give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U|=290,|A|=40,|B|=12,|C|=32,|A�¿B|=4,|A�¿C|=3,|B�¿C|=4,|A�¿B�¿C|=2
To determine the number of elements in each region marked I, II, III, IV, V, VI, VII, VIII, we can use the principle of inclusion-exclusion.
Here is a step-by-step explanation of how to calculate the numbers:
Step 1: Determine the number of elements in region I (A ∩ B ∩ C):
|A ∩ B ∩ C| = |A ∩ B| - |A ∩ B ∩ C| = 4 - 2 = 2
Step 2: Determine the number of elements in region II (A ∩ B - A ∩ B ∩ C):
|A ∩ B - A ∩ B ∩ C| = |A ∩ B| - |A ∩ B ∩ C| = 4 - 2 = 2
Step 3: Determine the number of elements in region III (B ∩ C - A ∩ B ∩ C):
|B ∩ C - A ∩ B ∩ C| = |B ∩ C| - |A ∩ B ∩ C| = 4 - 2 = 2
Step 4: Determine the number of elements in region IV (A ∩ B ∩ C' - A ∩ B ∩ C):
|A ∩ B ∩ C' - A ∩ B ∩ C| = |A ∩ B ∩ C'| - |A ∩ B ∩ C| = |A ∩ B| - |A ∩ B ∩ C| = 4 - 2 = 2
Step 5: Determine the number of elements in region V (A ∩ C - A ∩ B ∩ C):
|A ∩ C - A ∩ B ∩ C| = |A ∩ C| - |A ∩ B ∩ C| = 3 - 2 = 1
Step 6: Determine the number of elements in region VI (A ∩ C - A ∩ B ∩ C):
|B ∩ C - A ∩ B ∩ C| = |B ∩ C| - |A ∩ B ∩ C| = 4 - 2 = 2
Step 7: Determine the number of elements in region VII (A' ∩ B' ∩ C' - A ∩ B ∩ C):
|A' ∩ B' ∩ C' - A ∩ B ∩ C| = |U| - |A| - |B| - |C| + |A ∩ B| + |A ∩ C| + |B ∩ C| - |A ∩ B ∩ C| = 290 - 40 - 12 - 32 + 4 + 3 + 4 - 2 = 215
Step 8: Determine the number of elements in region VIII (A' ∩ B' ∩ C - A ∩ B ∩ C):
|A' ∩ B' ∩ C - A ∩ B ∩ C| = |U| - |A| - |B| + |A ∩ B| - |A ∩ B ∩ C| = 290 - 40 - 12 + 4 - 2 = 240
So, the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII are:
I: 2
II: 2
III: 2
IV: 2
V: 1
VI: 2
VII: 215
VIII: 240