1. Problem 13: Given sets \(\varepsilon = \{x : -20 \leq x \leq 20\}\), \(M = \{x : -20 < x < 15\}\), \(N = \{x : -10 < x \leq 10\}\), \(P = \{x : 9 \leq x < 18\}\), find: a. \(M'\) (complement of \(M\) in \(\varepsilon\)) b. \(N \cap P\) (intersection of \(N\) and \(P\)) c. \(n(N \cup P)\) (number of elements in union of \(N\) and \(P\)) d. \(N \cap M'\) (intersection of \(N\) and complement of \(M\)) 2. Problem 14: Given \(A = \{x : 10 \leq x \leq 80, x \text{ divisible by } 4\}\) and \(B = \{x : 10 \leq x \leq 80, x \text{ divisible by } 5\}\), find: a. \(n(A)\) b. \(n(B)\) c. \(n(A \cap B)\) 3. Problem 15: Given \(\varepsilon = \{x : x \text{ integer}, 4 \leq x \leq 15\}\), \(A = \{x : 30 < 5x < 45\}\), \(B = \{x : x \text{ factor of } 60\}\), find: a. List elements of: i. \(\varepsilon\) ii. \(A\) iii. \(B\) iv. \((A \cup B)'\) v. \(A' \cap B'\) vi. \((A \cap B)'\) b. Find \(n((A \cap B)')\) 4. Problem 16: Given \(\varepsilon = \{x : x \text{ positive integer}, 5 \leq x \leq 40\}\), \(P = \{x : x \text{ multiple of } 4\}\), \(Q = \{x : x \text{ perfect square}\}\) (no specific question given, so no calculation here). 5. Problem 19: Given \(\varepsilon = \{5,6,7,8,10,11,12,13,14,15\}\), \(P = \{x : x \text{ multiple of } 2\}\), \(Q = \{x : x \text{ multiple of } 3\}\), \(R = \{x : x \text{ multiple of } 5\}\), find: a. List elements of: i. \(P \cap R\) ii. \(Q \cap (P \cup R)\) b. Find \(n(Q' \cap R)\) 6. Problem 20: Given \(\varepsilon = \{7,8,9,10,11,12,13,14,15\}\), \(A = \{x : x \text{ multiple of } 3\}\), \(B = \{x : x \text{ odd}\}\), \(C = \{x : 10 \leq x \leq 13\}\), find: a. \(n(A \cup B)\) b. List elements of \(C'\) c. Find \(x\) such that \(x \in (B \cap C')\) and \(x \notin A\) 7. Problem 21: Given \(\varepsilon = \{3,4,5,6,7,8,10,11\}\), \(P = \{x : x \text{ factor of } 12\}\), \(Q = \{x : x \text{ odd integer}\}\), find: a. List elements of \(P\) b. List elements of \((P \cup Q)'\) --- ### Step-by-step solutions: **13a. Find \(M'\):** - \(M = \{x : -20 < x < 15\}\) means all \(x\) strictly between -20 and 15. - \(\varepsilon = \{x : -20 \leq x \leq 20\}\). - Complement \(M' = \varepsilon \setminus M = \{x : x \leq -20 \text{ or } x \geq 15\}\). - Since \(\varepsilon\) includes -20 and 20, \(M' = \{-20, 15, 16, ..., 20\}\). **13b. Find \(N \cap P\):** - \(N = \{x : -10 < x \leq 10\}\), i.e., \(-9, -8, ..., 10\). - \(P = \{x : 9 \leq x < 18\}\), i.e., \(9, 10, ..., 17\). - Intersection \(N \cap P = \{9, 10\}\). **13c. Find \(n(N \cup P)\):** - Union \(N \cup P = \{x : -9 \leq x \leq 10\} \cup \{9 \leq x < 18\} = \{-9, ..., 17\}\). - Count elements from -9 to 17 inclusive: total \(17 - (-9) + 1 = 27\). **13d. Find \(N \cap M'\):** - \(M' = \{x : x \leq -20 \text{ or } x \geq 15\}\) within \(\varepsilon\). - \(N = \{x : -9 \leq x \leq 10\}\). - No overlap because \(N\) is between -9 and 10, \(M'\) is outside that range. - So \(N \cap M' = \emptyset\). --- **14a. Find \(n(A)\):** - \(A = \{x : 10 \leq x \leq 80, x \text{ divisible by } 4\}\). - Multiples of 4 between 10 and 80 inclusive: 12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80. - Count: from 12 to 80 step 4, number of terms \(= \frac{80 - 12}{4} + 1 = \frac{68}{4} + 1 = 17 + 1 = 18\). **14b. Find \(n(B)\):** - \(B = \{x : 10 \leq x \leq 80, x \text{ divisible by } 5\}\). - Multiples of 5 between 10 and 80 inclusive: 10,15,20,25,30,35,40,45,50,55,60,65,70,75,80. - Count: \(\frac{80 - 10}{5} + 1 = \frac{70}{5} + 1 = 14 + 1 = 15\). **14c. Find \(n(A \cap B)\):** - Intersection are numbers divisible by