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sets

  1. 1. ทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตนทฤษฎีเซตเบื้องตน ((((((((EEEEEEEElllllllleeeeeeeemmmmmmmmeeeeeeeennnnnnnnttttttttaaaaaaaarrrrrrrryyyyyyyy SSSSSSSSeeeeeeeetttttttt TTTTTTTThhhhhhhheeeeeeeeoooooooorrrrrrrryyyyyyyy)))))))) F ก ““““ F F”””” F 2 F F F F F . . 2537 www.thai-mathpaper.net
  2. 2. F F ก F F ˈ F F 2 15 F F F F ˈ ก ก ก F ก F F (Modern Mathematics) F ก F ก ก ก F F ˈ F F Fก F F F ก ก F F ˈ F F F 8 ก . . 2549
  3. 3. ก 1 ก F F F F ก F ก ก F F ʿก ˈ ก F F F F ก ˈ F ก F F F F F กก F F 22 . . 2549 ก 2 ก F 2 F F F ก ก F F ก F ˈ ก ก F F F F ก ก Fก F F F F 1 . . 2550
  4. 4. 1 1 23 1.1 1 1.2 F 2 1.3 ก 4 1.4 ก ก 7 1.5 ก 11 1.6 F F 15 1.7 F ก ก F 17 1.8 F F ก ก F 22 2 ก F 25 31 2.1 F 26 2.2 ก F 28 ก 33
  5. 5. 1 ก ก กF F F ก F F F F F 1 1 ก F F F F F F F ก ก F F F 1.1 (Sets) ˈ (undefined term) ก F F ก F F F ก ˈ 2 ก 1) ก (Finite sets) ก ˈ ก F 2) F (Infinite Sets) F F ก ก ก ก ก F ก ก F F (Empty set) F ก F {} φ ก F (Relative universe) ก F ก F Fก F ก ก ก ก F F F Fก F ˈ F F F ก F ˈ ก ก F 2 ก 1) ก ก ก F ก ก F ก ก 2) ก ก F ก ก ˈ ก F ก ก F F F 1.1 ก F A = {1, 3, 5, 7, 9}, B = {1, 3, 5, } ก F F A ˈ ก ˈ ก ก ก F F B ก F F ˈ F F F ก ก F F F F F ก F ก ก ก
  6. 6. 2 F 1.1 F A B ˈ F F F ก F F A ˈ F B ก F ก ก A F ก ˈ ก B F ก ก ก ก F F A B F ก F F ก F A = {x | x = 2n + 1, n ∈ N n < 5} B = {y | y = 2k + 1, k ∈ N} 1.2 F (Subset) ก 1.1 ก F ก F F ˈ F F F ก F ⊂ F F F ก F ⊆ ˈ F F ⊂ ˈ F F F 1.2 ก F A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, C = {2, 4, 6, 8, 10, 12} F A ⊆ B F A  C B  C ก 1.2 ก F ก F F 1.2 F A B ˈ F F F ก F F A ˈ F F B ก F A ˈ F B F B F ˈ F A A ⊂ B ‹ ∀x [x ∈ A fl ∃x(x ∈ B fl x ∉ A)] A ⊆ B ‹ ∀x [x ∈ A fl x ∈ B]
  7. 7. F F 3 F ก F ก 2 F 1.3 ก F A = {2, 3, 5}, B = {2, 4, 6}, C = {2, 3, 5} ก F F A = C ก F A ⊆ C C ⊆ A F B ≠ C ก F ก B F F ˈ ก C ก ก ก C F F ˈ ก B F F ก F 1.1 (1) (2) (1) F ก F φ ⊆ A F ก F F (contrapositive) ก φ ⊆ A F F ก F ก F F φ ⊆ A ‹ ∀x [x ∈ φ fl x ∈ A] ‹ ∀x [x ∉ A fl x ∉ φ] F x ˈ ก ก F F x ∉ A F x ∉ A ก ˈ F ก x ∈ U ก F F x ∉ φ ก F F ก F F φ ⊆ A F ก (2) F ก F A ⊆ A F ก F ก F (direct proof) F x ∈ A F F F x ∈ A ก F F ก F F ∀x [x ∈ A fl x ∈ A] F F F ก F ˈ (Idempotence) F F A ⊆ A 1.1 (3) F F F F ˈ ʿก 1.1 F A ˈ , U ˈ ก F 1) φ ⊆ A 2) A ⊆ A 3) A ⊆ U 1.3 F A B ˈ F F F ก F F A = B ก F A ˈ F B B ˈ F A
