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Homework

Questions (P7.)

1. a
   1. A. True
   2. B. Order does not matter in sets
2. MISSISSIPPI
3. 1. \subseteq
   2. \in
   3. \subseteq
   4. \in
   5. \in x wrong \emptyset is a \subseteq of all sets
   6. \subseteq
4. 10. a) \{S_4, S_5, S_9\}
   11. b) ??
   12. c) quadrillion
   13. d)
       1. F
       2. T (if order does not matter)
       3. T
       4. F
       5. T
       6. T
       7. F
       8. F
       9. F
       10. T
       11. F
       12. F
       13. T
       14. F
       15. T
       16. T
5. 11. D_1=\{1\}, D_2=\{1,2\}, D_{10}=\{1,2,5\}
   12. b)
       1. T
       2. F
       3. T
       4. T
       5. T?
       6. F
       7. T
       8. F
       9. F
       10. F
       11. F
       12. T
   13. c) |D_{10}|=3, |D_{19}|=2
   14. D) |\mathcal{D}|=9

Questions	Answer
1. 1. (a) True or false? {red, white, blue} = {white, blue, red}. 2. (b) What is wrong with this statement: Red is the first element of the set {red, white, blue}?	a. True b. Order does not matter in sets
2. Which has the larger cardinality? The set of letters in the word MISSISSIPPI or the set of letters in the word FLORIDA ?	MISSIPPI
3. Fill in the blank with the appropriate symbol, ∈ or ⊆. 1. (a) {1, 2, 3} {1, 2, 3, 4} 2. (b) 3 {1, 2, 3, 4} 3. (c) {3} {1, 2, 3, 4} 4. (d) {𝑎} {{𝑎}, {𝑏}, {𝑎, 𝑏}} 5. (e) ∅ {{𝑎}, {𝑏}, {𝑎, 𝑏}} 6. (f) {{𝑎}, {𝑏}} {{𝑎}, {𝑏}, {𝑎, 𝑏}}
9. Let 𝑆1 = {𝑜, 𝑛, 𝑒}, 𝑆2 = {𝑡, 𝑤, 𝑜}, 𝑆3 = {𝑡, ℎ, 𝑟, 𝑒, 𝑒}, and so on. 1. (a) Find all 𝑘 ∈ {1, 2, . . . , 10} with |𝑆𝑘| = 4. 2. (b) Find distinct indices 𝑗, 𝑘 ∈ ℕ with 𝑆𝑗 = 𝑆𝑘. 3. (c) Find the smallest value of 𝑘 ∈ ℕ with 𝑎 ∈ 𝑆𝑘. 4. (d) Let 𝒮 = {𝑆𝑘}40 𝑘=1. Determine whether the following statements are true or false. 1. (i) 𝑆13 = {𝑛, 𝑒, 𝑖, 𝑡, ℎ, 𝑒, 𝑟} 2. (ii) {𝑛, 𝑒, 𝑡} ⊆ 𝑆20 3. (iii) 𝑆1 ∈ 𝒮 4. (iv) 𝑆3 ⊆ 𝒮 5. (v) ∅ ∈ 𝒮 6. (vi) ∅ ⊂ 𝒮 7. (vii) ∅ ⊆ 𝒮 8. (viii) 𝑆1 ⊆ 𝑆11 9. (ix) 𝑆1 ⊆ 𝑆21 10. (x) 𝑆1 ⊂ 𝑆21 11. (xi) {𝑛, 𝑖, 𝑒} ∈ 𝒮 12. (xii) {{𝑓, 𝑜, 𝑢, 𝑟}} ⊆ 𝒮 13. (xiii) 𝑢 ∈ 𝑆40 14. (xiv) 𝒫(𝑆9) ⊆ 𝒫(𝑆19) 15. (xv) {𝑠, 𝑖} ∈ 𝒫(𝑆6) 16. (xvi) 𝑤 ∈ 𝒫(𝑆2)
10. For 𝑘 ∈ {1, 2, . . . , 20}, let 𝐷𝑘 = {𝑥 ∣ 𝑥 is a prime number which divides 𝑘} and let 𝒟 = {𝐷𝑘 ∣ 𝑘 ∈ {1, 2, . . . , 20}}. 1. (a) Find 𝐷1, 𝐷2, 𝐷10, and 𝐷20. 2. (b) True or False: 1. (i) 𝐷2 ⊂ 𝐷10 2. (ii) 𝐷7 ⊆ 𝐷10 3. (iii) 𝐷10 ⊂ 𝐷20 4. (iv) ∅ ∈ 𝒟 5. (v) ∅ ⊂ 𝒟 6. (vi) 5 ∈ 𝒟 7. (vii) {5} ∈ 𝒟 8. (viii) {4, 5} ∈ 𝒟 9. (ix) {{3}} ⊆ 𝒟 10. (x) 𝒫(𝐷9) ⊆ 𝒫(𝐷6) 11. (xi) 𝒫({3, 4}) ⊆ 𝒟 12. (xii) {2, 3} ∈ 𝒫(𝐷12) 3. (c) Find |𝐷10| and |𝐷19|. 4. (d) Find |𝒟|

Unit 1

Krish Hw 2

1. A = {1, 2, 3}, B = {2, 3, 4} and C = {1, 2, 4}. List the elements of the specified set
   1. (a) A ∩ B; | {2,3}
   2. (b) A ∪ B; | {1,2,3,4}
   3. (c) C\A; | {1}
   4. (d) A ∪ (B ∩ C); | {2,1,3}
   5. (e) (A ∩ C) ∪ (B ∩ C); | {1,2,4}
   6. (f) A × B; | {(1,2),(1,3),(1,4),(2,3),(2,2),(2,4),(3,2),(3,3),(3,4)}
   7. (g) B × A; | {(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2) ,(4,3)}
   8. (h) (A × B) ∩ (B × A). | {(2,2),(3,3)}
2. M_2 = {2, 4, 6, 8, 10, · · · } and M_3 = {3, 6, 9, 12, 15, · · · }. Find:
   1. (a) M_2 ∩ M_3;
      1. M_6
   2. (b) M_3 \backslash M_2
      1. \{x|x=6k-3 \space\forall\space k \in\mathbb{N}\}