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E
P (E )
E
∅
X
Y
E
X ∩ Y = X ⇐⇒ X ⊆ Y
X ∩ Y = X =⇒ X ⊆ Y
X ∩ Y = X
x ∈ X X ∩ Y = X x ∈ X ∩ Y x ∈ X x ∈ Y
x ∈ Y
x ∈ X
∀x ∈ X, x ∈ Y
X ⊆ Y
X ∩ Y = X ⇐= X ⊆ Y
X ⊆ Y
X ∩ Y = X
x ∈ X ∩ Y x ∈ X x ∈ Y x ∈ X
∀x ∈ X ∩ Y, x ∈ X X ∩ Y ⊆ X
x ∈ X X ⊆ Y x ∈ Y x ∈ X x ∈ Y
x ∈ X ∩ Y ∀x ∈ X, x ∈ X ∩ Y X ⊆ X ∩ Y
X ∩ Y = X
X ∪ Y = X ⇐⇒ Y ⊆ X
X ∪ Y = X =⇒ Y ⊆ X
X ∪ Y = X
x ∈ Y x ∈ X ∪ Y X ∪ Y = X x ∈ X ∀x ∈ Y, x ∈ X Y ⊆ X
X ∪ Y = X ⇐= Y ⊆ X
Y ⊆ X
X ∪ Y = X
x ∈ X ∪ Y x ∈ X x ∈ Y Y ⊆ X x ∈ Y ⇒ x ∈ X
x ∈ X x ∈ Y ⇒ x ∈ X x ∈ X x ∈ X ∪ X = X ∀x ∈ X ∪ Y, x ∈ X Y ⊆ X
x ∈ X x ∈ X x ∈ Y x ∈ X ∪ Y ∀x ∈ Y, x ∈ X ∪ Y X ⊆ X ∪ Y
X ∪ Y = X
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f : P (E ) → P (E ) f (X ) = (X ∩ A) ∪ B A ⊆ E B ⊆ E
E
A = ∅ f (X ) X ∈ P (E )
f (X ) Def.
= (X ∪ A) ∩ B A=∅
= (X ∪ ∅) ∩ B (1.b)
= ∅ ∩ B (1.a)
= B
B = E
f (X ) Def.
= ((X ∩A)∪B B=E
= ((X ∩A)∪E Distributiva
= (X ∪E )∩(A∪E ) (1.b)
= E ∩E (1.a)
= E
f (∅) f (A) f (B) f (E )
f (∅) = (∅ ∩ A) ∪ B = ∅ ∪ B = B
f (A) = (A ∩ A) ∪ B = A ∪ B
f (B) = (B ∩ A) ∪ B = B (B ∩ A) ⊆ Bf (E ) = (E ∩ A) ∪ B = A ∪ B
f
∀X, X ∈ P (E ), X ⊆ X =⇒ f (X ) ⊆ f (X )
X ⊆ X
x ∈ f (X ) x ∈ (X ∩ A) ∪ B x ∈ (X ∩ A) ∪ B ⇔ x ∈ (X ∩ A)
x ∈ B ⇔ (x ∈ X x ∈ A) x ∈ B X ⊆ X x ∈ X ⇒ x ∈ X (x ∈ X
x ∈ A) x ∈ B ⇒ (x ∈ X
x ∈ A)
x ∈ B ⇔ x ∈ (X ∩ A) ∪ B
x ∈ f (X ) ∀x ∈ f (X ), x ∈ f (X )
f (X ) ⊆ f (X )
Y ∈ P (E )
Y
f
P (E )
B ⊆ Y ⊆ A ∪ B
f (Y ) = Y
⇒ ⇒ ⇒
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⇒ Y f P (E ) ∃X ∈ P (E ) Y = f (X )
x ∈ Y ∃X ∈ P (E ) Y = f (X ) f (X ) = (X ∩ A) ∪ B =
(X ∪ B) ∩ (A ∪ B) x ∈ (X ∪ B) ∩ (A ∪ B) x ∈ X ∪ B x ∈ A ∪ B
x ∈ A ∪ B ∀x ∈ Y , x ∈ A ∪ B Y ⊆ A ∪ B
