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1. 1. F 41101 Wila ( F) -1- (SETS)1. (SETS) F F F ก F F F ก F F F Fก FF ( ) F F F F กก F F9 F 3 F กF ก (element) F ก F∈ ก ∉ F ก F 1 ก F 1∈ N ∈ -1 F ก F -1 ∉ N Nก ก F2 1. ก ก ก F 1.1 ก F { a, e, i, o, u } 1.2 ก F { 1, 2, 3, 4, 5, 6, } 1.3 F { ก, , , , } 2. ก ก ก F A = {x/x F} B = { x / x2 = 100 } C = { x ∈ Ι / -2 < x < 2 }
2. 2. F 41101 Wila ( F) -2- กก 1 ก ก1. F ก ก F 1.1 F 0 9 1.2 F F 1.3 F 1.4 F F 1.5 ก 1.6 F 52. F A F กF F ก 16 F F 1. 10 ∈ A 2. 16 ∉ A 3. 4 ∈ A 1 4. ∉A 2 5. -2 ∈ A 6. 0∈A 7. 61 ∈ A 8. 5∈A
3. 3. F 41101 Wila ( F) -3-3. F F ก ก 1. 2. F MATHEATICS 3. Fก 4. F 5. { x / x กก F -9 } 6. { x / x = 2n n } 7. { x / x ∈ Ι } 8. { x / x ∈ Ι+ x<7 } 9. { x / x F ก ก x2 - 2x - 3 = 0 } 10. { x / x = 2y 1 y = 1, 2, 3, }4. F F ก ก 1. { ก F} 2. { -2 }
4. 4. F 41101 Wila ( F) -4-3. { 2, 4, 6, 8, }4. { -1, -3, -5, -7, -9, }5. { 10, 11, 12, 13, , 99 }6. { , -3, -2, -1, 0, 1, 2, 3, }7. { 7, 14, 21, 28, 35, , 98 }8. { 1, 4, 9, 16, 25, 36, 49, }9. { , , } 1 1 1 110. { 1, , , , , } 2 3 4 100
5. 5. F 41101 Wila ( F) -5-2. ก F ก ( finite sets ) ก Fก ก F F ก1. A = { 1, 2, 3, , 100 } ก ก A F ก 100 F ก F n(A) ก A n(A) = 1002. B { x / x -3 < x < 4 } ก B F กF -2, -1, 0, 1, 2, 3 ก B Fก 6 n(B) = 63. C { x / x ∈ Ι− x≥0 } F F กก F Fก F C F ก ก C=0 n(C) = 0 F ( empty set ) ก F ก ก F F F ก F∅ {} F F 1. { x / x F กF 0} 2. { x / x ∈ Ι 2x = 5 } F ( infinite sets ) F F ก F ก F F F 1. A = { 1, 2, 3, } 2. B = { x / x F 10 } 3. C = { x / x ∈ Ι x2 ≥ 100 }
6. 6. F 41101 Wila ( F) -6- กก 2 ก F F1. F F ก ก F F ก F ก F F ( ก FF ก ) ก F F ก F 1 { 1, 2, 3, , 10 } 2 {x/x } 3 { x / x ∈ Ι 1<x<2} 4 {x/x F } 5 { x / x ∈ N x > 10 } 6 { x / x ∈ Ι− x2 >0 } 7 { x / x ∈ Ι x2 = -1 } 8 {x/x∈R x = x} 9 {-50, -49, -48, ,48, 49, 50}10 {x/x }2. กF ก F F F F F F F F 1 {x/x∈Ν x=0} 2 {x/x∈Ι x2 < 0 } 3 {x/x∈Ι x2 ≤ 0 } 4 { x / x ∈ Ι+ 2x - 3 > 0 } 5 {x/x∈Ν 3x - 1 = 0 } 6 { {0} } 7 {∅} 8 {0} 9 {{ }} 10 {}
7. 7. F 41101 Wila ( F) -7-3. Fก A Fก Bก F ก ก กF ก ก A ก B ก ก B ก A A Fก B F A=B F Fก 1. { -1, -2,- 3 } = { -1, -2, -3 } 2. {a, b, c } = { b, a, c } 3. {3, -3 } = { x / x2 = 9 } ก Fก F ก F ก ก กF F ก equivalent set F A = { 1, 2, 3 } B = { 2, 4, 6 } F A B ก3 Fก F ก F ก กF A Fก ก B กก 3 Fก1. F ก กF F Fก Fก 1. A = {0, 1, 2, 3 } , B = {3, 0, 2, 1 } 2. A = {2, 1 } , B = { x / (x+2)(x+1) = 0 } 3. A = {0, 5, 10, 15 , 20 } , B = { x/ x = 5n , n = 1, 2, 3, 4, 5 } 4. A = {3, 3, 3, 2 , 2, 1} , B = { 1, 2, 3 }2. ก F A = {x / x F 1 7} B = { 1, 2, 3, 4, 5, 6 } C = { 2, 3, 4, 5, 6, 7 } F F F Fก A, B C 1. { 2, 4, 5, 3, 6 } 2. { 5, 6, 7, 2, 3, 4 } 3. { 1, 3, 2, 5, 6, 4} 4. { 2, 4, 3, 6, 7, 5 } 5. { 6, 5, 4, 3, 2 }
