give the number of elements in the region

Question

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

Answer 1

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