x ∈ B ∀X ∈ P (E ), x ∈ (X ∩ A) ∪ B = f (X ) ∃X ∈ P (E ) Y = f (X )
x ∈ Y ∀x ∈ B , x ∈ Y
B ⊆ Y
∃X ∈ P (E )
Y = f (X ) ⇒ B ⊆ Y ⊆ A ∪ B
⇒ B ⊆ Y ⊆ A ∪ B
f (Y ) = Y
x ∈ f (Y ) f x ∈ (Y ∩ A) ∪ B x ∈ Y ∩ A x ∈ B
x ∈ Y ∩ A x ∈ Y x ∈ A x ∈ Y
x ∈ B B ⊆ Y x ∈ Y
∀x ∈ f (Y ), x ∈ Y f (Y ) ⊆ Y
x ∈ Y B ⊆ Y Y = Y ∪ B Y = Y ∪ B ⊆ A ∪ B Y = (Y ∪ B) ∩ (A ∪ B) = (Y ∩ A) ∪ B = f (Y ) x ∈ f (Y ) ∀x ∈ Y, x ∈ f (Y ) Y ⊆ f (Y )
B ⊆ Y ⊆ A ∪ B ⇒ f (Y ) = Y
⇒ Y = f (Y )
Y ∈ P (E ) ∃X ∈ P (E ) Y = f (X ) X = Y
Y = f (Y ) ⇒ ∃X ∈ P (E ) Y = f (X )
f of = f
f of (X ) = f (f (X )) = f ((X ∩ A) ∪ B) = f ((X ∪ B) ∩ (A ∪ B)) = (((X ∪ B) ∩ (A ∪ B)) ∩
A)∪B = ((X ∪B)∩((A∪B)∩A))∪B (1.a)
= ((X ∪B)∩A)∪B = ((X ∪B)∪B)∩(A∪B) (1.b)
=(X ∪ B) ∩ (A ∪ B) = (X ∩ A) ∪ B = f (X )
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P (E ) X : f (X ) = A
B A
∃x ∈ B x ∈ A ∃x ∈ B x ∈ A ⇒ ∃x ∈ (X ∩ A) ∪ B x ∈ A ⇔ ∃x ∈ f (X ) x ∈ A ∃x ∈ f (X ) x ∈ A
f (X )
A ∀X ∈ P (E ), f (X ) = A
B ⊆ A f (X ) = (X ∩ A) ∪ B = (X ∪ B) ∩ (A ∪ B)
(1.b)= (X ∪ B) ∩ A
f (X ) = A ⇔ (X ∪ B) ∩ A = A (1.a)
⇔ A ⊆ X ∪ B
B ⊆ A
A ⊆ X ∪ B ⇔ A B ⊆ X
A ⊆ X ∪ B
x ∈ A B x ∈ A x ∈ B A ⊆ X ∪ B x ∈ X ∪ B x ∈ B ⇔ x ∈ X ∪ B x ∈ Bc ⇔ x ∈ (X ∪ B) ∩ Bc = (X ∩ Bc) ∪ (B ∩ Bc) =(X ∩ Bc) ∪ ∅ = X ∩ Bc x ∈ X x ∈ Bc x ∈ X ∀x ∈ A B, x ∈ X A B ⊆ X
A B ⊆ X
x ∈ A x ∈ A ⇒ x ∈ A∪B = (A∪B)∩E = (A∪B)∩(Bc∪B) =
(A ∩ Bc) ∪ B = (A B) ∪ B x ∈ A B x ∈ B A B ⊆ X x ∈ X x ∈ B x ∈ X ∪ B ∀x ∈ A, x ∈ X ∪ B A ⊆ X ∪ B
A ⊆ X ∪ B ⇔ A B ⊆ X
X : f (X ) = A
• B A • X : A B ⊆ X B ⊆ A
P (E ) X : f (X ) = B
f (X ) = B ⇔ (X ∩ A) ∪ B = B ⇔ X ∩ A ⊆ B X : f (X ) = B
X : X ∩ A ⊆ B
X ∩ A ⊆ B ⇔ X ⊆ B ∪ Ac
X ∩ A ⊆ B
x ∈ X = X ∩ E = X ∩ (A ∪ Ac) = (X ∩ A) ∪ (X ∩ Ac)
x ∈ X ∩ A
x ∈ X ∩ Ac
