1. Problem 13: Given sets 𝜀 = {x : -20 ≤ x ≤ 20}, M = {x : -20 < x < 15}, N = {x : -10 < x ≤ 10}, P = {x : 9 ≤ x < 18}, find: a. M' (complement of M in 𝜀) b. N ∩ P (intersection of N and P) c. n(N ∪ P) (number of elements in union of N and P) d. N ∩ M' (intersection of N and complement of M) 2. Problem 14: Given A = {x : 10 ≤ x ≤ 80 and x divisible by 4}, B = {x : 10 ≤ x ≤ 80 and x divisible by 5}, find: a. n(A) b. n(B) c. n(A ∩ B) 3. Problem 15: Given 𝜀 = {x : x integer 4 ≤ x ≤ 15}, A = {x : 30 < 5x < 45}, B = {x : x factor of 60}, find: a. List elements of: i. 𝜀 ii. A iii. B iv. (A ∪ B)' v. A' ∩ B' vi. (A ∩ B)' b. Find n((A ∩ B)') 4. Problem 16: Given 𝜀 = {x : positive integer 5 ≤ x ≤ 40}, P = {x : multiple of 4} 5. Problem 19: Given 𝜀 = {5,6,7,8,10,11,12,13,14,15}, P = multiples of 2, Q = multiples of 3, R = multiples of 5, find: a. List elements of: i. P ∩ R ii. Q ∩ (P ∪ R) b. Find n(Q' ∩ R) 6. Problem 20: Given 𝜀 = {7,8,9,10,11,12,13,14,15}, A = multiples of 3, B = odd numbers, C = {x : 10 ≤ x ≤ 13}, find: a. n(A ∪ B) b. List elements of C' c. Find x such that x ∈ (B ∩ C') and x ∉ A 7. Problem 21: Given 𝜀 = {3,4,5,6,7,8,10,11}, P = factors of 12, Q = odd integers, list elements of P --- ### Step-by-step solutions: **13a. Find M'** - M = {x : -20 < x < 15} means all x strictly between -20 and 15. - 𝜀 = {x : -20 ≤ x ≤ 20} includes -20 and 20. - Complement M' = elements in 𝜀 not in M. - So M' = {-20} ∪ [15, 20] **13b. Find N ∩ P** - N = (-10, 10], P = [9, 18) - Intersection is numbers x where x > -10 and x ≤ 10 and also 9 ≤ x < 18 - Overlap is [9, 10] **13c. Find n(N ∪ P)** - N ∪ P covers from > -10 to < 18, combining intervals (-10,10] and [9,18) - Union is (-10, 18) - Since x are real numbers, but assuming integers for counting: - Integers in N: from -9 to 10 inclusive → count = 10 - (-9) + 1 = 20 - Integers in P: from 9 to 17 inclusive → count = 17 - 9 + 1 = 9 - Overlap integers: 9 and 10 → 2 integers - So n(N ∪ P) = n(N) + n(P) - n(N ∩ P) = 20 + 9 - 2 = 27 **13d. Find N ∩ M'** - M' = {-20} ∪ [15, 20] - N = (-10, 10] - Intersection is empty set since N is between -10 and 10, M' is outside that range **14a. Find n(A)** - A = multiples of 4 between 10 and 80 inclusive - Multiples of 4: 12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80 - Count: 18 elements **14b. Find n(B)** - B = multiples of 5 between 10 and 80 inclusive - Multiples of 5: 10,15,20,25,30,35,40,45,50,55,60,65,70,75,80 - Count: 15 elements **14c. Find n(A ∩ B)** - Intersection are numbers divisible by both 4 and 5 → divisible by 20 - Multiples of 20 between 10 and 80: 20,40,60,80 - Count: 4 elements **15a.i. List 𝜀** - Integers 4 ≤ x ≤ 15: {4,5,6,7,8,9,10,11,12,13,14,15} **15a.ii. List A** - A = {x : 30 < 5x < 45} → divide inequalities by 5: - 6 < x < 9 - Integers satisfying: 7,8 **15a.iii. List B** - B = factors of 60 - Factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60 - Restrict to 𝜀: 4,5,6,10,12,15 **15a.iv. List (A ∪ B)'** - A ∪ B = {4,5,6,7,8,10,12,15} - Complement in 𝜀: elements in 𝜀 not in A ∪ B - 𝜀 = {4,5,6,7,8,9,10,11,12,13,14,15} - So (A ∪ B)' = {9,11,13,14} **15a.v. List A' ∩ B'** - A' = 𝜀 extbackslash A = {4,5,6,9,10,11,12,13,14,15} - B' = 𝜀 extbackslash B = {5,7,8,9,11,13,14} - Intersection: {5,9,11,13,14} **15a.vi. List (A ∩ B)'** - A ∩ B = elements in both A and B - A = {7,8}, B = {4,5,6,10,12,15} - Intersection is empty set - So (A ∩ B)' = 𝜀 = {4,5,6,7,8,9,10,11,12,13,14,15} **15b. Find n((A ∩ B)')** - Since A ∩ B = ∅, n((A ∩ B)') = n(𝜀) = 12 **19a.i. List P ∩ R** - P = multiples of 2 in 𝜀 = {6,8,10,12,14} - R = multiples of 5 in 𝜀 = {5,10,15} - Intersection: {10} **19a.ii. List Q ∩ (P ∪ R)** - Q = multiples of 3 = {6,9,12,15} - P ∪ R = {6,8,10,12,14,5,15} - Intersection: {6,12,15} **19b. Find n(Q' ∩ R)** - Q' = elements not multiples of 3 = {5,7,8,10,11,13,14} - R = {5,10,15} - Intersection: {5,10} - Count: 2 **20a. Find n(A ∪ B)** - A = multiples of 3 in 𝜀 = {9,12,15} - B = odd numbers in 𝜀 = {7,9,11,13,15} - Union = {7,9,11,12,13,15} - Count: 6 **20b. List elements of C'** - C = {10,11,12,13} - 𝜀 = {7,8,9,10,11,12,13,14,15} - C' = 𝜀 extbackslash C = {7,8,9,14,15} **20c. Find x such that x ∈ (B ∩ C') and x ∉ A** - B ∩ C' = odd numbers in C' = {7,9,15} - A = {9,12,15} - x ∉ A means exclude 9 and 15 - So x = 7 **21a. List elements of P** - P = factors of 12 in 𝜀 = {3,4,6} --- Final answers: 13a. M' = {-20} ∪ [15,20] 13b. N ∩ P = [9,10] 13c. n(N ∪ P) = 27 13d. N ∩ M' = ∅ 14a. n(A) = 18 14b. n(B) = 15 14c. n(A ∩ B) = 4 15a.i. 𝜀 = {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 ∪ B)' = {9,11,13,14} 15a.v. A' ∩ B' = {5,9,11,13,14} 15a.vi. (A ∩ B)' = 𝜀 15b. n((A ∩ B)') = 12 19a.i. P ∩ R = {10} 19a.ii. Q ∩ (P ∪ R) = {6,12,15} 19b. n(Q' ∩ R) = 2 20a. n(A ∪ B) = 6 20b. C' = {7,8,9,14,15} 20c. x = 7 21a. P = {3,4,6}