1. Draw a Venn diagram for the following pairs of sets and in each case give \( P \cup Q \) as a list of elements. (a) \( \mathrm{P}=\{1,3,10,11,4,2\} \) \[ \begin{aligned} \mathrm{Q} & =\{1,2,3,5,8\} \\ \mathrm{Q} & =\{a, b, c, d\} \end{aligned} \] (b) \( \mathrm{P}=\{a, c, e, g\} \) (c) P is the set of positive multiples of 3 less than 20 \( Q \) is the set of positive multiples of 4 less than 20 (d) P is the set of positive multiples of 2 less than 31 \( Q \) is the set of positive multiples of 6 less than 31 2. If \( \mathrm{E}=\{0,1,2, \ldots, 19,20\} \) (the universal set) and \( \mathrm{A}=\{6,7,9,10\} \), find (a) \( \mathrm{A}^{\prime} \) (b) \( n(A) \) (c) \( n(E) \) (d) \( n\left(A^{\prime}\right) \) 3. Let \( A \) and \( B \) be subsets of a universal set \( E \) and suppose \( n(E)=210, n(A)=100, n(B) \) \( =70 \) and \( \mathrm{n}(\mathrm{A} \cap \mathrm{B})=40 \). Calculate (a) \( n(A \cup B) \) (b) \( n\left(A^{\prime}\right) \) (c) \( n\left(A \cap B^{\prime}\right) \) (d) \( n\left(A^{\prime} \cap B\right) \) (e) \( \mathrm{n}\left(\mathrm{B}^{\prime}\right) \) (f) \( n\left(A^{\prime} \cap B^{\prime}\right) \) 4. In a class of 120 students, 80 indicated that they are registered for Accounting, 68 indicated that they are registered for Economics, and 42 indicated that they are registered for both courses. How many students are registered for (a) Accounting only? (b) none of the above courses? 5. E is the set of positive whole numbers less than 23 (the universal set), \( \mathrm{A}=\{2,4,6,8,10,12\} \) \( \mathrm{B}=\{2,5,8,11,14\} \) and \( \mathrm{C}=\{2,6,10,14,18,22\} \). (a) Draw a Venn diagram writing all the elements in the correct places. (b) Find (listing any set answers in set notation) (i) \( n(A) \) (ii) \( \mathrm{A} \cap \mathrm{B} \) (iii) \( \mathrm{B} \cap \mathrm{C} \) (iv) \( \mathrm{A} \cap \mathrm{B} \cap \mathrm{C} \) (v) \( n(A \cap B \cap C) \) (vi) \( \mathrm{A}^{\prime} \) (vii) \( A \cup B \) (viii) \( n(A \cup B) \) (ix) \( n\left[(A \cup B)^{\prime}\right] \) (x) \( \mathrm{A}^{\prime} \cap \mathrm{B} \) (xi) \( n\left[(A \cup B \cup C)^{\prime}\right] \) 6. Let \( \mathrm{A}, \mathrm{B} \) and C be subsets of a universal set E and suppose \( \mathrm{n}(\mathrm{E})=100, n(\mathrm{~A})=28, n(B) \) \( =30, \mathrm{n}(\mathrm{C})=34, \mathrm{n}(\mathrm{A} \cap \mathrm{B})=8, \mathrm{n}(\mathrm{A} \cap \mathrm{C})=10, \mathrm{n}(\mathrm{B} \cap \mathrm{C})=15 \) and \( \mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=5 \). Calculate (a) \( n(A \cup B \cup C) \) (b) \( n\left(A^{\prime} \cap B \cap C\right) \) (c) \( n(A \cap(B \cup C)) \) (d) \( n\left(A \cap(B \cup C)^{\prime}\right) \) (e) \( n\left(A^{\prime} \cap B^{\prime} \cap C^{\prime}\right) \) (f) \( n\left(A^{\prime} \cap(B \cup C)\right) \)