• x ∈ X ∩ A X ∩ A ⊆ B x ∈ X ∩ A ⇒ x ∈ B ⇒ x ∈ B x ∈ Ac x ∈ B ∪ Ac
• x ∈ X ∩ Ac x ∈ X ∩ Ac ⇒ x ∈ X x ∈ Ac ⇒ x ∈ Ac ⇒ x ∈ B x ∈ Ac ⇔ x ∈ B ∪ Ac x ∈ B ∪ Ac
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∀x ∈ X, x ∈ B ∪ Ac X ⊆ B ∪ Ac
X ⊆ B ∪ Ac
x ∈ X ∩ A x ∈ X ∩ A ⇔ x ∈ X x ∈ A ⇒ x ∈ B ∪ Ac x ∈ A ⇔ x ∈
(B ∪ Ac) ∩ A = (B ∩ A) ∪ (Ac ∩ A) = (B ∩ A) ∪ ∅ = B ∩ A ⇔ x ∈ B x ∈ A ⇒ x ∈ B
x ∈ B
∀x ∈ X ∩ A, x ∈ B X ∩ A ⊆ B
X ∩ A ⊆ B ⇔ X ⊆ B ∪ Ac
X : B ∩ Ac ⊆ X
A
B
f
f ∃C ∈ P (E ), ∀X ∈ P (E ), f (X ) = C
f
⇔ ∀X ∈ P (E ), f (X ) = B
f ∃C ∈ P (E ), ∀X ∈ P (E ), f (X ) = C
X = ∅ f (∅) = (∅ ∩ A) ∪ B = ∅ ∪ B = B C = B ∀X, f (X ) = B
∀X ∈ P (E ), f (X ) = B B ∈ P (E ) f ∃C ∈ P (E ), ∀X ∈ P (E ), f (X ) = C C = B
∀X ∈ P (E ), f (X ) = B ⇔ A ⊆ B
∀X ∈ P (E ), f (X ) = B
x ∈ A x ∈ A ⇒ x ∈ A x ∈ B ⇔ x ∈ A ∪ B = (A ∩ A) ∪ B =
f (A) = B ∀x ∈ A, x ∈ B A ⊆ B
A ⊆ B
X ∈ P (E ) f (X ) = (X ∩A)∪B = (X ∪B)∩(A∪B)
(1.b)= (X ∪B)∩B
(1.a)=
B ∀X ∈ P (E ), f (X ) = B
f
A ⊆ B
A B f
f
X, Y ∈ P (E ), X = Y ⇔ f (X ) = f (Y )
f (X ) = f (Y ) ⇔ X = Y
f ⇔ A = E B = ∅
f
f (X ) = f (Y ) ⇔ X = Y
f (E ) = (E ∩ A) ∪ B = A ∪ B = (A ∩ A) ∪ B = f (A)
f A = E
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f (∅) = (∅ ∩ A) ∪ B = ∅ ∪ B = B = B ∪ B = (B ∩ E ) ∪ B =
(B ∩ A) ∪ B = f (B) f B = ∅
A = E
B = ∅
∀X ∈ P (X ), f (X ) = (X ∩ A) ∪ B = (X ∩ E ) ∪ ∅ = X ∪ ∅ = X f (X ) = X f f (X ) = f (Y ) ⇔ X = Y
X = Y ⇔ X = Y
A B f
f
∀Y ∈ P (E ), ∃X ∈ P (E ), f (X ) = Y
f ⇔ A = E B = ∅
f ∀Y ∈ P (E ), ∃X ∈ P (E ), f (X ) = Y
Y = ∅ ∃X ∈ P (E ), f (X ) = (X ∩ A) ∪ B = ∅
B = ∅ B (X ∩A)∪B
(X ∩ A) ∪ B = ∅
Y = E
∃X ∈ P (E ), f (X ) = (X ∩ A) ∪ B =(X ∩ A) ∪ ∅ = X ∩ A = E A = E A ⊂ E
X ∩ A ⊂ E
X ∩ A = E
A = E
B = ∅
