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.