elements that are in at least two of the sets A, B, and C, we need to consider the elements in the intersections of these sets.

Given information:
- n(A∩B) = 10
- n(A∩C) = 5
- n(B∩C) = 5
- n(A∩B∩C) = 0 (the intersection of all three sets is empty, A∩B∩C = ∅)

We can calculate the number of elements in at least two of the sets by adding the elements in the intersections, but we need to be careful not to double-count the elements that are common to all three intersections.

Step 1: Add the elements in the intersections.
n(A∩B) + n(A∩C) + n(B∩C) = 10 + 5 + 5 = 20

Step 2: Subtract the elements that are common to all three intersections, if any.
Since n(A∩B∩C) = 0, there are no elements common to all three intersections, so we don't need to subtract anything.
{ otherwise the formula should be like:
n(A∩B) + n(A∩C) + n(B∩C) - 2*n(A∩B∩C) }

Therefore, the number of elements that are in at least two of the sets A, B, and C is 20.