∀X ∈ P (X ), f (X ) = (X ∩ A) ∪ B = (X ∩ E ) ∪ ∅ = X ∪ ∅ = X
f (X ) = X f ∀Y ∈ P (E ), ∃X ∈ P (E ), f (X ) =Y
X = Y
f f
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E A B E
A ∗ B = (A ∩ B) ∪ (Ac ∩ Bc),
Ac A E P (E )
A B E A ∗ B
A ∗ B = B ∗ A
A ∗ B
E
A ∗ B = (A ∩ B) ∪ (Ac ∩ Bc) = (B ∩ A) ∪ (Bc ∩ Ac) = B ∗ A
A B C E
A ∗ (B ∗ C ) = (A ∗ B) ∗ C
A ∗ (B ∗ C )
A B C B ∩ C Bc ∩ C c B ∗ C A ∩ (B ∗ C ) Ac ∩ (B ∗ C )c A ∗ (B ∗ C )
(A ∗ B) ∗ C
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A B C A ∩ B Ac ∩ Bc A ∗ B (A ∗ B) ∩ C (A ∗ B)c ∩ C c (A ∗ B) ∗ C
A ∗ (B ∗ C ) (A ∗ B) ∗ C
A E A ∗ E A ∗ A
∗
• A ∗ E = (A ∩ E ) ∪ (Ac ∩ E c) = A ∪ (Ac ∩ ∅) = A ∪ ∅ = A
• A ∗ A = (A ∩ A) ∪ (Ac ∩ Ac) = A ∪ Ac = E
A B C E
A ∗ B = Ac ∗ Bc
Ac∗Bc = (Ac∩Bc)∪((Ac)c∩(Bc)c) = (Ac∩Bc)∪(A∩B) = (A∩B)∪(Ac∩Bc) = A∗B
(A ∗ B)c = A ∗ (Bc) = (Ac) ∗ B
(A ∗ B)c = (Ac ∩ B) ∪ (A ∩ Bc)
(A ∗ B)c = ((A ∩ B) ∪ (Ac ∩ Bc))c = (A ∩ B)c ∩ (Ac ∩ Bc)c = (Ac ∪ Bc) ∩ ((Ac)c ∪(Bc)c) = (Ac ∪ B c) ∩ (A ∪ B ) = (Ac ∩ A) ∪ (Ac ∩ B ) ∪ (Bc ∩ A) ∪ (Bc ∩ B ) =∅ ∪ (Ac ∩ B) ∪ (Bc ∩ A) ∪ ∅ = (Ac ∩ B) ∪ (Bc ∩ A) = (Ac ∩ B) ∪ (A ∩ Bc)
(Ac ∪ B) ∩ (A ∪ Bc) = A ∗ (Bc)
(Ac
∩ B) ∪ (A ∩ Bc
) = (A ∩ Bc
) ∪ (Ac
∩ B) = (A ∩ Bc
) ∪ (Ac
∩ (Bc
)c
) = A ∗ (Bc
)
(Ac ∪ B) ∩ (A ∪ Bc) = (Ac) ∗ B
(Ac ∪ B) ∩ (A ∪ Bc) = (Ac ∪ B) ∩ ((Ac)c ∪ Bc) = (Ac) ∗ B
(A ∗ B)c = A ∗ (Bc) = (Ac) ∗ B
E
A ∗ X = B
X = A ∗ B A ∗ X = B ⇔ A ∗ B = X
A ∗ X = B ⇒ A ∗ B = X
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A ∗ X = B
A ∗ B ⊆ X
A ∗ X = B X ∪ (A ∗ B)c = E
X ∪ (A ∗ B)c = X ∪ ((Ac ∪ Bc) ∩
(A ∪ B)) = (X ∪ (Ac ∪ Bc)) ∩ (X ∪ (A ∪ B)) = ((X ∪ Ac) ∪ Bc) ∩ ((X ∪ A) ∪ B)
(X ∪ Ac) ∪ Bc (X ∪ A) ∪ B
(X ∪ Ac) ∪ Bc
X A∗X = (A∩X )∪(Ac∩X c) = (A∪X c)∩(X ∪Ac) = B