  8. 8. 4 F ʿก 1.2 F 1.1 (3) 1.3 ก (Power set) F ก ก ก ก ˈ F F 1.4 ก F A = {0, 1, 2} P(A) ก A = {0, 1, 2} F F {0}, {1}, {2}, {0, 1}, {0, 2}, {0, 1, 2}, φ ˈ F A 1.4 F F P(A) = {{0}, {1}, {2}, {0, 1}, {0, 2}, {0, 1, 2}, φ} F 1.5 ก F A = {0, 1}, B = {2, 3} P(A) ∪ P(B), P(A ∪ B), P(A) ∩ P(B) P(A ∩ B) ก A = {0, 1}, B = {2, 3} F F P(A) = {{0}, {1}, {0, 1}, φ} P(B) = {{2}, {3}, {2, 3}, φ} P(A) ∪ P(B) = {{0}, {1}, {0, 1}, φ} ∪ {{2}, {3}, {2, 3}, φ} = {{0}, {1}, {2}, {3}, {0, 1}, {2, 3}, φ} P(A) ∩ P(B) = {{0}, {1}, {0, 1}, φ} ∩ {{2}, {3}, {2, 3}, φ} = {φ} ก A ∪ B = {0, 1, 2, 3} A ∩ B = φ F F P(A ∪ B) = {{0}, {1}, {2}, {3}, {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 3}, {2, 3}, {0, 1, 2}, {0, 1, 3}, {0, 2, 3}, {1, 2,3 }, {0, 1, 2, 3}, φ} P(A ∩ B) = P(φ) = {φ} 1.4 ก A F P(A) P(A) = {B | B ⊆ A}
  9. 9. F F 5 F ก F ก ก F 1) F ก F F A ⊆ B F P(A) ⊆ P(B) F A ⊆ B ก F C ∈ P(A) F F C ⊆ A ก A ⊆ B F F C ⊆ B C ∈ P(B) P(A) ⊆ P(B) 2) F ก F P(A) ∪ P(B) ⊆ P(A ∪ B) ก F C ∈ P(A) ∪ P(B) F F C ∈ P(A) C ∈ P(B) ก C ∈ P(A) F F C ⊆ A ก C ∈ P(B) F F C ⊆ B C ⊆ A ∪ B C ∈ P(A ∪ B) P(A) ∪ P(B) ⊆ P(A ∪ B) F ก 3) F ก F P(A) ∩ P(B) = P(A ∩ B) (⊆) F C ∈ P(A) ∩ P(B) F F C ∈ P(A) C ∈ P(B) C ⊆ A C ⊆ B C ⊆ A ∩ B C ∈ P(A ∩ B) P(A) ∩ P(B) ⊆ P(A ∩ B) (⊇) F C ∈ P(A ∩ B) F F C ⊆ A ∩ B C ⊆ A C ⊆ B C ∈ P(A) C ∈ P(B) 1.2 F A, B ˈ F F 1) F A ⊆ B F P(A) ⊆ P(B) 2) P(A) ∪ P(B) ⊆ P(A ∪ B) 3) P(A) ∩ P(B) = P(A ∩ B)
  10. 10. 6 F C ∈ P(A) ∩ P(B) P(A ∩ B) ⊆ P(A) ∩ P(B) ก F F P(A) ∩ P(B) = P(A ∩ B) ก F 1.2 F F ก ก ก ก F F F ก ก ก ก ก F 1.4 กF F F ก F F 1.2 F ʿก 1.3 1. F A = {1, 2, {1}, {2}, {1, 2}} 2. ก F F P(A ∪ B) ⊈ P(A) ∪ P(B) 3. ก F A, B ˈ F ˈ F ก F U F B ⊂ A F P(A B) = P(A) P(B) F F ˈ F F F F F F ก F F 4. ก F A, B ˈ F ˈ F ก F U A ⊆ B P(A) ⊗ P(B) = {C1 ∩ C2 | C1 ∩ C2 ≠ φ, C1 ∈ P(A) C2 ∈ P(B)} F P(A) ⊗ P(B) ⊆ P(B)
