Give the numbers of elements in the region marked I in the figure. $| U | = 500 , | A | = 160 , | B | = 170 , | C | = 65 , | A \cap B | = 20$ $$| A \cap C | = 55 , | B \cap C | = 20 , | A \cap B \cap C | = 20$$
A) 150 elements
B) 0 elements
C) 160 elements
D) 10 elements
E) 105 elements

Question

Give the numbers of elements in the region marked I in the figure. $| U | = 500 , | A | = 160 , | B | = 170 , | C | = 65 , | A \cap B | = 20$ $$| A \cap C | = 55 , | B \cap C | = 20 , | A \cap B \cap C | = 20$$ 
A) 150 elements 
B) 0 elements 
C) 160 elements 
D) 10 elements 
E) 105 elements

Quizplus · Accepted Answer

To find the number of elements in region I, we need to subtract the elements that are not in region I from the total number of elements in set U. 
Elements in set A but not in region I: |A|-|A∩B∩C|-|A∩C|=160-20-55=85
Elements in set B but not in region I: |B|-|A∩B∩C|-|B∩C|=170-20-20=130
Elements in set C but not in region I: |C|-|A∩B∩C|-|A∩C|-|B∩C|=65-20-55-20=−30
(Note: The negative value for elements in set C but not in region I indicates that there is an overlap, and some of these elements are actually in region I.)
Therefore, the total number of elements in region I is:
|I|=|U|-|A|-|B|-|C|+|A∩B|+|A∩C|+|B∩C|+|A∩B∩C|
|I|=500-160-170-65+20+55+20+20
|I|=105. 
Thus, the answer is (E) 105 elements.

Give the Numbers of Elements in the Region Marked I ∣U∣=500,∣A∣=160,∣B∣=170,∣C∣=65,∣A∩B∣=20| U | = 500 , | A | = 160 , | B | = 170 , | C | = 65 , | A \cap B | = 20∣U∣=500,∣A∣=160,∣B∣=170,∣C∣=65,∣A∩B∣=20

Give the Numbers of Elements in the Region Marked I $| U | = 500 , | A | = 160 , | B | = 170 , | C | = 65 , | A \cap B | = 20$