x ∈ B ⇔ x ∈ (A ∪ X c) x ∈ (X ∪ Ac) ⇒ x ∈ (X ∪ Ac)
B ⊆ X ∪ Ac
(X ∪ Ac) ∪ Bc ⊇ B ∪ Bc = E E X,A,B ⊆ E
E
(X ∪ Ac) ∪ Bc ⊆ E
(X ∪ Ac) ∪ Bc = E
(X ∪ A) ∪ B
X
A ∗ X = B A ∗ X = B ⇔ (A ∗ X )c = (Ac ∪
X c)∩(A∪X ) = Bc x ∈ Bc ⇔ x ∈ Ac∪X c x ∈ A ∪X ⇒ x ∈ A ∪X = X ∪A Bc ⊆ X ∪ A
(X ∪ A) ∪ B ⊇ Bc ∪ B = E E X,A,B ⊆ E
E (X ∪ A) ∪ B ⊆ E
(X ∪ A) ∪ B = E
((X ∪ Ac) ∪ Bc) ∩ ((X ∪ A) ∪ B) =E ∩ E = E X ∪ (A ∗ B)c = E
X ∪ (A ∗ B)c = E ⇒A ∗ B ⊆ X
X ∪ (A ∗ B)c = E
x ∈ A∗B A∗B = E ∩(A∗B) = (X ∪(A∗B)c)∩(A∗B) = X ∩(A∗B) x ∈ X ∩ (A ∗ B) x ∈ X x ∈ A ∗ B x ∈ X
A ∗ X = B ⇒ X ⊆ A ∗ B
X ∩ (A ∗ B)c = ∅
X ∩ (A ∗ B)c = X ∩ ((A ∩ Bc) ∪
(Ac ∩ B)) = (X ∩ (A ∩ Bc)) ∪ (X ∩ (Ac ∩ B)) = ((X ∩ A) ∩ Bc) ∪ ((X ∩ Ac) ∩ B)
(X ∩ A) ∩ Bc (X ∩ Ac) ∩ B
(X ∩ A) ∩ Bc
x ∈ X ∩ A x ∈ X ∩ A ⇒ x ∈ A ∩ X x ∈ Ac ∩ X c ⇔ x ∈(A ∩ X ) ∪ (Ac ∩ X c) = A ∗ X = B X ∩ A ⊆ B
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(X ∩ A) ∩ Bc ⊆ B ∩ Bc = ∅
∅ ⊆ (X ∩A)∩Bc (X ∩A)∩Bc =∅
(X ∩ Ac) ∩ B
x ∈ X ∩ Ac x ∈ X ∩ Ac ⇒ x ∈ (X ∩ Ac) (X c ∩ A) ⇔ x ∈
(X ∩Ac)∪(X c∩A) (2.b)
= (X ∗A)c = (A∗X )c = Bc X ∩Ac ⊆ Bc
(X ∩ Ac) ∩ B ⊆ Bc ∩ B = ∅
∅ ⊆ (X ∩Ac)∩B (X ∩Ac)∩B =∅
((X ∩ A) ∩ Bc) ∪ ((X ∩ Ac) ∩ B) =
∅ ∪ ∅ = ∅ X ∩ (A ∗ B)c = ∅
X ∩ (A ∗ B)c = ∅ ⇒X ⊆ A ∗ B
X ∩ (A ∗ B)c = ∅
x ∈ X = X ∩ E = X ∩ ((A ∗ B) ∪ (A ∗ B)c) = (X ∩ (A ∗ B)) ∪ (X ∩ (A ∗ B)c) =(X ∩ (A ∗ B)) ∪ ∅ = (X ∩ (A ∗ B)) x ∈ X x ∈ A ∗ B x ∈ A ∗ B X ⊆ A ∗ B
A∗B ⊆ X X ⊆ A ∗B
X = A ∗ B
A ∗ X = B ⇐ A ∗ B = X
X
A ∗ X = A ∗ (A ∗ B) (1.b)
= (A ∗ A) ∗ B (1.c)
= E ∗ B = (E ∩ B) ∪ (E c ∩ Bc) = B ∪ (∅ ∩ Bc) =B ∪ ∅ = B
A ∗ X = B ⇔ A ∗ B = X
A ∗ X = B X = A ∗ B
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E F
E
F
F
E
E
F
f : E → F g : F → E
h = g ◦ f, R = E \ g(F ), F = M ∈ P (E ) | R ∪ h(M ) ⊆ M , A =M ∈F
M.