  11. 11. F F 7 1.4 ก ก ก ก ก ˈ F F F 4 F กF 1) 2) F ก 3) F (complement of set) 4) F F ก ก F ก F F 1.4 ก F A = {1, 2, 3, 4}, B = {1, 3, 5}, C = {2, 4, 6} A ∪ B, A ∪ C, B ∪ C, A ∩ B, B ∩ C, A ∩ C, A ∪ B ∪ C A ∩ B ∩ C ก F F F A ∪ B = {1, 2, 3, 4} ∪ {1, 3, 5} = {1, 2, 3, 4, 5} A ∪ C = {1, 2, 3, 4} ∪ {2, 4, 6} = {1, 2, 3, 4, 6} B ∪ C = {1, 3, 5} ∪ {2, 4, 6} = {1, 2, 3, 4, 5, 6} A ∩ B = {1, 2, 3, 4} ∩ {1, 3, 5} = {1, 3} B ∩ C = {1, 3, 5} ∩ {2, 4, 6} = φ A ∩ C = {1, 2, 3, 4} ∩ {2, 4, 6} = {2, 4} 1.4 F A, B ˈ U ˈ ก F 1) ก A B F A ∪ B ก F ก A B A ∪ B = {x | x ∈ A x ∈ B} F ก ʽ ∀x [x ∈ A ∪ B ‹ x ∈ A / x ∈ B] 2) F ก A B F A ∩ B ก F ก A B A ∩ B = {x | x ∈ A x ∈ B} F ก ʽ ∀x [x ∈ A ∩ B ‹ x ∈ A - x ∈ B] 3) F A F A F A′ ก F ก U F ˈ ก A A′ = {x | x ∈ U x ∉ A} F ก ʽ ∀x [x ∈ A′ ‹ x ∈ U - x ∉ A] 4) F A B F A B F A ∩ B′ ก F ก A F F ˈ ก B A B = A ∩ B′ = {x | x ∈ A x ∉ B} F ก ʽ ∀x [x ∈ A B ‹ x ∈ A - x ∉ B]
  12. 12. 8 F A ∪ B ∪ C = {1, 2, 3, 4} ∪ {1, 3, 5} ∪ {2, 4, 6} = {1, 2, 3, 4, 5, 6} A ∩ B ∩ C = {1, 2, 3, 4} ∩ {1, 3, 5} ∩ {2, 4, 6} = φ F ˈ F ก ก ก ก F ก F A, B, C ˈ 1) (⊆) F x ∈ A ∪ A F F x ∈ A x ∈ A x ∈ A F ก (⊇) F ก ⊆ A ∪ A = A 2) F F F F ˈ ʿก 3) (⊆) F x ∈ A ∪ B F F x ∈ A x ∈ B x ∈ B x ∈ A x ∈ B ∪ A (⊇) F ก ⊆ 4) F F F F ˈ ʿก 5) (⊆) F x ∈ (A ∪ B) ∪ C 1.3 ก F F A, B, C ˈ ก F U 1) A ∪ A = A 2) A ∩ A = A 3) A ∪ B = B ∪ A 4) A ∩ B = B ∩ A 5) (A ∪ B) ∪ C = A ∪ (B ∪ C) 6) (A ∩ B) ∩ C = A ∩ (B ∩ C) 7) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 8) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 9) φ′ = U U′ = φ 10) (A′)′ = A 11) (A ∪ B)′ = A′ ∩ B′ 12) (A ∩ B)′ = A′ ∪ B′ 13) F A ⊆ B F B′ ⊆ A′
  13. 13. F F 9 F F x ∈ A ∪ B x ∈ C x ∈ A x ∈ B ∪ C x ∈ A ∪ (B ∪ C) (⊇) F ก ⊆ 6) F F F F ˈ ʿก 7) (⊆) F x ∈ A ∪ (B ∩ C) F F x ∈ A x ∈ B ∩ C x ∈ A x ∈ B x ∈ C x ∈ A x ∈ B x ∈ A x ∈ C x ∈ A ∪ B x ∈ A ∪ C F F x ∈ (A ∪ B) ∩ (A ∪ C) (⊇) F ก ⊆ 8) F F F ˈ ʿก 9) F F F ˈ ʿก 10) (⊆) F x ∈ (A′)′ F F x ∉ A′ x ∈ A (⊇) F ก ⊆ 11) (⊆) F x ∈ (A ∪ B)′ F F x ∉ A ∪ B x ∉ A x ∉ B x ∈ A′ x ∈ B′ x ∈ A′ ∩ B′ (⊇) F ก ⊆ 12) F F F ˈ ʿก 13) F A ⊆ B B′  A′ F x ∈ B′ ก B′  A′ F F x ∉ A′ x ∈ A F F x ∈ B ( ก F ) x ∉ B′ F F F ก F ก F A ⊆ B F B′ ⊆ A′ F ก
  14. 14. 10 F 1) 2.3 (7) (8) F กก ก Fก (De Morgan s Law) 2) ก F 1.3 F F ก F F ก F ก F F F F F ˈ ʿก F ก ก ก 2 3 ก F ก F F F F ก F 2.4 (1) 2.4 (2) F F ก F F F F ˈ ʿก F ก F F F F (Venn Euler s Diagram) ก F ก F ก ˈ 3 F F F ก F A ∪ B F กF F ก F F F ก A ∪ B ( F n(A ∪ B)) F F ก n(A) + n(B) A ∩ B F กF F ˈ F F ก A ∩ B F ก ก F ก ก A ∩ B ( F n(A ∩ B)) F ก A B ก ก F ก A ∩ B ก F F n(A ∪ B) = n(A) + n(B) n(A ∩ B) F ก 1.4 ก F A, B, C ˈ ก ก F F n(A), n(B), n(C) ˈ ก A, B, C F F F 1) n(A ∪ B) = n(A) + n(B) n(A ∩ B) 2) n(A ∪ B ∪ C) = n(A) + n(B) + n(C) n(A ∩ B) n(A ∩ C) n(B ∩ C) + n(A ∩ B ∩ C)
