set

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

3. 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

5. 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

7. 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

9. 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

10. 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]

11. 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

12. 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}

13. 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)

14. 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)

15. 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]

16. 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′

17. 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

18. 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)

19. 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)

20. 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}

21. 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
∈
′∩

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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 ∪

31. 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

33. 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

34. 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)

35. 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

36. 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

37. 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

38. 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

39. 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

41. 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.