both 4 and 5, i.e., divisible by 20. - Multiples of 20 between 10 and 80 inclusive: 20,40,60,80. - Count: 4. --- **15a.i. List \(\varepsilon\):** - Integers from 4 to 15 inclusive: \(\{4,5,6,7,8,9,10,11,12,13,14,15\}\). **15a.ii. List \(A = \{x : 30 < 5x < 45\}\):** - Solve inequalities: \(30 < 5x < 45 \Rightarrow 6 < x < 9\). - Since \(x\) integer, \(x = 7,8\). - So \(A = \{7,8\}\). **15a.iii. List \(B = \{x : x \text{ factor of } 60\}\) within \(\varepsilon\):** - Factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60. - Within \(4 \leq x \leq 15\), factors are \(4,5,6,10,12,15\). - So \(B = \{4,5,6,10,12,15\}\). **15a.iv. Find \((A \cup B)'\):** - \(A \cup B = \{4,5,6,7,8,10,12,15\}\). - Complement in \(\varepsilon\) is elements in \(\varepsilon\) not in \(A \cup B\): \(\{9,11,13,14\}\). **15a.v. Find \(A' \cap B'\):** - \(A' = \varepsilon \setminus A = \{4,5,6,9,10,11,12,13,14,15\}\). - \(B' = \varepsilon \setminus B = \{7,8,9,11,13,14\}\). - Intersection \(A' \cap B' = \{9,11,13,14\}\). **15a.vi. Find \((A \cap B)'\):** - \(A \cap B = \{7,8\} \cap \{4,5,6,10,12,15\} = \emptyset\). - So \((A \cap B)' = \varepsilon\) (all elements). **15b. Find \(n((A \cap B)')\):** - Since \(A \cap B = \emptyset\), \(n((A \cap B)') = n(\varepsilon) = 12\). --- **19a.i. Find \(P \cap R\):** - \(P = \{x : x \text{ multiple of } 2\} = \{6,8,10,12,14\}\) within \(\varepsilon\). - \(R = \{x : x \text{ multiple of } 5\} = \{10,15\}\). - Intersection \(P \cap R = \{10\}\). **19a.ii. Find \(Q \cap (P \cup R)\):** - \(Q = \{x : x \text{ multiple of } 3\} = \{6,12,15\}\). - \(P \cup R = \{6,8,10,12,14,15\}\). - Intersection \(Q \cap (P \cup R) = \{6,12,15\}\). **19b. Find \(n(Q' \cap R)\):** - \(Q' = \varepsilon \setminus Q = \{5,7,8,10,11,13,14\}\). - \(R = \{10,15\}\). - Intersection \(Q' \cap R = \{10\}\). - So \(n(Q' \cap R) = 1\). --- **20a. Find \(n(A \cup B)\):** - \(A = \{x : x \text{ multiple of } 3\} = \{9,12,15\}\). - \(B = \{x : x \text{ odd}\} = \{7,9,11,13,15\}\). - Union \(A \cup B = \{7,9,11,12,13,15\}\). - Count \(= 6\). **20b. List elements of \(C'\):** - \(C = \{10,11,12,13\}\). - \(\varepsilon = \{7,8,9,10,11,12,13,14,15\}\). - Complement \(C' = \varepsilon \setminus C = \{7,8,9,14,15\}\). **20c. Find \(x \in (B \cap C')\) and \(x \notin A\):** - \(B \cap C' = \{7,9,11,13,15\} \cap \{7,8,9,14,15\} = \{7,9,15\}\). - \(A = \{9,12,15\}\). - Elements in \(B \cap C'\) but not in \(A\) are \(\{7\}\). - So \(x = 7\). --- **21a. List elements of \(P\):** - Factors of 12: \(1,2,3,4,6,12\). - Within \(\varepsilon = \{3,4,5,6,7,8,10,11\}\), factors are \(3,4,6\). - So \(P = \{3,4,6\}\). **21b. List elements of \((P \cup Q)'\):** - \(Q = \{x : x \text{ odd integer}\} = \{3,5,7,11\}\). - \(P \cup Q = \{3,4,5,6,7,11\}\). - Complement in \(\varepsilon\) is \(\{8,10\}\). --- Final answers: - 13a. \(M' = \{-20, 15, 16, ..., 20\}\) - 13b. \(N \cap P = \{9, 10\}\) - 13c. \(n(N \cup P) = 27\) - 13d. \(N \cap M' = \emptyset\) - 14a. \(n(A) = 18\) - 14b. \(n(B) = 15\) - 14c. \(n(A \cap B) = 4\) - 15a.i. \(\varepsilon = \{4,5,6,7,8,9,10,11,12,13,14,15\}\) - 15a.ii. \(A = \{7,8\}\) - 15a.iii. \(B = \{4,5,6,10,12,15\}\) - 15a.iv. \((A \cup B)' = \{9,11,13,14\}\) - 15a.v. \(A' \cap B' = \{9,11,13,14\}\) - 15a.vi. \((A \cap B)' = \varepsilon\) - 15b. \(n((A \cap B)') = 12\) - 19a.i. \(P \cap R = \{10\}\) - 19a.ii. \(Q \cap (P \cup R) = \{6,12,15\}\) - 19b. \(n(Q' \cap R) = 1\) - 20a. \(n(A \cup B) = 6\) - 20b. \(C' = \{7,8,9,14,15\}\) - 20c. \(x = 7\) - 21a. \(P = \{3,4,6\}\) - 21b. \((P \cup Q)' = \{8,10\}\)