  15. 15. F F 11 ʿก 1.4 1. F 1.3 F F F F 2. F F F A F F ก 2n(A) n(A) ˈ ก A 1.5 ก F F F ก ˈ ก F F F ก ก ก ˈ F ˈ F ก ก ก ก กF ก 1.5 ˈ ก F ก F F A = {Aa | a ∈ K} F 1.5 ก F K = {1, 5, 6} F a ∈ K ก F Aa = {1, 2, 3, , a} F F 1) A 2) A1 3) A5 4) A6 5) A1 ∪ A6 6) A5 ∩ A6 1) ก A = {Aa | a ∈ K} F 1, 5, 6 ∈ K A = {A1, A5, A6} 2) ก Aa = {1, 2, 3, , a} F F A1 = {1} 3) ก Aa = {1, 2, 3, , a} F F A5 = {1, 2, 3, 4, 5} 1.5 ก F K ˈ F a ∈ K F Aa ⊆ U ก K F ˈ ก A ˈ Aa a ∈ K F (family of sets)
  16. 16. 12 F 4) ก Aa = {1, 2, 3, , a} F F A6 = {1, 2, 3, 4, 5, 6} 5) A1 ∪ A6 = {1} ∪ {1, 2, 3, 4, 5, 6} = {1, 2, 3, 4, 5, 6} = A6 6) A5 ∩ A6 = {1, 2, 3, 4, 5} ∩ {1, 2, 3, 4, 5, 6} F F A5 ∩ A6 = {1, 2, 3, 4, 5} = A5 ก F F F 1.6 ก F 1.5 F a a K A ∈ ∪ a a K A ∈ ∩ a a K A ∈ ∪ = A1 ∪ A5 ∪ A6 = A6 = {1, 2, 3, 4, 5, 6} a a K A ∈ ∩ = A1 ∩ A5 ∩ A6 = A1 = {1} 1.6 ก F K ≠ φ ˈ A = {Aa | a ∈ K} ˈ F ก ก A ก ก F a a K A ∈ ∪ = {x | x ∈ Aa a ∈ K} 1.7 ก F K ≠ φ ˈ A = {Aa | a ∈ K} ˈ F ก A ก ก F a a K A ∈ ∩ = {x | x ∈ Aa ก a ∈ K}
  17. 17. F F 13 ก F ก F F F ก F F 1) F 3) F F F F F ˈ ʿก ก F K ˈ A = {Aa | a ∈ K} ˈ B ˈ 1) (⊆) F x ∈ B ∪ a a K A ∈       ∪ F F x ∈ B x ∈ a a K A ∈ ∪ ก x ∈ a a K A ∈ ∪ F F x ∈Aa a ∈ K x ∈ B x ∈ Aa a ∈ K ก x ∈ B ∪ Aa a ∈ K x ∈ ( )a a K B A ∈ ∪∪ (⊇) F y ∈ ( )a a K B A ∈ ∪∪ F F y ∈ B ∪ Aa a ∈ K y ∈ B y ∈ Aa a ∈ K F F y ∈ B y ∈ a a K A ∈ ∪ 1.5 ก F A = {Aa | a ∈ K} ˈ B ˈ F F F 1) B ∪ a a K A ∈       ∪ = ( )a a K B A ∈ ∪∪ 2) B ∩ a a K A ∈       ∩ = ( )a a K B A ∈ ∩∩ 3) B ∩ a a K A ∈       ∪ = ( )a a K B A ∈ ∩∪ 4) B ∪ a a K A ∈       ∩ = ( )a a K B A ∈ ∪∩ 5) a a K A ∈ ′      ∪ = ( )a a K A ∈ ′∩
  18. 18. 14 F y ∈ B ∪ a a K A ∈       ∪ B ∪ a a K A ∈       ∪ = ( )a a K B A ∈ ∪∪ F ก 3) (⊆) F x ∈ B ∩ a a K A ∈       ∪ F F x ∈ B x ∈ a a K A ∈ ∪ ก x ∈ a a K A ∈ ∪ F F x ∈Aa a ∈ K x ∈ B x ∈ Aa a ∈ K ก x ∈ B ∩ Aa a ∈ K x ∈ ( )a a K B A ∈ ∩∪ (⊇) F y ∈ ( )a a K B A ∈ ∩∪ F F y ∈ B ∩ Aa a ∈ K y ∈ B y ∈ Aa a ∈ K F F y ∈ B y ∈ a a K A ∈ ∪ y ∈ B ∩ a a K A ∈       ∪ B ∩ a a K A ∈       ∪ = ( )a a K B A ∈ ∩∪ F ก ʿก 1.5 1. ก ก F F 1) 1 1 n n- ,   ก n 2) ( ε, ε) ก ε 2. ก F 1.5 F F F F
