Sets：练习题

The universal set \( U = \{1, 2, 3, \dots, 30\} \).
Set \( S \) is the set of square numbers in \( U \) and set \( C \) is the set of cube numbers in \( U \).
Find the value of \( n(S \cup C) \).

At a language school, a group of 50 students are surveyed about the languages they study. 30 students study Spanish (S) and 25 study French (F). If 10 students study both Spanish and French, how many students study neither of these two languages?

Let the universal set \( U = \{1, 2, 3, \dots, 12\} \).
Set \( A = \{ x \mid x \text{ is a factor of 12} \} \) and set \( B = \{ x \mid x \text{ is a multiple of 3} \} \).
Find the number of elements in the set \( A \cap B' \).

Let \( U = \{x \mid x \in \mathbb{Z}^+, x < 15\} \) and \( A = \{x \mid x \in U \text{ and } x \text{ is prime}\} \). Find \( n(A') \).

In a class of 30 students, 15 students play football (\( F \)) and 12 students play tennis (\( T \)).
If 5 students play both football and tennis, find the value of \( n(F \cup T)' \).

If set \( X \) is defined as the set of letters in the word 'BANANA', find the value of \( n(X) \).

In a universal set \( U \) where \( n(U) = 50 \), let \( n(A) = 22 \), \( n(B) = 27 \), and \( n(A \cup B) = 40 \). Find the value of \( n(A \cap B)' \).

In a group of 60 people, set \( A \) and set \( B \) are such that \( n(A) = 3x \), \( n(B) = 2x \), and \( n(A \cap B) = x - 5 \). If the number of people in neither set is 15, find the value of \( x \).

The universal set \( U = \{x \mid x \text{ is an integer and } 1 \le x \le 20\} \).
Set \( A = \{x \mid x \text{ is a multiple of 5}\} \).
Set \( B = \{x \mid x \text{ is a factor of 20}\} \).

a) List the elements of set \( A \).
b) List the elements of set \( B \).
c) Find \( n(A \cup B) \).
d) List the elements of \( A \cap B' \).

In a group of 60 tourists, 35 speak French (F) and 28 speak German (G). Let \( x \) be the number of tourists who speak both languages. There are 8 tourists who speak neither language.

a) Write down an expression, in terms of \( x \), for the number of tourists who speak only French.
b) Form an equation in \( x \) and solve it to find the number of tourists who speak both languages.
c) Describe the Venn diagram that represents this information.
d) One tourist is picked at random from the group. Find the probability that they speak German.