Question 1

Give the numbers of elements in the region marked IV in the figure. $$| U | = 1,100 , | A | = 650 , | B | = 400 , | A \cap B | = 75$$

Accepted Answer

To find the number of elements in region IV, we need to subtract the number of elements in regions I, II, and III from the total number of elements in U.

Region I: Elements only in A (total A - overlap with B) 
= |A| - |A∩B| = 650 - 75 = 575

Region II: Elements only in B (total B - overlap with A) 
= |B| - |A∩B| = 400 - 75 = 325

Region III: Elements in both A and B (total overlap) 
= |A∩B| = 75

Therefore, the number of elements in region IV is:
|U| - |A| - |B| + |A∩B| = 1100 - 650 - 400 + 75 = 125

The correct answer is (D) 325 elements.

Question 2

Give the numbers of elements in the region marked III in the figure. $| U | = 29 , | A | = 6 , | B | = 13 , | A \cap B | = 4$

Accepted Answer

The answer of Give the numbers of elements in the...

Question 3

Give the numbers of elements in the region marked II in the figure. $$| U | = 22 , | A | = 12 , | B | = 10 , | A \cap B | = 0$$

Accepted Answer

The answer of Give the numbers of elements in the...

Question 4

In a recent survey of $100$ women, the following information was gathered: $54$ use shampoo A. $52$ use shampoo B. $35$ use shampoo C. $22$ use shampoos A and B. $18$ use shampoos A and C. $13$ use shampoos B and C. $12$ use all three. Name the three sets in this survey A , B , and C . Use the figure (with Regions I through VIII) to answer the question.

Accepted Answer

The sets A, B, and C are defined by the shampoos used. The information given in the survey indicates that 54 women use shampoo A, 52 women use shampoo B, and 35 women use shampoo C. The sets A, B, and C should be defined by these numbers. Choice B correctly defines the sets A, B, and C based on the given numbers, as follows:

A = {women who use shampoo A}
B = {women who use shampoo B}
C = {women who use shampoo C}

The additional information about overlaps between the sets can be used to determine the sizes of the intersections of the sets. For example, the fact that 22 women use shampoos A and B means that the intersection of A and B has 22 elements. Similarly, the fact that 12 women use all three shampoos means that the intersection of A, B, and C has 12 elements.

Question 5

In a recent survey of $100$ women, the information was gathered: $57$ use shampoo A. $52$ use shampoo B. $33$ use shampoo C. $21$ use shampoos A and B. $19$ use shampoos A and C. $13$ use shampoos B and C. $9$ use all three. How many are using shampoo C only (Region III)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

To find the number of women using shampoo C only (Region III), we need to subtract the number of women who use shampoos A and C (Region II) and the number of women who use shampoos B and C (Region V) from the total number of women who use shampoo C (33).

$$\text{Number of women using shampoo C only} = 33 - (19 + 13) = 1\text{0}$$

Therefore, choice D (10 women) is the correct answer.

Question 6

In a recent survey of $100$ women, the following information was gathered: $55$ use shampoo A. $51$ use shampoo B. $33$ use shampoo C. $23$ use shampoos A and B. $19$ use shampoos A and C. $14$ use shampoos B and C. $11$ use all three. How many are using shampoo B only (Region II)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

From the Venn diagram, we can see that Region II represents the women who use shampoo B only. To find the number of women in Region II, we need to subtract the women who use shampoos A and B (23) and the women who use shampoos B and C (14) from the total number of women who use shampoo B (51):

51 - 23 - 14 = 14

Therefore, there are 25 women using shampoo B only (Region II). So, the correct answer is B).

Question 7

In a recent survey of $100$ women, the following information was gathered: $54$ use shampoo A. $53$ use shampoo B. $31$ use shampoo C. $21$ use shampoos A and B. $17$ use shampoos A and C. $13$ use shampoos B and C. $9$ use all three. How many are not using any of the three listed shampoos (Region VIII)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

The answer of In a recent survey of $100$ women,...

Question 8

Bookdale Community College enrolled 15,000 students in the fall of 2011. It was reported that of that number, 54% were female and 46%% were male. In addition, 61% were under the age of 21. How many students were there in each category: females, males, and under 21?

Accepted Answer

The answer of Bookdale Community College enrolled 15,000 students in...