  19. 19. F F 15 1.6 F F F F (Venn Euler Diagram) ˈ F F F ˈ F ก F F F ˈ ก กF ก F 2 F F F (John Venn, 1834 1923) F F (Leonard Euler, 1707 1783) F F ก F F F ก F ก F ก F ˈ F ก F F 1.7 ก F U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {3, 4, 5, 6, 9} U A ก F ก F F 1.8 ก F U ˈ ก F A, B ˈ F ก F F F ก F F 1) A ∪ B 2) A ∩ B 1) F F กF A ∪ B 1 2 3 4 5 6 7 8 9 U 3 4 5 6 9 U
  20. 20. 16 F 2) F F กF A ∩ B ʿก 1.6 1. F F ก F F 1) A = B 2) A B F ก F ก 3) A ˈ F F B 2. F A B = {2, 4, 6}, B A = {0, 1, 3} A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8} F A ∩ B ˈ F F F 1) {0, 1, 4, 5, 6, 7} 2) {1, 2, 4, 5, 6, 8} 3) {0, 1, 3, 5, 7, 8} 4) {0, 2, 4, 5, 6, 8} U
  21. 21. F F 17 1.7 F ก ก F F ˈ ก F ˆ F ก ก F ˈ F F F ʾ F F 1.9 A B F F 1) F A ∩ B = φ F A ⊆ B′ B ⊆ A′ 2) A (A ∩ B) = A B 3) (A ∪ B) A = B 4) F A ∩ B = A F A ⊆ B 1) ก F ( F F F ˈ ʿก ) 2) ก A (A ∩ B) = A ∩ (A ∩ B)′ = A ∩ (A′ ∪ B′) = (A ∩ A′) ∪ (A ∩ B′) = (A A) ∪ (A B) = φ∪ (A B) = A B F 2 ก F 3) ก (A ∪ B) A = (A ∪ B) ∩ A′ = (A ∩ A′) ∪ (B ∩ A′) = φ∪ (B A) = B A ≠ B F 3 4) F A ∩ B = A F x ∈ A ∩ B x ∈ A ก x ∈ A ∩ B F x ∈ A x ∈ B A ⊆ B F 4 ก F F 3 F ก F
  22. 22. 18 F F 1.10 F A = {1, 2, 3, , 9} S = {B | B ⊆ A (1 ∈ B 9 ∈ B)} F ก S F ก F ก S Fก F F S ก F F B A 1, 9 ∈ A ก F B1 = {B | B ⊆ A 1 ∈ B}, B2 = {B | B ⊆ A 9 ∈ B} F S = B1 ∪ B2 n(S) = n(B1 ∪ B2) = n(B1) + n(B2) n(B1 ∩ B2) -----(1.6.1) ก n(B1) = 1 ⋅ 29 1 = 1 ⋅ 28 ก F F n(B2) = 1 ⋅ 28 n(B1 ∩ B2) = 1 ⋅ 29 2 = 1 ⋅ 27 F F n(S) = 1 ⋅ 28 + 1 ⋅ 28 1 ⋅ 27 = 2 ⋅ 28 81 2 2⋅ = 83 2 2⋅ = 384 F 1.11 ก F S = {n ∈ I+ | n ≤ 1000, . . . n 100 F ก 1} ก S F ก F (n, 100) = 1 ก 100 = 22 ⋅ 52 F n F F 2 5 ˈ ก n F F 2 ⋅ 5 = 10 F ก F F ก n ≤ 1000 F 10 F ก 1000 10 = 100 ก n ≤ 1000 F 10 F F ก 1000 100 = 900----(1.6.2) F ก n ≤ 1000 F 2 F F ก 1000 500 = 500----(1.6.3) F ก n ≤ 1000 F 5 F F ก 1000 200 = 800----(1.6.4) n(S) = 800 + 500 900 = 400 F 1.12 ก F ก F U = {1, 2, 3, 4, 5} A, B, C ˈ F n(A) = n(B) = n(C) = 3 n(A ∩ B) = n(B ∩ C) = n(A ∩ C) = 2 F A ∪ B ∪ C = U F F F 1) n(A ∪ B) = 4 2) n(A ∪ (B ∩ C)) = 3 3) n(A ∩ (B ∪ C)) = 2 4) n(A ∩ B ∩ C) = 1