N = N ∪ {0}
X
Y
u : X −→ Y U, V ⊆ X
U ⊆ V ⇒ u(U ) ⊆ u(V ) u(U ∪ V ) = u(U ) ∪ u(V )
u
u(U ∩ V ) = u(U ) ∩ u(V )
E ∈ F A ∈ F
E ∈ F R ∪ h(E ) ⊆ E
x ∈ R ∪ h(E ) = E \ g(f ) ∪ h(E )
x ∈ E \ g(f )
x ∈ h(E ) ⇒ x ∈ E
x ∈ E ⇔ x ∈ E x ∈ E R ∪ h(E ) ⊆ E
F E ∈ F ⇔ R ∪ h(E ) ⊆ E E ∈ F
A ∈ F R ∪ h(A) ⊆ A
x ∈ R ∪ h(A) = R ∪ h(
M ∈F M ) h
x ∈ R ∪
M ∈F h(M )
x ∈ M ∈F (R ∪ h(M )) M ∈ F ⇔ R ∪ h(M ) ⊆M
x ∈
M ∈F (R ∪ h(M )) ⇒ x ∈
M ∈F M = A
x ∈ A
R ∪ h(A) ⊆ A
A ∈ F ⇔ R ∪ h(A) ⊆ A
A ∈ F
∀M ∈ F , R ∪ h(M ) ∈ F
∀M ∈ F , R∪h(M ) ∈ F ∀M ∈ F , R∪h(R∪h(M )) ⊆R ∪ h(M ) F
M ∈ F R ∪ h(M ) ⊆ M
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x ∈ R ∪ h(R ∪ h(M )) x ∈ R x ∈ h(R ∪ h(M )) R ∪ h(M ) ⊆ M
x ∈ R x ∈ h(M ) x ∈ R ∪ h(M ) R ∪ h(R ∪ h(M )) ⊆ R ∪ h(M ) ∀M ∈ F , R ∪ h(R ∪ h(M )) ⊆ R ∪ h(M ) ∀M ∈ F , R ∪ h(M ) ∈ F
A =
n≥0 h
n(R)
A ⊇
n≥0 hn(R)
∀n ∈ N, A = (n
i=0 hi(R)) ∪ hn+1(A)
n = 0 (n
i=0 hi(R)) ∪ hn+1(A) = R ∪ h(A) = A
A ∈ F R ∪ h(A) ⊆ A A ∈ F R ∪ h(A) ∈ F A =
M ∈F M =
M ∈F M ∩ (R ∪ h(A)) = A ∩ (R ∪ h(A))
A = A ∩ (R ∪ h(A)) ⇔ A ⊆ R ∪ h(A) A ⊆ R ∪ h(A)
n A = (n
i=0 hi(R)) ∪ hn+1(A)
n + 1 A = (n+1
i=0 hi(R)) ∪ h(n+1)+1(A)
A = (n
i=0 hi(R)) ∪ hn+1(A) = (
ni=0 h
i(R)) ∪ hn+1(R ∪ h(A)) = (n
i=0 hi(R)) ∪ hn+1(R) ∪
hn+1(h(A)) = (n+1
i=0 hi(R)) ∪ h(n+1)+1(A))
A ⊇ n≥0 hn(R)
x ∈
n≥0 h
n(R)
x ∈
n≥0 hn(R) ⇒ ∃ j ∈ N, x ∈ h j(R) ⇒ x ∈ (
ji=0 h
i(R)) ∪
h j+1 ∀n ∈ N, A = (
ni=0 h
i(R))
A = ( j
i=0 hi(R))∪h j+1 x ∈ A
A ⊇
n≥0 hn(R)
A ⊆
n≥0 h
n(R)
n≥0 h
n(R) ∈ F n≥0 h
n(R) = R ∪
n≥1 hn(R) = R ∪
n≥0 h(h
n(R)) = R ∪ h(
n≥0 hn(R))
R ∪ h(
n≥0 h
n(R)) ⊆
n≥0 h
n(R)
n≥0 h
n(R) ∈ F ⇔ R ∪ h(
n≥0 h
n(R)) ⊆
n≥0 hn(R) n≥0 hn(R) ∈ F
n≥0 h
n(R) ∈ F
A =
M ∈F M =
M ∈F M ∩
n≥0 hn(R) =
A ∩
n≥0 hn(R)
A = A ∩
n≥0 h
n(R) ⇔ A ⊆
n≥0 hn(R)
A ⊆
n≥0 hn(R)
A ⊇
n≥0 h
n(R) A ⊆
n≥0 h
n(R)
