Give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U| = 300, |A| = 30, |B| = 18, |C| = 16 |A�¿B| = 4, |A�¿C| = 3, |B�¿C| = 6, |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| = 300, |A| = 30, |B| = 18, |C| = 16 |A�¿B| = 4, |A�¿C| = 3, |B�¿C| = 6, |A�¿B�¿C| = 2
| I | =____
| II | =____
| III | =____
| IV | =____
| V | =____
| VI | =____
| VII | =____
| VIII | =____
To find the number of elements in each region, we can use the principle of inclusion-exclusion. The principle states that the size of the union of two or more sets can be calculated by summing the sizes of individual sets and subtracting the sizes of their intersections.
Using this principle, we can calculate the number of elements in each region as follows:
Region I: |A ∩ B ∩ C|
Region II: |(A ∩ B) - (A ∩ B ∩ C)|
Region III: |B ∩ C|
Region IV: |(A ∩ C) - (A ∩ B ∩ C)|
Region V: |(A ∪ B ∪ C) - (A ∩ B ∩ C)|
Region VI: |(A ∩ B) - (A ∩ B ∩ C)|
Region VII: |(A ∪ B ∪ C) - (A ∩ B ∩ C)|
Region VIII: |U - (A ∪ B ∪ C)|
Given the values provided:
|A ∩ B ∩ C| = 2
|(A ∩ B) - (A ∩ B ∩ C)| = |A ∩ B| - |A ∩ B ∩ C| = 4 - 2 = 2
|B ∩ C| = 6
|(A ∩ C) - (A ∩ B ∩ C)| = |A ∩ C| - |A ∩ B ∩ C| = 3 - 2 = 1
|(A ∪ B ∪ C) - (A ∩ B ∩ C)| = |A ∪ B ∪ C| - |A ∩ B ∩ C| = |A| + |B| + |C| - |A ∩ B ∩ C| = 30 + 18 + 16 - 2 = 62
|(A ∩ B) - (A ∩ B ∩ C)| = 2 (already calculated)
|(A ∪ B ∪ C) - (A ∩ B ∩ C)| = 62 (already calculated)
|U - (A ∪ B ∪ C)| = |U| - |A ∪ B ∪ C| = 300 - 62 = 238
Therefore, the numbers of elements in the regions are:
|I| = 2
|II| = 2
|III| = 6
|IV| = 1
|V| = 62
|VI| = 2
|VII| = 62
|VIII| = 238
To find the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII, we can use the principle of inclusion-exclusion.
Let's break down the given information:
|U| = 300 represents the total number of elements in the universal set U.
|A| = 30 represents the number of elements in set A.
|B| = 18 represents the number of elements in set B.
|C| = 16 represents the number of elements in set C.
|A�¿B| = 4 represents the number of elements in the intersection of A and B.
|A�¿C| = 3 represents the number of elements in the intersection of A and C.
|B�¿C| = 6 represents the number of elements in the intersection of B and C.
|A�¿B�¿C| = 2 represents the number of elements in the intersection of A, B, and C.
Now, let's calculate the numbers of elements in the given regions:
I: This region represents the elements in A but not in B or C. To calculate the number of elements in region I, we subtract the elements in the intersections from the elements in set A. So, the number of elements in region I is |A| - |A�¿B�¿C|.
Therefore, |I| = 30 - 2 = 28.
II: This region represents the elements in B but not in A or C. To calculate the number of elements in region II, we subtract the elements in the intersections from the elements in set B. So, the number of elements in region II is |B| - |A�¿B�¿C|.
Therefore, |II| = 18 - 2 = 16.
III: This region represents the elements in C but not in A or B. To calculate the number of elements in region III, we subtract the elements in the intersections from the elements in set C. So, the number of elements in region III is |C| - |A�¿B�¿C|.
Therefore, |III| = 16 - 2 = 14.
IV: This region represents the elements in A and B, but not in C. To calculate the number of elements in region IV, we subtract the elements in the intersection of A and B from set B. So, the number of elements in region IV is |B�¿C| - |A�¿B�¿C|.
Therefore, |IV| = 6 - 2 = 4.
V: This region represents the elements in A and C, but not in B. To calculate the number of elements in region V, we subtract the elements in the intersection of A and C from set A. So, the number of elements in region V is |A�¿C| - |A�¿B�¿C|.
Therefore, |V| = 3 - 2 = 1.
VI: This region represents the elements in B and C, but not in A. To calculate the number of elements in region VI, we subtract the elements in the intersection of B and C from set C. So, the number of elements in region VI is |B�¿C| - |A�¿B�¿C|.
Therefore, |VI| = 6 - 2 = 4.
VII: This region represents the elements in A, B, and C. To calculate the number of elements in region VII, we look at the intersection of A, B, and C, which is |A�¿B�¿C|.
Therefore, |VII| = |A�¿B�¿C| = 2.
VIII: This region represents the elements that do not belong to any of the sets A, B, or C. To calculate the number of elements in region VIII, we subtract the elements in sets A, B, and C from the total number of elements in the universal set U. So, the number of elements in region VIII is |U| - |A| - |B| - |C|.
Therefore, |VIII| = 300 - 30 - 18 - 16 = 236.
The numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure are:
| I | = 28
| II | = 16
| III | = 14
| IV | = 4
| V | = 1
| VI | = 4
| VII | = 2
| VIII | = 236