  23. 23. F F 19 ก n(A ∪ B ∪ C) = n(A) + n(B) + n(C) n(A ∩ B) n(A ∩ C) n(B ∩ C) + n(A ∩ B ∩ C) F F 5 = 3 + 3 + 3 2 2 2 + n(A ∩ B ∩ C) n(A ∩ B ∩ C) = 5 (3 + 3 + 3) + (2 + 2 + 2) = 2 F F 4 F 1.13 F A, B C ˈ n(A ∪ B) = 16, n(A) = 8, n(B) = 14, n(C) = 5 n(A ∩ B ∩ C) = 2 F n((A ∩ B) × (C A)) ˈ F F ก F F 1) 6 2) 12 3) 18 4) 24 ก ก n(A ∪ B) = n(A) + n(B) n(A ∩ B) F F F n(A ∩ B) = (8 + 14) 16 = 6 n((A ∩ B) × (C A)) = n(A ∩ B) ⋅ n(C A) = 6 ⋅ n(C A) = 6 ⋅ (n(C) n(A ∩ C)) = 6 ⋅ (5 n(A ∩ C)) -----(1.6.5) ก F F ก F F ก (1.6.5) F F ก F F ก F 5 n(A ∩ C) F n(A ∩ C) F F F F ˈ F F F F ก 0, 1, 2, 3, 4 F F F F F F F x = n(A ∩ C) ก F F F F F F x ≥ 2 F n(A ∩ C) F ก 2 F n((A ∩ B) × (C A)) F ก 6 ⋅ (5 2) = 18 U A B C 2 4 x 2
  24. 24. 20 F F 1.14 ก F A, B, C ˈ F n(B) = 42, n(C) = 28, n(A ∩ C) = 8, n(A ∩ B ∩ C) = 3 n(A ∩ B ∩ C′) = 2, n(A ∩ B′ ∩ C′) = 20 n(A ∪ B ∪ C) = 80 F n(A′ ∩ B ∩ C) F ก F F 1) 5 2) 7 3) 10 4) 13 F F F F F F F ก F x = n(A′ ∩ B ∩ C) F F 80 = 20 + 2 + 3 + 5 + (37 x) + x + (20 x) = 87 x x = 87 80 = 7 F F F ก F F F F F F n(A′ ∩ B ∩ C) = 7 U A B C 3 5 2 x 20 37 x 20 x U A B C 3 5 2 7 20 30 13
  25. 25. F F 21 F 1.15 ก F A, B, C ˈ A ∩ B ⊆ B ∩ C F n(A) = 25, n(C) = 23, n(B ∩ C) = 7 n(A ∩ C) = 10 n(A ∪ B ∪ C) = 49 F n(B) F ก F F 1) 11 2) 14 3) 15 4) 18 F F F F F n(A ∩ B ∩ C) = x n(A ∩ B ∩ C′) = y ก n(A ∪ B ∪ C) = n(A) + n(B) + n(C) n(A ∩ B) n(A ∩ C) n(B ∩ C) + n(A ∩ B ∩ C) -----(1.6.6) F F ก F F ก (1.6.6) F F 49 = 25 + n(B) + 23 (x + y) 10 7 + x = 31 y + n(B) n(B) = 49 31 + y = y + 18 -----(1.6.7) ก A ∩ B ⊆ B ∩ C F (A ∩ B) (B ∩ C) = φ n((A ∩ B) (B ∩ C)) = n(φ) = 0 ก n(A ∩ B) n(A ∩ B ∩ C) = 0 n(A ∩ B) = n(A ∩ B ∩ C) = x ก F F F F x + y = x y = 0 F y = 0 ก (1.6.7) F F n(B) = 0 + 18 = 18 U A B C x 7 x10 x y 15 y 6 + x
  26. 26. 22 F 1.8 F F ก ก F Fก F F ก ก F ก F F ก F ก F F 1.16 F p - p, p / p F F F ก ก p ก F (Idempotent) F A F A ∪ A A ∩ A F ก F ก A F ก ก ˈ ก ก F 1.17 ก ก (commutative law) ก ก F ก ก ก ก F กF A ∪ B = B ∪ A, A ∩ B = B ∩ A, A ∪ (B ∪ C) = (A ∪ B) ∪ C ก Fก F ก F 1.18 ก Fก F A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∩ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (B ∩ C) ก Fก ก Fก ก F ก F F 1.19 F F A ก F F (A′)′ = A ก Fก ก F F F ∼(∼p) ≡ p ก F F F F ก ก F F ก F ∪ / ∩ - = ‹ ′ ∼ F ก ก fl F ก ก F F F ก fl F F F F F F ก ก F p fl q ≡ ∼p / q F ก fl F ก ก ก ′ ก ก ∪
