Question 1: If set A = {1, 2, 3, 4} and set B = {3, 4, 5, 6}, write the set notation for the elements common to both sets.

Answer: A ∩ B = {3, 4}

Question 2: Given sets A = {x | x is even and 2 ≤ x ≤ 8} and B = {x | x is a multiple of 3 between 1 and 10}, write the sets in roster notation.

Answer: A = {2, 4, 6, 8}; B = {3, 6, 9}

Question 3: Construct set notation for all students who study Mathematics or Science, given that 20 students study Math, 15 study Science, and 5 study both.

Answer: |M ∪ S| = |M| + |S| - |M ∩ S| = 20 + 15 - 5 = 30

Question 4: In a class, 18 students like football, 12 like basketball, and 6 like both. Represent this information using set notation for students who like either sport.

Answer: |F ∪ B| = 18 + 12 - 6 = 24

Question 5: A survey shows 25 people prefer tea, 30 prefer coffee, and 10 prefer both. Write the set notation for those who prefer only tea.

Answer: People who prefer only tea = |T| - |T ∩ C| = 25 - 10 = 15

Question 6: Calculate the number of students who like neither sport if 40 students like football, 25 like basketball, and 10 like both, in a class of 60.

Answer: Total liking at least one sport = 40 + 25 - 10 = 55; Neither = 60 - 55 = 5

Question 7: Two sets A and B are such that |A| = 8, |B| = 10, and |A ∩ B| = 3. Find |A ∪ B| using set notation.

Answer: A ∪ B = |A| + |B| - |A ∩ B| = 8 + 10 - 3 = 15

Question 8: In a company, 50 employees work in Department X, 40 in Department Y, and 10 work in both. Write the set notation for employees working in either department.

Answer: Employees in either department = 50 + 40 - 10 = 80

Question 9: A shop sells apples and oranges. 80 customers buy apples, 60 buy oranges, and 20 buy both. Express the set of customers who buy only apples using set notation.

Answer: Customers who buy only apples = |A| - |A ∩ O| = 80 - 20 = 60

Question 10: Identify a common mistake when students write the union of two sets as simply adding their sizes without subtracting the intersection, and explain why this is wrong.

Answer: Common mistake: |A| + |B| without subtracting |A ∩ B| leads to overcounting elements in both sets; correct formula is |A ∪ B| = |A| + |B| - |A ∩ B|.