Sets and Venn Diagrams：練習題

Let the universal set \( U \) be the set of all positive integers less than or equal to 12. Let \( A \) be the set of factors of 12 and \( B \) be the set of even numbers in \( U \). What is \( n(A \cap B) \)?

In a youth club of 40 members, 22 play football, 18 play basketball, and 6 play both sports. How many members play neither football nor basketball?

In a survey of 120 university students, it was found that:
- 65 take French (
- 55 take German (
- 50 take Spanish (
- 25 take both French and German
- 20 take both German and Spanish
- 22 take both French and Spanish
- 8 take all three languages
How many students take at least two of these languages?

Given the universal set \( U = \{x : x \text{ is an integer and } 1 \le x \le 10\} \) and the set \( A = \{2, 3, 5, 7\} \), find the complement set \( A' \).

Let \( U \) be the universal set with \( n(U) = 60 \). Let \( P \) and \( Q \) be two subsets such that \( n(P) = 32 \), \( n(Q) = 28 \), and \( n(P \cap Q') = 18 \). Find \( n(P' \cap Q') \).

Given that the universal set \( \mathcal{E} \) contains 40 elements, \( n(A) = 24 \), and \( n(A \cup B) = 33 \), find the value of \( n(A' \cap B') \).

In a survey of 80 students, 48 study History, 40 study Geography, and 12 study neither. How many students study only History?

In a group of 100 people, 70 speak English, 45 speak French, and 20 speak Spanish. It is known that 25 speak both English and French, 15 speak both English and Spanish, and 10 speak both French and Spanish. If 5 people speak all three languages, how many people in the group speak none of these three languages?

In a survey of 40 students in a class, 22 students play basketball (\(B\)) and 18 students play football (\(F\)). There are 6 students who play neither sport.

(a) Find the number of students who play both sports.
(b) Find the number of students who play only football.
(c) If a student is chosen at random from the class, find the probability that the student plays basketball given that they play football.

A survey was conducted on 50 students regarding their participation in three extracurricular activities: Music (M), Art (A), and Sport (S). The results are as follows:
- 28 students participate in Music.
- 25 students participate in Art.
- 20 students participate in Sport.
- 12 students participate in both Music and Art.
- 11 students participate in both Art and Sport.
- 10 students participate in both Music and Sport.
- 8 students participate in all three activities.

Calculate the number of students who participate in exactly one activity.