Sets: Practice Questions

In a group of \( 40 \) students, \( 25 \) like coffee, \( 18 \) like tea, and \( 7 \) like neither. Find the number of students who like both coffee and tea.

In a group of 100 students, 50 study Spanish (\(S\)), 40 study French (\(F\)), and 35 study German (\(G\)). The following information is also known:
- 15 students study both Spanish and French.
- 12 students study both French and German.
- 10 students study both Spanish and German.
- 5 students study all three languages.

Calculate the number of students who study exactly two of these languages.

Let set A = \(\{2, 4, 6, 8, 10\}\) and set B = \(\{1, 2, 3, 4, 5\}\). Find the set represented by \((A \cap B') \cup (B \cap A')\).

Let the universal set be \(U = \{x \in \mathbb{Z} | 1 \le x \le 20\}\). Define three sets:
\(A = \{x | x \text{ is a multiple of 3}\}\)
\(B = \{x | x \text{ is a multiple of 4}\}\)
\(C = \{x | x \text{ is a multiple of 6}\}\)

Which of the following statements is correct?

In the following Venn diagram, three sets \( A \), \( B \), and \( C \) intersect within a universal set \( U \). If a region is shaded such that it is entirely inside circle \( A \) but strictly outside both circle \( B \) and circle \( C \), write the set notation that represents this shaded region.

A survey of \( 100 \) people found that \( 60 \) like tea (\(T\)), \( 45 \) like coffee (\(C\)), and \( 30 \) like milk (\(M\)). If \( 20 \) people like both \( T \) and \( C \), \( 15 \) like both \( C \) and \( M \), \( 10 \) like both \( T \) and \( M \), and \( 5 \) people like all three, find the number of people who like none of these drinks.

The universal set \( U = \{x | x \text{ is an integer, } 1 \le x \le 10\} \). Set \( A = \{2, 3, 5, 7\} \) and set \( B = \{x | x \text{ is an even number}\} \). Find the value of \( n(A \cap B') \).

Consider the sets \( A \) and \( B \) such that \( n(U) = 30 \), \( n(A) = 15 \), and \( n(B) = 18 \).
Let \( n(A \cap B) = x \).

(a) Find the smallest possible value of \( x \).
(b) Find the largest possible value of \( x \).
(c) If \( x = 10 \), find \( n(A \cup B)' \).
(d) Given that \( A \subseteq B \), write down the value of \( x \).

Given the set notation expressions below, simplify each one using set properties and identities:
(a) \( (A \cap B) \cup (A \cap B') \)
(b) \( (A \cup B) \cap (A \cup B)' \)
(c) \( A \cap (A \cup B) \)
(d) Use a Venn diagram with three sets to illustrate the region defined by \( A' \cap (B \cup C) \) and explain which areas are included.