150 men live in a certain town:

85 are married
70 have a cell phone
75 own a car
55 are married and have a cell phone
35 have a cellphone and a car
40 are married and have car
30 are married, have a car and have a cellphone. How many men are single and do not have a car? How many men are single and do not have either a car or a cellphone?


I know the answer is 20. I've used an equation where I added and subtracted each entity. I also did the Venn Diagram and Subsets. I'm looking for different ways to solve this question. If you were answering this question. What method would you use?

There are several methods to solve this question, including using set theory, Venn diagrams, or creating a table. One additional method you can use is called the Principle of Inclusion-Exclusion.

To solve this question using the principle of inclusion-exclusion, you can start by calculating the total number of men who have a car or a cellphone. Then, subtract the number of men who have both a car and a cellphone to avoid double-counting. Finally, subtract this number from the total number of men to find the number of men who are single and do not have a car.

Step-by-step solution using the principle of inclusion-exclusion:

1. Total men: 150
2. Men married (A): 85
3. Men with a cellphone (B): 70
4. Men with a car (C): 75
5. Men married and have a cellphone (A ∩ B): 55
6. Men with a cellphone and a car (B ∩ C): 35
7. Men married and have a car (A ∩ C): 40
8. Men married, have a car, and have a cellphone (A ∩ B ∩ C): 30

By using the principle of inclusion-exclusion formula:

Total men with car or cellphone = A + B + C - (A ∩ B) - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)
Total men with car or cellphone = 150 - (85 + 70 + 75 - 55 - 35 - 40 + 30)
Total men with car or cellphone = 150 - 70
Total men with car or cellphone = 80

To find the number of men who are single and do not have a car (S ∩ ¬C):

Single men without a car = Total men - (A ∩ C)
Single men without a car = 150 - 40
Single men without a car = 110

Therefore, there are 110 men who are single and do not have a car.

To find the number of men who are single and do not have either a car or a cellphone (S ∩ ¬(C ∪ B)):

Single men without a car or cellphone = Total men - (A ∩ (C ∪ B))
Single men without a car or cellphone = 150 - (30 + 40 + 20)
Single men without a car or cellphone = 150 - 90
Single men without a car or cellphone = 60

Therefore, there are 60 men who are single and do not have either a car or a cellphone.

To solve this question, you can approach it using the principle of inclusion-exclusion, which is another common method for solving problems involving sets.

Let's define the following sets:
A - Men who are married
B - Men who have a cell phone
C - Men who own a car

We are given the following information:
|A| = 85
|B| = 70
|C| = 75
|A ∩ B| = 55
|B ∩ C| = 35
|A ∩ C| = 40
|A ∩ B ∩ C| = 30

Using the principle of inclusion-exclusion, we can find the number of men who are single and do not have a car. We can start by summing the sizes of sets A, B, and C, and then subtracting the sizes of their intersections:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|

Substituting the given values:

|A ∪ B ∪ C| = 85 + 70 + 75 - 55 - 35 - 40 + 30
|A ∪ B ∪ C| = 130

This means that there are 130 men who are either married, have a cellphone, or own a car. To find the number of men who are single and do not have a car, we subtract this value from the total number of men in the town:

Total number of men = 150
Number of men without a car = Total number of men - |A ∪ B ∪ C| = 150 - 130 = 20

Therefore, there are 20 men who are single and do not have a car.

To find the number of men who are single and do not have either a car or a cellphone, we need to consider the intersection of sets A, B, and C as well:

Number of men without a car and cellphone = |A ∪ B ∪ C| - |A ∩ B ∩ C| = 130 - 30 = 100

Hence, there are 100 men who are single and do not have either a car or a cellphone.