8. 8. F 41101 Wila ( F) -8-4. F (Subsets) A Bก F ก ก A ก B 1. A B F A⊂B 2. F ก F F A F ก B F A F B F A⊄B ก ก 1. ก 2. F ก 3. F A ⊂ B B ⊂ C F A ⊂ C 4. A = B ก F A⊂B B⊂A 5. F A กn A 2n F F A = {1, 2, 3} 1 B = {1, 2, 3, 4, 5} C = {6, 7} D = {x∈I|5 < x < 8} E = {8, 9, 10} F = {8, 10, 12, 14} FF A⊂B ก ก A 1, 2, 3 ก B F C⊂D D = {6, 7} E⊄F 9∈E F9∉F Fก F 2 ก A = {0, 1} A ก2 A 22 = 4 A F กF Φ , {0}, {1}, {0,1}
9. 9. F 41101 Wila ( F) -9- F 3 A = { a, b , c } A ก3 A 23 = 8 A F กF Φ , {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, { a, b , c } F ก A F Bก F ก A F F F ก B( F ก FA⊄Bก ก 2 ก F2ก 1. A F B (proper subset) A⊂B FB⊄A F A = {1, 2, 4} B = {1, 2, 4, 8} 2. A F F B (improper subset) A⊂B B⊂A A=B F A = {2, 4, 6, 8} B = {8, 6, 4, 2} F A (Power set of A) F A F ก F P(A) A P(A) = { x | x ⊂ A }F ก F A ก กn A 2n ก P(A) F ก 2n F 1 ก A = {0, 1} F F n(A) = 2 n(P(A)) Fก A = 22 = 4 P(A) = { φ, {0}, {1}, {0,1} } F 2 F A = {1, {2, 3}} F F n(A) = 2 n(P(A)) Fก A F ก 22 = 4 P(A) = {φ, {1},{{ 2, 3}}, {1, {2, 3,}}}
10. 10. F 41101 Wila ( F) - 10 - F 3 A = { a, b , c } F F n(A) = 2 n(P(A)) Fก A = 23 = 8 A F กF {φ, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, { a, b , c } } F ก ก ก F 1. x ∈ P(A) ก F x⊂A 2 . φ ∈ P(A) 3. A ∈ P(A) 4. P(φ) = {φ} 5. P(A) ≠ φ 6. A ⊂ B ก F P(A) ⊂ P(B) 7. F n(A) = k F n(P(A)) = 2k k n[P(P(A))] = ( 2 ) 2 8. F A F F P(A) F กก 4 F F ก F ก F1) A={ }2. B = {1}3. C = {a , b}4. D = { φ , { φ }}5. E = {x , y , z}6. A = {{1} , {1,2} }7. B = { 0 , {{0}} , {0, 1} }8. A = {φ } *****
11. 11. F 41101 Wila ( F) - 11 -5. F F (Venn Euler Diagram) Fก ก ก ก F F ก F ก ก F F John Venn Leonhard Eulerก ก F (U) F F A,B,C U F ก ก F A B U F F A B ก F 1 ก. F A⊂B 1 . A B F ก F ก 1 . A B กF ก 1 . A=B F F ก A,B,C,U F F F1. ก U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {9 , 7 , 5 , 3 , 1} B = {7 , 5 , 3 , 2} F F2. ก U= { 1, 2, 3, 4, 5, 6, 7, 8, 9 } A= { 1, 2, 3, 4} B= { 1, 3, 5, 7, 9} C= { 2, 3, 4, 5, 6 , 7 }