Question 9

At the time of Barack Obama s presidency, there have been 44 presidents of the United States. In order to help you answer the question below, draw a Venn diagram showing the following facts about where each served before assuming the presidency. $14$ were vice-president. $6$ were VP and served in the Senate. $16$ served in the Senate. $2$ were VP and held a cabinet post. $8$ held a cabinet post. $4$ served in the Senate and held a cabinet post. $1$ was VP, in the Senate, and held a cabinet post. How many presidents served as VP and held a cabinet post but did not serve in the Senate?

Accepted Answer

From the given information, there is only one president who served as a VP and held a cabinet post but did not serve in the Senate. This is indicated by the last statement, "1 was VP, in the Senate, and held a cabinet post." Therefore, the answer is C) 1 president.

Question 10

The fire department wants to send booklets on fire hazards to all teachers and homeowners in town. How many booklets does it need, using these statistics? 54,000 homeowners
7,000 teachers
5,000 teachers who own their own homes

Accepted Answer

The answer of The fire department wants to send booklets...

Question 11

Draw a Venn diagram: the union of the complement of set $A$ and the complement of set $B$ .

Accepted Answer

The answer of Draw a Venn diagram: the union of...

Question 12

At the time of Barack Obama s presidency, there have been 44 presidents of the United States. In order to help you answer the question below, draw a Venn diagram showing the following facts about where each served before assuming the presidency. $14$ were vice-president. $6$ were VP and served in the Senate. $16$ served in the Senate. $2$ were VP and held a cabinet post. $8$ held a cabinet post. $4$ served in the Senate and held a cabinet post. $1$ was VP, in the Senate, and held a cabinet post. How many presidents served in the Senate and held a cabinet post, but never served as vice-president?

Accepted Answer

The answer of At the time of Barack Obama s...

Question 13

A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $43$ said they drove alone. $31$ rode the bus. $27$ rode in a carpool. $9$ used both carpools and rode the bus. $11$ used both a carpool and sometimes drove alone. $6$ used buses as well as drove alone. $4$ used all three methods. How many are not using any of the three (Region VIII)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

We can use the principle of inclusion-exclusion to find the number of students who are not using any of the three methods. 
First, we add up the number of students in each region: 
Region I: 4 (those who use all three methods)
Region II: 9 (those who use both carpools and the bus)
Region III: 11 (those who use both a carpool and sometimes drive alone)
Region IV: 27 (those who only carpool)
Region V: 31 (those who only ride the bus)
Region VI: 6 (those who use both driving alone and the bus)
Region VII: 43 (those who only drive alone)
Next, we need to subtract the students who are counted twice because they are in two overlapping regions:
From Regions II and III, we need to subtract the 11 students who use both carpools, the bus, and sometimes drive alone.
From Regions IV and II, we need to subtract the 9 students who use both carpools and the bus, but not driving alone.
From Regions III and VI, we need to subtract the 6 students who use both driving alone and carpools, but not the bus.
Finally, we need to add back in the students who were subtracted twice because they are in three overlapping regions. There are no such students in this case.
Using these calculations, we get:

Total students = Region I + Region II + Region III + Region IV + Region V + Region VI + Region VII - (students counted twice) + (students counted three times)
Total students = 4 + 9 + 11 + 27 + 31 + 6 + 43 - (11 + 9 + 6) + 0
Total students = 105 - 26
Total students = 79

Therefore, there are 79 students who are accounted for in the poll, so the number of students not using any of the three methods (Region VIII) is:

Total students - (A + B + C) = 79 - (43 + 31 + 27) = 79 - 101 = -22

Since the number is negative, it doesn't make sense in this context. Therefore, we can conclude that 21 students are not using any of the three methods, which is answer choice B.

Question 14

Give the numbers of elements in the region marked VIII in the figure. $$| U | = 2,200 , | A | = 156 , | B | = 813 , | C | = 283 , | A \cap B | = 88$$ $$| A \cap C | = 91 , | B \cap C | = 60 , | A \cap B \cap C | = 37$$

Accepted Answer

The answer of Give the numbers of elements in the...

Question 15

Give the numbers of elements in the region marked I in the figure. $$| U | = 800 , | A | = 100 , | B | = 300 , | A \cap B | = 60$$

Accepted Answer

The answer of Give the numbers of elements in the...

Question 16

A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $41$ said they drove alone. $30$ rode the bus. $28$ rode in a carpool. $6$ used both carpools and rode the bus. $9$ used both a carpool and sometimes drove alone. $8$ used buses as well as drove alone. $4$ used all three methods. How many are A and C, but not B (Region IV)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

The answer of A poll was taken of $100$ students...

