เซต

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
N
ก
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 }

F 41101 Wila ( F) -2-

กก 1
ก ก

1. 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 5

2. 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

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 }

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 1
10. { 1, , , , , }
2 3 4 100

F 41101 Wila ( F) -5-

2. ก F

ก ( finite sets ) ก Fก ก F

F ก
1. A = { 1, 2, 3, , 100 } ก
ก A F ก 100
F ก F n(A) ก A n(A) = 100
2. B { x / x -3 < x < 4 }
ก B F กF -2, -1, 0, 1, 2, 3
ก B Fก 6 n(B) = 6
3. 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 }

F 41101 Wila ( F) -6-

กก 2
ก F F
1. 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 {}

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 Fก
1. 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 }

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}

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⊄B

ก 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,}}}

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 F
1) 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 }
*****

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 Euler
ก 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 F
1. ก U = {0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9}
A = {9 , 7 , 5 , 3 , 1}
B = {7 , 5 , 3 , 2}
F F

2. 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 }

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 }

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

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 ∪ φ
2. 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

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

F 41101 Wila ( F) - 16 -

F ก
F A,B,C ก FU
1. A ∩B = B ∩A
2. (A ∩ B) ∩ C = A ∩ (B ∩ C)
3. A ∩B = A ก F A⊂B
4. A∩ φ = φ
5. A ∩U = A
6. A ∩A = A
7. A ∩B⊂ A A ∩B ⊂ B
8. A ∩B ⊂ A ∪B
9. 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

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∩U

2. 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

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

F 41101 Wila ( F) - 19 -
3) U/ = φ
4) A ∩ A/ = φ
5) A∪A/ = U
6) 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
F

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/ 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)/

.

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 }

F 41101 Wila ( F) - 21 -

กก 9
F F

1. 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) C
2. 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)

F 41101 Wila ( F) - 22 -

10. ก

กก 10
ก
ก 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) = ..

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 F
ก 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 F

1. 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)

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∩B
1. 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)

F 41101 Wila ( F) - 25 -
กก 12
ก

1. 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 F

6. 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 )

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 F

8. 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 F

9. 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 F

10. กก ก 48 ก ก 3
F กF F ก
29 22 F 21 7
10 F 12 F
ก ก Fก F

11. 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

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