Give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U| = 220, |A| = 20, |B| = 14, |C| = 12 |A∩B| = 2, |A∩C| = 2, |B∩C| = 6, |A∩B∩C| = 1
| I | =
| II | =
| III | =
| IV | =
| V | =
| VI | =
| VII | =
| VIII |
We don't see the figure, so we don't know where to put the answers!
To determine the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure, we can use the principle of inclusion-exclusion. This principle states that for any three sets A, B, and C, the formula for determining the size of their union is:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
Using the given information, we can fill in the formula and calculate the values for each region:
| I | = |A ∩ B ∩ C| = 1
| II | = |B ∩ C| - |A ∩ B ∩ C| = 6 - 1 = 5
| III | = |A ∩ C| - |A ∩ B ∩ C| = 2 - 1 = 1
| IV | = |B ∩ C| - |A ∩ B ∩ C| = 6 - 1 = 5
| V | = |B ∪ C| - |B ∩ C| = |B| + |C| - |B ∩ C| = 14 + 12 - 6 = 20
| VI | = |A ∪ C| - |A ∩ C| = |A| + |C| - |A ∩ C| = 20 + 12 - 2 = 30
| VII | = |A ∪ B| - |A ∩ B| = |A| + |B| - |A ∩ B| = 20 + 14 - 2 = 32
| VIII | = |U| - |A ∪ B ∪ C| = |U| - (|A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|) = 220 - (20 + 14 + 12 - 2 - 2 - 6 + 1) = 169
Hence, the numbers of elements in the regions are as follows:
| I | = 1
| II | = 5
| III | = 1
| IV | = 5
| V | = 20
| VI | = 30
| VII | = 32
| VIII | = 169