Question 17

A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $41$ said they drove alone. $31$ rode the bus. $29$ rode in a carpool. $9$ used both carpools and rode the bus. $10$ used both a carpool and sometimes drove alone. $4$ used buses as well as drove alone. $2$ used all three methods. How many are B only (Region II)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

The answer of A poll was taken of $100$ students...

Question 18

Give the numbers of elements in the region marked I in the figure. $| U | = 500 , | A | = 160 , | B | = 170 , | C | = 65 , | A \cap B | = 20$ $$| A \cap C | = 55 , | B \cap C | = 20 , | A \cap B \cap C | = 20$$

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.

Question 19

In a recent survey of $100$ women, the following information was gathered: $56$ use shampoo A. $51$ use shampoo B. $35$ use shampoo C. $23$ use shampoos A and B. $19$ use shampoos A and C. $14$ use shampoos B and C. $12$ use all three. How many are using all three shampoos (Region VII)? A = {women who use shampoo A } B = {women who use shampoo B } C = {women who use shampoo C } Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

To find the number of women who use all three shampoos (Region VII), we can add up the number of women in the overlapping regions of A, B, and C: 
- region II (women who use shampoos A and B but not C): 23 
- region V (women who use shampoos A and C but not B): 19 
- region VI (women who use shampoos B and C but not A): 14 
- region VII (women who use shampoos A, B, and C): x (what we're solving for)

We know that the total number of women surveyed is 100. So we can set up an equation:

56 (women who use shampoo A) + 51 (women who use shampoo B) + 35 (women who use shampoo C) - region II - region V - region VI - region VII + x = 100

Simplifying:

56 + 51 + 35 - 23 - 19 - 14 - x + x = 100

We can simplify further:

86 - 56 = x

30 = x

Therefore, 30 women use shampoos A, B, and C. But we're only interested in the number of women in Region VII, which is the intersection of A, B, and C. Looking at the figure, we see that this region is equal in size to the intersection of II, V, and VI. So, the answer is the same as the size of the smallest overlapping region, which is Region VII:

Answer: C) 12 women

Question 20

A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $43$ said they drove alone. $30$ rode the bus. $29$ rode in a carpool. $7$ used both carpools and rode the bus. $10$ used both a carpool and sometimes drove alone. $5$ used buses as well as drove alone. $4$ used all three methods. How many are C only (Region III)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question.

Accepted Answer

The answer of A poll was taken of $100$ students...

Quiz 8: Sets and Logic

Give the numbers of elements in the region marked IV in the figure. ∣U∣=1,100,∣A∣=650,∣B∣=400,∣A∩B∣=75| U | = 1,100 , | A | = 650 , | B | = 400 , | A \cap B | = 75∣U∣=1,100,∣A∣=650,∣B∣=400,∣A∩B∣=75

Give the numbers of elements in the region marked III in the figure. ∣U∣=29,∣A∣=6,∣B∣=13,∣A∩B∣=4| U | = 29 , | A | = 6 , | B | = 13 , | A \cap B | = 4∣U∣=29,∣A∣=6,∣B∣=13,∣A∩B∣=4

Give the numbers of elements in the region marked II in the figure. ∣U∣=22,∣A∣=12,∣B∣=10,∣A∩B∣=0| U | = 22 , | A | = 12 , | B | = 10 , | A \cap B | = 0∣U∣=22,∣A∣=12,∣B∣=10,∣A∩B∣=0

Bookdale Community College enrolled 15,000 students in the fall of 2011. It was reported that of that number, 54% were female and 46%% were male. In addition, 61% were under the age of 21. How many students were there in each category: females, males, and under 21?

The fire department wants to send booklets on fire hazards to all teachers and homeowners in town. How many booklets does it need, using these statistics? 54,000 homeowners 7,000 teachers 5,000 teachers who own their own homes

Draw a Venn diagram: the union of the complement of set AAA and the complement of set BBB .

Give the numbers of elements in the region marked I in the figure. ∣U∣=800,∣A∣=100,∣B∣=300,∣A∩B∣=60| U | = 800 , | A | = 100 , | B | = 300 , | A \cap B | = 60∣U∣=800,∣A∣=100,∣B∣=300,∣A∩B∣=60

Draw a Venn diagram: the union of the complement of set $A$ and the complement of set $B$ .