A =
n≥0 hn(R)
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B A E A = f (A) B = g−1(B) R ∪ h(A) = A B A F
∀n ∈ N, A = (n
i=0 hi(R)) ∪ hn+1(A)
n = 0
A = R ∪ h(A)
(A)c = B
(A)c = (f (A))c = (f (
n≥0 h
n(R)))c = (
n≥0 f (hn(R)))c =
n≥0(f (hn(R)))c
B = g−1(B) = g−1(Ac) = g−1((R ∪ h(A))c) = g−1(Rc ∩ h(A)c) =g−1(Rc) ∩ g−1(h(A)c)
Rc = (E \ g(F ))c = (E ∩ (g(F ))c)c = E c ∪ ((g(F ))c)c = ∅ ∪ g(F ) = g(F ) g−1(Rc) = g−1(g(F ))
{x ∈ F | g(x) ∈ g(F )} g ∀x ∈ F, ∃! y ∈ E g(x) = y y ∈ {g(x) ∈ E | x ∈ F } = g(F ) ∀x ∈ F, ∃! y ∈ g(F ) g(x) = y {x ∈ F | g(x) ∈ g(F )} = {x ∈ F } = F g−1(Rc) = F
g−1(Rc)∩g−1(h(A)c) = F ∩g−1(h(A)c) = g−1(h(A)c) = g−1((h(
n≥0 hn(R)))c) =
g−1((
n≥0 h(hn(R)))c) = g−1(
n≥0 h
n+1(R)c) =
n≥0 g−1(hn+1(R)c)
B =
n≥0 g−1(hn+1(R)c)
(f (A))c = B ⇔ n≥0(f (h
n(R)))c =n≥0 g
−1
(hn+1
(R)c
)
n≥0(f (h
n(R)))c =
n≥0 g−1(hn+1(R)c)
∀n ∈ N, (f (hn(R)))c = g−1(hn+1(R)c)
n ∈ N a ∈ g−1(hn+1(R)c)
a ∈ {x | g(x) ∈ hn+1(R)c} = {x | g(x) ∈ hn+1(R)} = {x | g(x) ∈ h(hn(R))} = {x | g(x) ∈g(f (hn(R)))} a ∈ {x | g(x) ∈ g(f (hn(R)))} ⇔ g(a) ∈ g(f (hn(R))) = {g(x) | x ∈f (hn(R))} ⇔ x ∈ f (hn(R)) g(x) = g(a) g x ∈ f (hn(R))
g(x) = g(a) ⇔ x ∈ f (hn(R)) x = a ⇔ a ∈ f (hn(R)) ⇔ a ∈ (f (hn(R)))c a ∈ (f (hn(R)))c
∀n ∈ N, (f (hn(R)))c = g−1(hn+1(R)c)
(f (A))c = B
f : A → A g : B → B f g
f g
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f x, y ∈ A f (x) = f (y) ⇔
x = y
x, y ∈ A x, y ∈ E f f f (x) = f (x)
f (y) = f (y) f f (x) = f (y) ⇔ x = y f (x)
f (x) f (y)
f (y)
f (x) = f (y) ⇔ x = y
f
∀y ∈ A, ∃x ∈ A f (x) = y
y ∈ A y ∈ f (A) y ∈ {f (x) | x ∈ A}
y ∈ {f (x) | x ∈ A} ⇔ ∃x ∈ A f (x) = y ∃x ∈ A f (x) = y
y ∈ A
∀y ∈ A, ∃x ∈ A f (x) = y
g
x, y ∈ B g(x) = g (y) ⇔ x = y
x, y ∈ B x, y ∈ F g g g(x) = g(x)
g(y) = g(y) g g(x) = g(y) ⇔ x = y g(x)