  27. 27. F F 23 ʿก 1.8 1. F F p, q, r ก A, B, C 2. ก ก F ก ก ก F 3. ก Fก ก F 4. ก Fก ก F
  28. 28. 2 ก F ก F F F ก ˈ ก F ก F F ก F ˈ 2 ก (finite sets) F (infinite sets) ก F ก ก ก ก ก ก F Fก ก F ก ก ก F F กก F F (countable sets) ก F F F (uncountable sets) F ก F F F ก F ˈ ก ก F F F F ก F ˈ F F ก F F ก ก F F F Fก F F F ก F ก ก F ก ก F F F ก F F 5 F ˆ กF F F F ก F ก F ก F กF ก F F F 2.1 ก F A = {1, 2, 3, , 10}, B = {1, 3, 5, 7, 9}, C = {1, 2, 3, } ก F F A ˈ ก ก ก ก ก B ˈ ก F C ˈ F ก ก F ก F F ก ก F F C ˈ F F (countable infinite set) F F 2.2 ก F D = {x | x ∈ I 2|x}, E = {x | x ∈ Q}, F = {x | x ∈ R} ก F F D ˈ F F E F F ก ˈ F F F
  29. 29. 26 F 2.1 F F 2.3 ก F A = {1, 2, 3} B = {a, b, c} F F A ∼ B ก F f = {(1, a), (2, b), (3, c)} F F 1. f ˈ ˆ กF F ( F F F ˈ ʿก ) 2. f ˈ ˆ กF ก A B F F F Rf ⊆ B B ⊆ Rf ก Rf ⊆ B ˈ F F F F B ⊆ Rf ก F F y ∈ B y = a y = b y = c ก y = a F F x = 1 ∈ A (1, a) ∈ f a ∈ Rf ก F F b, c ∈ Rf F F y ∈ Rf B ⊆ Rf 2.1 F F A ∼ B ก ก ก F F F ก F ก F ˈ F ก F F F F F 2.4 F = ˈ F (R) ก F a, b, c ∈ R F F = F 2.2 1) ก a = a = F F 2) ก a = b F b = a = F 2.1 ก F A B ˈ ก F F A ก B ( F ก F A ∼ B) ก F F (one to one correspondence) ก A B 2.2 F (equivalent relation) F F 1) F (reflexivity) 2) (symmetry) 3) F (transitivity)
  30. 30. F F 27 3) F a = b F F a b = 0 ก b = c a c = 0 a = c = F F ก F 1) 3) F F = ˈ F F 1. ∼ F (reflexivity) ก F A ∈ X ก IA : A → A F F F A ∼ A ∼ F X 2. ∼ (symmetry) ( F F F ˈ ʿก ) 3. ∼ F (transitivity) ก F A, B, C ∈ X A ∼ B B ∼ C F f : A → B, g : B → C ก 2.2 F F 5 F F gof : A → C A ∼ C F ∼ F X ก F 1 3 F F ∼ ˈ F X F ก ʿก 2.1 1. ก F ก F F F F F F F F 2 F 2. ก F X ˈ (family of sets) A, B ∈ X F F ∼ X 3. ก F N = ก Z = F N ∼ Z 2.1 ก F X ˈ (family of sets) F ∼ ˈ F X
  31. 31. 28 F 2.2 ก F F F Fก F ก F F F F F F ก ก ก Fก Fก F F กก F ก ก ก F ก ʽ F F F ก F ก ก ก 2.2 ก F F F F A ˈ F F A ก 0 F A F ˈ F F A ก n F 2.4 ก F A = {1, 2, 3, , 10} F A ˈ ก F ก A 2.2 F F F F A ∼ {1, 2, 3, , 10} ก F IA : A → {1, 2, 3, , 10} A ∼ {1, 2, 3, , 10} F F A ก 10 F 2.5 ก F A, B ˈ A ∼ B F F F A ˈ ก F B ˈ ก ก F A, B ˈ A ∼ B F A ˈ ก 2.2 F F ก n F A ∼ {1, 2, 3, , n} 2.1 F f: A → {1, 2, 3, , n} ก A ∼ B F F g: A → B ก g ˈ ˆ กF F 2.3 F F 5 F F g 1 : B → A ก g 1 ˈ ˆ กF F A F f ˈ ˆ กF F {1, 2, 3, , n} 2.2 F F 5 F F fog 1 : B → {1, 2, 3, , n} ก fog 1 ˈ ˆ กF F {1, 2, 3, , n} F F B ∼ {1, 2, 3, , n} ก n B ˈ ก 2.2 ก F A ˈ F A ˈ F F A ∼ {1, 2, 3, , n} ก n F ก F F A ˈ ก