12. 12. F 41101 Wila ( F) - 12 - กก 5 F F (Venn Euler Diagram)F ก A,B,C,U F F F 1) U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {1 , 3 , 5 , 7 , 9} B = {3 , 5 , 7} 2) U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {2 , 3 , 5 , 7} B = {4 , 6 , 8} 3) U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {0 , 1 , 2 , 3 , 5 , 7} B = {1 , 3 , 5 , 6 , 7} C = {0 , 3 , 5 , 6 , 9} 4) U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {2 , 3 , 5 , 7} B = {1 , 3 , 5 , 7 , 9} C = {2 , 4 ,6} 5) U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , ..,10} A = {0 , 1 , 2} B = {0 , 1 , 2 , 3 , 4 , 5} C = {2 , 4 , 6 , 8 , 10} 6) U = { a, b, c, d, e, f, g, h, I, j, k, l } A = { b, c } B = { a, b, c, d } C = { a, c, e, ,f,, h, l }
13. 13. F 41101 Wila ( F) - 13 -6. (Union) A ก B ก ก A ก B A ก B F A∪B A ∪ B = { x ∈U | x ∈ A x∈B } F ก F U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {x ∈U | x } B = {x ∈U | x } F A = {9 , 7 , 5 , 3 , 1} B = {7 , 5 , 3 , 2} A ∪ B = {9 , 7 , 5 , 3 , 2 , 1} A ∪B F F F A,B,C ก FU 1. A ∪B = B ∪A 2. (A ∪ B) ∪ C = A ∪ (B ∪ C) 3. A ∪B = B ก F A⊂B 4. A∪ φ = A 5. A ∪U = U 6. A ∪A = A 7. A ⊂ A ∪B B ⊂ A ∪B 8. F B ⊂ A C ⊂ A F (B ∪ C) ⊂ A 9. A ∪B = φ ก F A= φ B= φ
14. 14. F 41101 Wila ( F) - 14 - 10. P(A) ∪ P(B) ⊂ P(A ∪ B) 11. P(A) ⊂ P(B) ก F P⊂B ..................................... กก 6 (Union)1. ก F U = { a , b , c , d , e , f , g , h , i , j , k} A = {a , b , c , d} , B = {b , d , e , f , g} C = {c , d , e , h , i , j} F ก ก F 1. B ∪ C 2. A ∪ B 3. A ∪ C 4. A ∪ B ∪ C 5. A ∪ B ∪ φ2. ก U = { x∈ I | - 10 < x < 10 } ∈ A = { x∈U | - 5 < x ≤ 4 } ∈ B = { x∈U | - 3 ≤ x < 6 } ∈ C = { x∈U | - 4 < x < 4 } ∈ 1. A ={x | - 5 < x ≤ 4} 2. B ={x | - 3 ≤ x < 6} 3. C = {x | - 4 < x < 4} 4. B ∪ C 5. A ∪ B ∪ C
15. 15. F 41101 Wila ( F) - 15 -7. F ก (Intersection) A F ก ก B ก ก A ก B F ก A B F A ∩B ก FF A ∩ B = {x ∈U | x ∈ A x ∈ B} F 1 1. F A = { 0, 1, 2, 3 } B = { 0,3, 5 } F A ∩ B = { 0, 3 } 2. F A = { a, b, c } B = { d,e, f } F A ∩B = { } F 2 ก F U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {x ∈U | x F 3 } B = {x ∈U | x F 2 } F A = {0 , 3 , 6 , 9} B = {0 , 2 , 4 , 6 , 8} A ∩ B = {0 , 6} A ∩B F F
16. 16. F 41101 Wila ( F) - 16 - F ก F A,B,C ก FU1. A ∩B = B ∩A2. (A ∩ B) ∩ C = A ∩ (B ∩ C)3. A ∩B = A ก F A⊂B4. A∩ φ = φ5. A ∩U = A6. A ∩A = A7. A ∩B⊂ A A ∩B ⊂ B8. A ∩B ⊂ A ∪B9. F A ⊂ B A ⊂ C F A ⊂ (B ∩ C)10. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)11. P(A) ∩ P(B) = P(A ∩ B)12. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)13. A ∩B = A∪B ก F A=B
