Give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U| = 200, |A| = 25, |B| = 18, |C| = 32 |A∩B| = 4, |A∩C| = 2, |B∩C| = 7, |A∩B∩C| =2
Give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U| = 200, |A| = 25, |B| = 18, |C| = 32 |A∩B| = 4, |A∩C| = 2, |B∩C| = 7, |A∩B∩C| =2
Will use bold U for universal set.
There are eight sections. They are
1. U - (A U B U C) --- in none of the sets.
2. A - (B U C) --- in A only.
3. B - (A U C) --- in B only.
4. C - (A U C) --- in C only.
5. (A ∩ B) - C --- in A and B only.
6. (A ∩ C) - B --- in A and C only.
7. (B ∩ C) - A --- in B and C only.
8. A ∩ B ∩ C --- in all three sets.
Region 1:
By the inclusion exclution principle, it is:
|U| - |A| - |B| - |C| + |A ∩ B| + |A ∩ C| + |B ∩ C| - |A ∩ B ∩ C|
= 200 - 25 - 18 - 32 + 4 + 2 + 7 - 2 = 136
The rest of the elements are in A U B U C so |A U B U C| = 200 - 136 = 64
Region 2:
|A U B U C| - |B U C| = |A U B U C| - |B| - |C| + |B ∩ C|
= 64 - 18 - 32 + 7 = 21
Region 3:
|A U B U C| - |A U C| = |A U B U C| - |A| - |C| + |A ∩ C|
= 64 - 25 - 32 + 2 = 9.
Region 4:
|A U B U C| - |A U B| = |A U B U C| - |A| - |B| + |A ∩ B|
= 64 - 25 - 18 + 4 = 25
Region 5:
|A ∩ B| - |A ∩ B ∩ C| = 4 - 2 = 2
Region 6:
|A ∩ C| - |A ∩ B ∩ C| = 2 - 2 = 0
Region 7:
|B ∩ C| - |A ∩ B ∩ C| = 7 - 2 = 5
Region 8:
|A ∩ B ∩ C| = 2
Comments
I assume that the problem comes with a Venn Diagram marked up with I to VIII.
I'd need to see it.