g(x) g(y)
g(y)
g(x) = g (y) ⇔ x = y
g ∀y ∈ B, ∃x ∈ B g(x) = y
y ∈ B y ∈ Ac = (R∪h(A))c = Rc∩h(A)c Rc = g(F )
y ∈ g(F )∩h(A)c y ∈ g(F ) y ∈ h(A)c y ∈g(F ) y ∈ g(F ) = {g(x) | x ∈ F } y ∈ {g(x) | x ∈ F } ⇔ ∃x ∈ F
g(x) = y
∃x ∈ F
g(x) = y
g(x) = y ⇒ ∃a ∈ B
g(x) = a
a = y ⇔ x ∈ {x | g(x) ∈ B } = g−1(B) = B
x ∈ B ∃x ∈ B g(x) = y x ∈ B g(x) = g (x) ∃x ∈ B g(x) = y y ∈ B ∀x ∈ B, ∃x ∈ B g(x) = y
f
g
φ : E −→ F
x −→
f (x) x ∈ A(g)−1(x)
x ∈ B
g
(g)−1
φ ∀x ∈ E , ∃! y ∈ F φ(x) = y
x ∈ E = A ∪ Ac x ∈ A x ∈ Ac
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x ∈ A φ(x) = f (x) f ∃! y ∈ A f (x) = y A ⊆ F ∃! y ∈ F φ(x) = y
x ∈ Ac φ(x) = (g)−1(x) (g)−1 ∃! y ∈ B
(g)−1(x) = y B ⊆ F ∃! y ∈ F φ(x) = y
A ∩ Ac = ∅ x
∃! y ∈ F φ(x) = y
x ∀x ∈ E , ∃! y ∈ F φ(x) = y φ
φ
φ
φ
x, y ∈ E , φ(x) = φ(y) ⇔ x = y
x, y ∈ E = A ∪ Ac = A ∪ B
• x ∈ A y ∈ A φ(x) = f (x) φ(y) = f (y) φ(x) = φ(y) ⇔ f (x) =f (y) f f (x) = f (y) ⇔ x = y φ(x) = φ(y) ⇔ x = y
• x ∈ B y ∈ B φ(x) = (g)−1(x) φ(y) = (g)−1(y) φ(x) = φ(y) ⇔
(g
)
−1
(x
) = (g
)
−1
(y
)
(g
)
−1
(g
)
−1
(x
) = (g
)
−1
(y
) ⇔x = y
φ(x) = φ(y) ⇔ x = y
• x ∈ A y ∈ B A ∩ B = A ∩ Ac = ∅ x = y φ(x) = f (x) φ(y) = (g)−1(y) f (g)−1
φ(x) ∈ A φ(y) ∈ B A ∩ B = A ∩ (A)c = ∅
φ(x) = φ(y) x = y φ(x) = φ(y)
x = y ⇔ φ(x) = φ(y)
φ(x) = φ(y) ⇔ x = y
• x ∈ B y ∈ A x ∈ A y ∈ B
x
y y
x
x, y ∈ E , φ(x) = φ(y) ⇔ x = y φ
φ ∀y ∈ F, ∃x ∈ E φ(x) = y
y ∈ F y ∈ F = A ∪ (A)c y ∈ A y ∈ (A)c = B
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• y ∈ A f
y ∈ A ∃x ∈ A f (x) = y x ∈ A φ(x) = f (x) ∃x ∈ A φ(x) = y A ⊆ E ∃x ∈ E φ(x) = y
• x ∈ B (g)−1 y ∈ B ∃x ∈ B (g)−1(x) = y x ∈ B φ(x) = (g)−1(x) ∃x ∈ B φ(x) = y B ⊆ E ∃x ∈ E φ(x) = y
∃x ∈ E φ(x) = y y ∀y ∈ F, ∃x ∈ E φ(x) = y φ
φ
E F
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