Let's solve the problems step by step. ### Problem 1: Venn Diagrams and Unions #### (a) Given: - \( P = \{1, 3, 10, 11, 4, 2\} \) - \( Q = \{1, 2, 3, 5, 8\} \) **Step 1:** Identify the union \( P \cup Q \). The union of two sets includes all unique elements from both sets. \[ P \cup Q = \{1, 2, 3, 4, 5, 8, 10, 11\} \] #### (b) Given: - \( P = \{a, c, e, g\} \) - \( Q = \{a, b, c, d\} \) **Step 1:** Identify the union \( P \cup Q \). \[ P \cup Q = \{a, b, c, e, g, d\} \] #### (c) Given: - \( P \) is the set of positive multiples of 3 less than 20: \( P = \{3, 6, 9, 12, 15, 18\} \) - \( Q \) is the set of positive multiples of 4 less than 20: \( Q = \{4, 8, 12, 16\} \) **Step 1:** Identify the union \( P \cup Q \). \[ P \cup Q = \{3, 4, 6, 8, 9, 12, 15, 16, 18\} \] #### (d) Given: - \( P \) is the set of positive multiples of 2 less than 31: \( P = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\} \) - \( Q \) is the set of positive multiples of 6 less than 31: \( Q = \{6, 12, 18, 24, 30\} \) **Step 1:** Identify the union \( P \cup Q \). \[ P \cup Q = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30\} \] ### Problem 2: Set Operations Given: - \( E = \{0, 1, 2, \ldots, 20\} \) - \( A = \{6, 7, 9, 10\} \) #### (a) Find \( A' \) **Step 1:** Identify the complement of \( A \) in \( E \). \[ A' = E - A = \{0, 1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\} \] #### (b) Find \( n(A) \) **Step 1:** Count the elements in \( A \). \[ n(A) = 4 \] #### (c) Find \( n(E) \) **Step 1:** Count the elements in \( E \). \[ n(E) = 21 \] #### (d) Find \( n(A') \) **Step 1:** Count the elements in \( A' \). \[ n(A') = 17 \] ### Problem 3: Set Operations with Given Values Given: - \( n(E) = 210 \) - \( n(A) = 100 \) - \( n(B) = 70 \) - \( n(A \cap B) = 40 \) #### (a) Find \( n(A \cup B) \) **Step 1:** Use the formula for the union of two sets. \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) = 100 + 70 - 40 = 130 \] #### (b) Find \( n(A') \) **Step 1:** Use the total number of elements. \[ n(A') = n(E) - n(A) = 210 - 100 = 110 \] #### (c) Find \( n(A \cap B') \) **Step 1:** Use the formula for the intersection. \[ n(A \cap B') = n(A) - n(A \cap B) = 100 - 40 = 60 \] #### (d) Find \( n(A' \cap B) \) **Step 1:** Use the formula for the intersection. \[ n(A' \cap B) = n(B) - n(A \cap B) = 70 - 40 = 30 \] #### (e) Find \( n(B') \) **Step 1:** Use the total number of elements. \[ n(B') = n(E) - n(B) = 210 - 70 = 140 \] #### (f) Find \( n(A' \cap B') \) **Step 1:** Use the formula for the complement. \[ n(A' \cap B') = n(E) - n(A \cup B) = 210 - 130 = 80 \] ### Problem 4: Students Registered for Courses Given: - Total students = 120 - Registered for Accounting = 80 - Registered for Economics = 68 - Registered for both = 42 #### (a) Find students registered for Accounting only. **Step 1:** Use the formula for students registered for only one course. \[ \text{Accounting only} = n(A) - n(A \cap E) = 80 - 42 = 38 \] #### (b) Find students registered for none of the courses. **Step 1:** Use the formula for students registered for at least one course. \[ n(A \cup E) = n(A) + n(E) - n(A \cap E) = 80 + 68 - 42 = 106 \] \[ \text{None} = \text{Total} - n(A \cup E) = 120 - 106 = 14 \] ### Problem 5: Venn Diagram and Set Operations Given: - \( E \) is the set of positive whole numbers less than 23: \( E = \{1, 2, \ldots, 22\} \) - \( A = \{2, 4, 6, 8, 10, 12\} \) - \( B = \{2, 5, 8, 11, 14\} \) - \( C