17. 17. F 41101 Wila ( F) - 17 - กก 7 F ก 1. ก F U = { a , b , c , d , e , f , g , h , i , j , k} A = {a , b , c , d} , B = {b , d , e , f , g} C = {c , d , e , h , i , j} 1. B∩C 2. A∩B 3. A∩C 4. A∩B∩C 5. A∩B∩U2. ก U= {x | - 10 < x < 10} A= {x | - 5 < x ≤ 4} B= {x | - 3 ≤ x < 6} C= {x | - 4 < x < 4} 1. A ={x | - 5 < x ≤ 4} 2. B ={x | - 3 ≤ x < 6} 3. C = {x | - 4 < x < 4} 4. A∩B 5. A∩B∩C
18. 18. F 41101 Wila ( F) - 18 -8. F (Complement of sets) F A ก F U ก F ก ก U F F ก A F A F A/ ก F A/ = {A ∈U | x ∉ A } F 1 ก F U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} A = {x ∈U | x } FF A = {2 , 3 , 5 , 7} A/ = {0 , 1 , 4 , 6 , 8 , 9} (A/)/ = {x∈U|x∉A/} = {2 , 3 , 5 , 7} A ∩ A/ = φ A∪A/ = {2 , 3 , 5 , 7} ∪ {0 , 1 , 4 , 6 , 8 , 9} = U U/ = {x∈U F x ∉U } = φ φ/ = {x∈U F x∉φ} = U A/ ก ก F F A,B ก FU 1) (A/)/ = A 2) φ/ = U
19. 19. F 41101 Wila ( F) - 19 -3) U/ = φ4) A ∩ A/ = φ5) A∪A/ = U6) A⊂B ก F B/ ⊂ A/7) A ∩B = φ ก F A ⊂ B/ B ⊂ A/8) ก Fก (de Morgan, s Law) (A∪B)/ = A/ ∩ B/ (A ∩ B)/ = A/ ∪B/ กก 8 F1. ก F U = { a , b , c , d , e , f , g , h , i , j , k} A = {a , b , c , d} , B = {b , d , e , f , g} C = {c , d , e , h , i , j} 1. B/ 5. (A ∩ C)/ 2. A/ 6. (A ∪ B ∪ C)/ 3. C/ 7. (A ∪ C)/ ∩ B 4. (A ∩ B)/ 8. (A ∩ B ∩ C)/2. ก U = {x | - 10 < x < 10} A = {x | - 5 < x ≤ 4} B = {x | - 3 ≤ x < 6} C = {x | - 4 < x < 4} 1. (A ∪ B)/ 4. A/ 2. (C ∪ A)/ 5. C/ 3. (C ∪ A)/ 6. (A ∪ B ∪ C)/ .
20. 20. F 41101 Wila ( F) - 20 -9. F F F F A B ก ก A F ก B F A B F A B ก FF A B = {x/x∈ A x ∉ B} A B F F F ก A = { 0, 2, 4, 6, 8, 10 } B = { 1, 2, 3, 4, 5, 6 } F ก A B ก F กF 2, 4, 6 F A - B = {0, 8, 10 } 0 ∈ A F 0∉B 8 ∈ A F 0∉B 10 ∈ A F 0 ∉ B ก F B - A = {1, 3, 5 }
21. 21. F 41101 Wila ( F) - 21 - กก 9 F F1. ก U= { 1, 2, 3, 4, 5, 6, 7, 8, 9 } A= { 1, 2, 4, 6} B= { 1, 3, 5, 7, 9} C= { 2, 3, 4, 5, 6 } F F F F F- F 1.1 A-B 1.2 A-C 1.3 B-C 1.4 (A∪B) C 1.5 (A∩B) C2. ก U = {x | - 10 < x < 10} A = {x | - 5 < x ≤ 4} B = {x | - 3 ≤ x < 6} C = {x | - 4 < x < 4} F F F 2.1 A-B 2.2 C-A 2.3 B-C 2.4 (A∪B) C 2.5 (A∩B) C 2.6 B - (A∩ C)
22. 22. F 41101 Wila ( F) - 22 -10. ก กก 10 กก A ,B ก n(A) ก A n(B) ก B n(A ∪ B) ก A∪B n(A ∩ B) ก A∩B(1.) ก A = { a, b, c, d, e } B = { a, d, e, g, i, j, k } F F F ก F 1.1 n(A) = .. n(B) = .. 1.2 A∩B = .. n(A ∩ B) = .. 1.3 A∪B = .. n(A ∪ B) = . 1.4 A B = .. n(A B) = ... 1.5 B A= . n(B A) = กก ก F F ก ก F n(A ∪ B) = . n(A B) = ..
