1. Consider the universal set \( U = \{x : x \text{ is multiple of 2 and } 0 < x \leq 30\} \)
\( A = \{x : x \text{ is a multiple of 6}\} \)
\( B = \{x : x \text{ is a multiple of 8}\} \)
(i) List all elements of sets \(A\) and \(B\) in tabular form
\[A = \{6, 12, 18, 24, 30\}\]
\[B = \{8, 16, 24\}\]
(ii) Find \(A \cap B\)
\[A \cap B = \{24\}\]
(iii) Draw a Venn diagram
(Drawing is required, cannot be done here directly.)
2. Let \(U = \{x : x \text{ is an integer and } 0 < x \leq 150\}\)
\(G = \{x : x = 2^m \text{ for integer } m \text{ and } 0 \leq m \leq 12\}\)
\(H = \{x : x \text{ is a square}\}\)
(i) List all elements of sets \(G\) and \(H\) in tabular form
\[G = \{1, 2, 4, 8, 16, 32, 64, 128\}\]
\[H = \{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144\}\]
(ii) Find \(G \cup H\)
\[G \cup H = \{1, 2, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 128, 144\}\]
(iii) Find \(G \cap H\)
\[G \cap H = \{1, 4, 16, 64\}\]
3. Consider the sets \(P = \{x : x \text{ is a prime number and } 0 < x \leq 20\}\)
\(Q = \{x : x \text{ is a divisor of 210 and } 0 < x \leq 20\}\)
(i) Find \(P \cap Q\)
\[P = \{2, 3, 5, 7, 11, 13, 17, 19\}\]
\[Q = \{1, 2, 3, 5, 6, 7, 10, 14, 15\}\]
\[P \cap Q = \{2, 3, 5, 7\}\]
(ii) Find \(P \cup Q\)
\[P \cup Q = \{1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19\}\]
4. Verify the commutative properties of union and intersection for the following pairs of sets
(i) \(A = \{1, 2, 3, 4, 5\}\), \(B = \{4, 6, 8, 10\}\)
Union:
\[A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\}\]
\[B \cup A = \{1, 2, 3, 4, 5, 6, 8, 10\}\]
\[A \cup B = B \cup A\]
Intersection:
\[A \cap B = \{4\}\]
\[B \cap A = \{4\}\]
\[A \cap B = B \cap A\]
(ii) \(A = \mathbb{N}, B = \mathbb{Z}\)
Union:
\[A \cup B = \mathbb{Z}\]
\[B \cup A = \mathbb{Z}\]
\[A \cup B = B \cup A\]
Intersection:
\[A \cap B = \mathbb{N}\]
\[B \cap A = \mathbb{N}\]
\[A \cap B = B \cap A\]
(iii) \(A = \{x \, | \, x \in \mathbb{R} \land x \geq 0\}, B = \mathbb{R}\)
Union:
\[A \cup B = \mathbb{R}\]
\[B \cup A = \mathbb{R}\]
\[A \cup B = B \cup A\]
Intersection:
\[A \cap B = A\]
\[B \cap A = A\]
\[A \cap B = B \cap A\]
5. Verify De Morgan’s Laws for these sets \( U = \{a, b, c, d, e, f, g, h, i, j\} \):
\(A = \{a, b, c, d, g, h\}\), \(B = \{c, d, e, f, j\}\)
(i) \((A \cup B)^c = A^c \cap B^c\)
\[A \cup B = \{a, b, c, d, e, f, g, h, j\}\]
\[(A \cup B)^c = \{i\}\]
\[A^c = \{e, f, i, j\}, \, B^c = \{a, b, g, h, i\}\]
\[A^c \cap B^c = \{i\}\]
\[(A \cup B)^c = A^c \cap B^c\]
(ii) \((A \cap B)^c = A^c \cup B^c\)
\[A \cap B = \{c, d\}\]
\[(A \cap B)^c = \{a, b, e, f, g, h, i, j\}\]
\[A^c = \{e, f, i, j\}, \, B^c = \{a, b, g, h, i\}\]
\[A^c \cup B^c = \{a, b, e, f, g, h, i, j\}\]
\[(A \cap B)^c = A^c \cup B^c\]
6. If \(U = \{1, 2, 3, \dots, 20\}\), \(A = \{1, 3, 5, \dots, 19\}\), verify the following:
(i) \(A \cup A’ = U\)
\[A = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}, \, A’ = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}\]
\[A \cup A’ = \{1, 2, 3, \dots, 20\} = U\]
(ii) \(A \cap U = A\)
\[A \cap U = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\} = A\]
(iii) \(A \cap A’ = \phi\)
\[A \cap A’ = \phi\]
7. In a class of 55 students, 34 like to play cricket and 30 like to play hockey. Each student likes to play at least one of the two games. How many students like to play both games?
\[\text{Total students} = 55, \, |C| = 34, \, |H| = 30, \, |C \cup H| = 55\]
\[|C \cap H| = |C| + |H| – |C \cup H|\]
\[|C \cap H| = 34 + 30 – 55 = 9\]
\[\text{Students who like both games: } 9\]
8. In a group of 500 employees:
\(250\) can speak Urdu,
\(150\) can speak English,
\(50\) can speak Punjabi,
\(40\) can speak Urdu and English,
\(30\) can speak English and Punjabi,
\(10\) can speak Urdu and Punjabi,
How many can speak all three languages?
Let \(x\) be the number of people who can speak all three languages.
\[|U| + |E| + |P| – |U \cap E| – |E \cap P| – |U \cap P| + |U \cap E \cap P| = 500\]
\[250 + 150 + 50 – 40 – 30 – 10 + x = 500\]
\[420 + x = 500\]
\[x = 80\]
\[\text{Employees who can speak all three languages: } 80\]
9. In sports events:
\(19\) people wear blue shirts,
\(15\) wear green shirts,
\(3\) wear blue and green shirts,
\(4\) wear a cap and blue shirts,
\(2\) wear a cap and green shirts,
Total number of people with either a blue or green shirt or cap is \(25\).
How many people are wearing caps?
Let \(x\) be the number of people wearing caps.
\[|B| + |G| + |B \cap G| + x = 25\]
\[19 + 15 – 3 + x = 25\]
\[31 + x = 25\]
\[x = 25 – 31 = -6\]
10. In a training session:
\(17\) participants have laptops,
\(11\) have tablets,
\(9\) have laptops and tablets,
\(6\) have laptops and books,
\(4\) have both tablets and books,
\(8\) participants have all three items.
Total participants with laptops, tablets, or books is \(35\). How many participants have books?
Using the inclusion-exclusion principle:
\[|L| + |T| + |B| – |L \cap T| – |T \cap B| – |L \cap B| + |L \cap T \cap B| = 35\]
\[17 + 11 + |B| – 9 – 4 – 6 + 8 = 35\]
\[17 + 11 + |B| – 11 + 8 = 35\]
\[25 + |B| = 35\]
\[|B| = 35 – 25 = 10\]
\[\text{Participants with books: } 10\]
