Give the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, VIII in the figure.
|U| = 240, |A| = 15, |B| = 14, |C| = 24 |A�¿B| = 4, |A�¿C| = 2, |B�¿C| = 7, |A�¿B�¿C| = 1
| I | = 1
This is about the 5th time you've posted this with no indication of how I,II,III,... relate to A,B,C,...
Based on the given information, let's calculate the numbers of elements in regions I, II, III, IV, V, VI, VII, and VIII step-by-step.
Step 1: Find the number of elements in region II
|A̅ ∩ B| = |A| - |A ∩ B| = 15 - 4 = 11
|II| = |A̅ ∩ B ∩ C̅| = |A̅ ∩ B| - |A̅ ∩ B ∩ C| = 11 - 1 = 10
Step 2: Find the number of elements in region III
|A̅ ∩ C̅| = |A| - |A ∩ C| = 15 - 2 = 13
|III| = |A̅ ∩ B̅ ∩ C| = |A̅ ∩ C̅| - |A̅ ∩ B̅ ∩ C| = 13 - 1 = 12
Step 3: Find the number of elements in region IV
|B̅ ∩ C̅| = |B| - |B ∩ C| = 14 - 7 = 7
|IV| = |A ∩ B̅ ∩ C| = |B̅ ∩ C̅| - |A ∩ B̅ ∩ C| = 7 - 1 = 6
Step 4: Find the number of elements in region V
|IV| = |A ∩ B̅ ∩ C| = 6
Step 5: Find the number of elements in region VI
|III| = |A̅ ∩ B̅ ∩ C| = 12
Step 6: Find the number of elements in region VII
|II| = |A̅ ∩ B ∩ C̅| = 10
Step 7: Find the number of elements in region VIII
|I| = |A ∩ B ∩ C| = 1
In summary, the numbers of elements in the regions marked I, II, III, IV, V, VI, VII, and VIII are as follows:
|I| = 1
|II| = 10
|III| = 12
|IV| = 6
|V| = 6
|VI| = 12
|VII| = 10
|VIII| = 1
To determine the number of elements in each region in the figure, we can use the principle of inclusion-exclusion.
Let's break down the regions step by step:
Region I: The number of elements in region I is given as |I| = 1.
Region II: To find the number of elements in region II, we need to exclude region I from the set A. Since |A| = 15, |I| = 1, the number of elements in region II is |A - I| = |A| - |I| = 15 - 1 = 14.
Region III: Similarly, to find the number of elements in region III, we need to exclude region I from the set B. Therefore, the number of elements in region III is |B - I| = |B| - |I| = 14 - 1 = 13.
Region IV: To find the number of elements in region IV, we need to exclude regions I, II, and III from the set U. Since |U| = 240, we need to subtract the elements in regions I, II, and III from it. Therefore, the number of elements in region IV is |U - (I ∪ II ∪ III)| = |U| - (|I| + |II| + |III|) = 240 - (1 + 14 + 13) = 240 - 28 = 212.
Region V: To find the number of elements in region V, we need to exclude regions I and III from the set C. Therefore, the number of elements in region V is |C - (I ∪ III)| = |C| - (|I| + |III|) = 24 - (1 + 13) = 24 - 14 = 10.
Region VI: To find the number of elements in region VI, we need to exclude regions I, II, and V from the set A. Therefore, the number of elements in region VI is |A - (I ∪ II ∪ V)| = |A| - (|I| + |II| + |V|) = 15 - (1 + 14 + 10) = 15 - 25 = 0 (since the result is negative, we take it as 0).
Region VII: To find the number of elements in region VII, we need to exclude regions I, III, and V from the set B. Therefore, the number of elements in region VII is |B - (I ∪ III ∪ V)| = |B| - (|I| + |III| + |V|) = 14 - (1 + 13 + 10) = 14 - 24 = 0 (since the result is negative, we take it as 0).
Region VIII: To find the number of elements in region VIII, we need to exclude regions I, II, III, V, VI, and VII from the set U. Therefore, the number of elements in region VIII is |U - (I ∪ II ∪ III ∪ V ∪ VI ∪ VII)| = |U| - (|I| + |II| + |III| + |V| + |VI| + |VII|) = 240 - (1 + 14 + 13 + 10 + 0 + 0) = 240 - 38 = 202.
Therefore, the numbers of elements in regions I-VIII are as follows:
| I | = 1
| II | = 14
| III | = 13
| IV | = 212
| V | = 10
| VI | = 0
| VII | = 0
| VIII | = 202