  32. 32. F F 29 ก F F F F A ˈ F F B F ˈ F F F ˈ ก ก ก ก 2.2 F ก F F ก F F ก ก F F ก F ก ก F F ก ก ก ก F F ก F F F ก F ˈ F 2.6 ก F A = {1, 2, 3} f : A → {1, 2, 3} f ˈ ˆ กF F {1, 2, 3} F F F F B A F กF {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, φ F F F F B F ˆ กF F {1, 2, 3} F ( ?) F ˆ กF F {1, 2, 3, , m} ก m m < n F B = {1} F h : {1} → {1} F F 1 < 3 B = {2} F h : {2} → {1} F F 1 < 3 B = {3} F h : {3} → {1} F F 1 < 3 B = {1, 2} F h : {1, 2} → {1, 2} F F 2 < 3 B = {1, 3} F h : {1, 3} → {1, 2} F F 2 < 3 B = {2, 3} F h : {2, 3} → {1, 2} F F 2 < 3 B = φ F h : φ→ φ F F 0 < 3 F ก F B ˈ F F A F F f : A → B ก A ˈ ก F F A ∼ {1, 2, 3, , n} ก n F g : A → {1, 2, 3, , n} F gof 1 : B → {1, 2, 3, , n} ˈ F {1, 2, 3, , n} F F F ก 2.2 F F F ก F ˈ 2.2 ก F A ˈ F f : A → {1, 2, 3, , n} ก n F B ˈ F F A F F F F F g : B → {1, 2, 3, , n} F F B ≠ φ F F h : B → {1, 2, 3, , m} ก m m < n 2.3 F A ˈ ก F A F F ก F F
  33. 33. 30 F ˆ F ก ก ก Fก ˆ ก F F F F F ก F m, n ˈ ก A m ≠ n F m < n F F f : A → {1, 2, 3, , m} g : A → {1, 2, 3, , n} F F gof 1 : {1, 2, 3, , m} → {1, 2, 3, , n} ˈ ˆ กF F {1, 2, 3, , n} F F F {1, 2, 3, , m} ⊂ {1, 2, 3, , n} gof 1 ก F ˈ ˆ กF F ก F F ก F ก 2.2 F F F F ⊂ ˈ F F 2.5 ก F F F ก F F F F F ก ก ก ก F F ก F F F F 2.7 C F 2.1 ˈ F F 2.8 D, E, F F 2.2 ˈ F F Fก F F F ก F A ≠ φ ˈ F A ˈ ก F ก F B ˈ ก F A ˈ ก F f : A → {1, 2, 3, , n} ก n 2.4 ก A F 2.3 ก F F ก ก F F 2.5 ก F B ≠ φ n ˈ ก F F F ก 1) ˆ กF f : {1, 2, 3, , n} → B 2) ˆ กF F g : B → {1, 2, 3, , n} 3) B ˈ ก ก ก n 2.6 ก F A ≠ φ ˈ F B ˈ F B ⊆ A F A ˈ F
  34. 34. F F 31 ก B ⊆ A f : B → {1, 2, 3, , n} ก n F F B ˈ ก F F F F A ˈ F F ก ʿก 2.2 1. F F F ก ก F ˈ ก 2. F F A ∪ B ˈ ก ก F A B F ก ˈ ก 3. F F N ˈ F
  35. 35. F F 33 ก ก ก ก ก F. ก F. F 2. ก : ก F , 2542. . F Ent 47. ก : F F ก F, 2547. ˆ ก F F . ก ก F 1 ก F F ˅ 2001. ก : , 2544. F . ก ก F 1 ก F F ˅ 2004. ก : , 2547. . F. F 1. ก : ก F, 2533. ก . ก F .4 ( 011, 012). F ก. ก : ʽ ก F F, 2539.

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