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. Step 1: Understand the universal set \(\varepsilon\) includes all \(x\) such that \(-20 \leq x \leq 20\). 3. Step 2: Find \(M'\) which is all \(x\) in \(\varepsilon\) not in \(M\). Since \(M = \{x : -20 < x < 15\}\), \(M' = \{x : x \leq -20 \text{ or } x \geq 15\}\). Given \(\varepsilon\), \(M' = \{-20, 15, 16, \ldots, 20\}\). 4. Step 3: Find \(N \cap P\). \(N = \{x : -10 < x \leq 10\}\), \(P = \{x : 9 \leq x < 18\}\). Intersection is \(\{x : 9 \leq x \leq 10\}\). 5. Step 4: Find \(N \cup P\). Union covers \(x\) from \(-10 < x \leq 10\) and \(9 \leq x < 18\), so union is \(\{x : -10 < x < 18\}\). 6. Step 5: Count elements in \(N \cup P\). Since \(x\) are integers in \(\varepsilon\), elements are integers from \(-9\) to \(17\) inclusive, total \(17 - (-9) + 1 = 27\). 7. Step 6: Find \(N \cap M'\). \(M' = \{-20, 15, \ldots, 20\}\), \(N = \{-9, \ldots, 10\}\), intersection is \(\{15, \ldots, 10\}\) which is empty because 15 > 10, so \(N \cap M' = \emptyset\). --- 8. Problem 14: Given \(A = \{x : 10 \leq x \leq 80, x \text{ divisible by } 4\}\), \(B = \{x : 10 \leq x \leq 80, x \text{ divisible by } 5\}\), find: a. \(n(A)\) b. \(n(B)\) c. \(n(A \cap B)\) 9. Step 1: List elements divisible by 4 between 10 and 80: 12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80. Count: 18. 10. Step 2: List elements divisible by 5 between 10 and 80: 10,15,20,25,30,35,40,45,50,55,60,65,70,75,80. Count: 15. 11. Step 3: Find intersection divisible by both 4 and 5, i.e., divisible by 20: 20,40,60,80. Count: 4. --- 12. Problem 15: \(\varepsilon = \{x : x \text{ integer}, 4 \leq x \leq 15\}\), \(A = \{x : 30 < 5x < 45\}\), \(B = \{x : x \text{ factor of } 60\}\). 13. Step 1: List \(\varepsilon = \{4,5,6,7,8,9,10,11,12,13,14,15\}\). 14. Step 2: Solve \(30 < 5x < 45\) implies \(6 < x < 9\), so \(A = \{7,8\}\). 15. Step 3: Factors of 60 within \(\varepsilon\) are \(B = \{4,5,6,10,12,15\}\). 16. Step 4: Find \((A \cup B)'\) complement in \(\varepsilon\). \(A \cup B = \{4,5,6,7,8,10,12,15\}\), so complement is \(\{9,11,13,14\}\). 17. Step 5: 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 is \(\{9,11,13,14\}\). 18. Step 6: Find \((A \cap B)'\). \(A \cap B = \emptyset\) (no common elements), so complement is \(\varepsilon\) itself, count \(n((A \cap B)') = 12\). --- 19. Problem 16 is incomplete, skipping. --- 20. Problem 19: \(\varepsilon = \{5,6,7,8,10,11,12,13,14,15\}\), \(P = \{x : x \text{ multiple of } 2\} = \{6,8,10,12,14\}\), \(Q = \{x : x \text{ multiple of } 3\} = \{6,12,15\}\), \(R = \{x : x \text{ multiple of } 5\} = \{5,10,15\}\). 21. a.i. \(P \cap R = \{10\}\). 22. a.ii. \(Q \cap (P \cup R)\). \(P \cup R = \{5,6,8,10,12,14,15\}\), intersection with \(Q\) is \(\{6,12,15\}\). 23. b. Find \(n(Q' \cap R)\). \(Q' = \varepsilon \setminus Q = \{5,7,8,10,11,13,14\}\), intersect with \(R = \{5,10,15\}\) gives \(\{5,10\}\), so count is 2. --- 24. Problem 20: \(\varepsilon = \{7,8,9,10,11,12,13,14,15\}\), \(A = \{x : x \text{ multiple of } 3\} = \{9,12,15\}\), \(B = \{x : x \text{ odd}\} = \{7,9,11,13,15\}\), \(C = \{x : 10 \leq x \leq 13\} = \{10,11,12,13\}\). 25. a. \(n(A \cup B)\). \(A \cup B = \{7,9,11,12,13,15\}\), count is 6. 26. b. \(C' = \varepsilon \setminus C = \{7,8,9,14,15\}\). 27. c. Find \(x \in (B \cap C')\) and \(x \notin A\). \(B \cap C' = \{7,9,15\}\), removing \(A = \{9,12,15\}\) leaves \(\{7\}\). Final answer: \(x = 7\). --- 28. Problem 21: \(\varepsilon = \{3,4,5,6,7,8,10,11\}\), \(P = \{x : x \text{ factor of } 12\} = \{3,4,6\}\), \(Q = \{x : x \text{ odd integer}\} = \{3,5,7,11\}\). 29. a. List elements of \(P = \{3,4,6\}\). --- Final answers summarized: 13a. \(M' = \{-20, 15, 16, 17, 18, 19, 20\}\) 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. \(n(Q' \cap R) = 2\) 20a. \(n(A \cup B) = 6\) 20b. \(C' = \{7,8,9,14,15\}\) 20c. \(x = 7\) 21a. \(P = \{3,4,6\}\)