23. 23. F 41101 Wila ( F) - 23 -2. ก A = { 5, 6, 7, 7, 9, 10 } B = { 2, 3, 4, 5 } ก F 2.1 n(A ∪ B) 2.2 n(A B)3. ก n(A) = 19 , n(B) = 40 n(A ∩ B) = 12 n(A ∪ B) กก 11 ก F ก F F Fก A ,B C ก(1.) F A = { 0, 1, 2, 3, 4, 5 }, B = { 2, 6, 8, 10, 12 } C = {2, 3 , 4, 6 ,7, 8, 9} F F ก ก F1. n(A) = 4. n(A ∩ B) = .2. n(B) = 5. n(A ∩ C) =3. n(C) = 6. n(B ∩ C) = . ... 7. n(A ∩ B ∩ C) = .. 8. n(A ∩ B - C) = . 9. n(A ∩ C - B) = .. 10. n( B - A ∪ C) = 11 n(C - A ∪ B ) = . 12. n(A ∪ B ∪ C) = ... ก ก A∪ B∪ C F n(A ∪ B ∪ C) =(2.). ก n(A) = 30 , n(B) = 45, n(C) = 56 n(A ∩ B) = 10, n(A ∩ C) = 27, n(B ∩ C) = 14 n(A ∩ B ∩ C) = 5 n(A ∪ B ∪ C)
24. 24. F 41101 Wila ( F) - 24 - กก ก 10 11 ก ก ก ก กF ก ก ก A ,B ก n(A) ก A n(B) ก B n(A ∪ B) ก A∪B n(A ∩ B) ก A∩B1. n(A ∪ B) = n(A) + n(B) - n(A ∩ B)2. ก A∩B = φ n(A ∪ B) = n(A) + n(B)3. n(A B) = n(A) - n(A ∩ B)4. n(A′) = n(U) - n(A) ′ n(U) = n(A) + n(A′)′5. ก A ,B , C ก n(A ∪ B ∪ C) = n(A) + n(B) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
25. 25. F 41101 Wila ( F) - 25 - กก 12 ก1. ก n(A) = 19 , n(B) = 40 n(A ∩ B) = 12 n(A ∪ B)2. ก n(A) = 25 , n(B) = 36 n(B - A) = 10 n(A ∪ B)3. F n(U) = 80 ก n(A) = 37 , n(B) = 43 n(A ∩ B) = 10 F F F F ก ก F 1. n(A ∪ B) 2. n(A ∩ B) ′ 3. n(A ∪ B)′ 4. n(B - A) 5. n(A B)4. ก n(U) = 150, n(A) = 60 , n(B) = 42, n(C) = 48 , n(A ∩ B) = 15, n(A ∩ C) = 21, n(B ∩ C) = 26 n(A ∩ B ∩ C) = 13 n(A ∪ B ∪ C)5. กก F 180 F F ก 95 92 ก F 125 ก 52 ก ก F 43 ก F 57 180 F F ก F F F ก F Fก F F6. ก F 48 ก F ก ก ก F F 20 F ก 15 F 25 F F F 10 ก 3 ก F ก ก (5 )
26. 26. F 41101 Wila ( F) - 26 -7. ก F A, B, C n(A∪B) = 92,n(A∪C) = 79, n(B∪C) =75, n(A∩B∩C) = 32, n((A∩B) - C) = 18, n((A∩C) - B) = 6, n((B∩C) - A) = 2 n(A∪B∪C) F ก F8. กก ก ก ก F F F F F 100 ก F 41 F F ก 10 ก F 32 ก F F F F ก F Fก F9. F ก 80 ก 3 ก F ก ก F ก F F 1 F ก 30 F ก F ก 20 ก F F F ก ก 18 ก F F F ก ก F ก ก 3 Fก F10. กก ก 48 ก ก 3 F กF F ก 29 22 F 21 7 10 F 12 F ก ก Fก F11. กก F ก F ก F ก ก F ก F 200 F F 130 F ก 100 F F 110 F ก 60 F ก F 55 F F 45 F ก ก