5.3 " 4 | 9 ½ + Ë_ l à º-Cantor Theorem

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1 / 13

&

ño 2(Cantor Theorem) e”

__ |9½+Ë X \ @/ #Œ

cardX < cardP(X ).

7

£x"î

â

ĺ (1) : X = φ

cardφ = 0 < 1 = cardP(φ).

5.3 " 4 | 9 ½ + Ë_  l à º-Cantor Theorem

&

ño 2(Cantor Theorem) e”

__ |9½+Ë X \ @/ #Œ

cardX < cardP(X ).

7

£x"î

â

ĺ (1) : X = φ

cardφ = 0 < 1 = cardP(φ).

2 / 13

â

ĺ (2) : X 6= φ

†

<Êú g : X → P(X )\¦ 6£§õ °ú s &ñ_ :

g (x ) = {x } ∀x ∈ X

g H éߖsÙ¼–Ð, X ∼ {{x} | x ∈ X } ⊆ P(X ). "f cardX ≤ cardP(X )

s

. s]j cardX 6= cardP(X ), 7£¤, X 6∼ P(X )e”`¦ ˜Ðs.

X ∼ P(X ), 7£¤, „éߖ f : X → P(X ) ”>rFôÇ“¦ &ñ “¦,

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½+Ë S = {x ∈ X | x 6∈ f (x)}`¦ ÒqtyŒ• .

5.3 " 4 | 9 ½ + Ë_  l à º-Cantor Theorem

S ∈ P(X )s“¦ f H „sl M:ëH\, f (e) = S

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¦ëߖ7ᤠH e ∈ X ”>rFôÇ.

â

ĺ (2-1) e ∈ S

e ∈ f (e)sl M:ëH\, S_ &ñ_\ _ #Œ e 6∈ Ss“¦, sכ “Ér Ô

¦0px .

â

ĺ (2-2) e 6∈ S

e 6∈ f (e)sl M:ëH\, S_ &ñ_\ _ #Œ e ∈ Ss“¦, sכ “Ér Ô

¦0px .



"f, cardX 6= cardP(X )s. ÕªQÙ¼–Ð cardX < cardP(X ) s

. _{4 / 13}

2, 3  : yŒ• K.

5.4 l à º_  » ! l r

kü< ls Ä»ôÇlú 7£¤, k, l ∈ N ∪ {0}“ âĺ k + lõ kl“Ér #Q

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"

 _p\¦ t“¦ e”`¦?

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ñ_ 2

"

f–Ð ™è“ ¿º |9½+Ë A, B_ lú\¦yŒ•yŒ• a, b ½+É M:,

card(A ∪ B)\¦ a, b_ lú_ ½+Ës “¦a + b–Ð
·p.

6 / 13

&

ñ_2H ¸ú˜ &ñ_÷&#Q e”.

aü< b lú . (C-1)\ _K, cardX = a, cardY = b`¦ ë

ߖ7ᤠH |9½+Ë X ü< Y  ”>rFôÇ. ëߖ€• X ∩ Y 6= φs€, A = X × {0}, B = Y × {1}s Z~. ÕªQ€ A ∩ B = φ, A ∼ X , B ∼ YsÙ¼–Ð, cardA = a, cardB = bs. ÕªQÙ¼–Ð a + b = card(A ∪ B).

¢

¸ôÇ,a + bH Ä»{9 . =
 €, cardA1= a, cardB1= b“

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f–Ð ™è“ ¿º |9½+Ë A₁, B₁s ”>rFôÇ€, A₁ ∼ A, B₁ ∼ Bs.

Õ

ªQ€ 4©œ_ &ño 6\ _ #Œ A1∪ B₁ ∼ A ∪ B.

Õ

ªQÙ¼–Ðcard(A₁∪ B₁) = card(A ∪ B).

5.4 l à º_  » ! l r

0

A_ ?/6 x“Ér &ño 3_ 7£x"îe”.

&

ño 3

a, b lú . ÕªQ€

(a) 6£§`¦ ëߖ7ᤠH "f–Ð ™è“ ¿º |9½+Ë A, B ”>rFôÇ:

cardA = a, cardB = b.

(b) cardA = cardA₁, cardB = cardB₁, A ∩ B = φ, A1∩ B₁ = φs€,

cardA ∪ B) = card(A₁∪ B₁).

8 / 13

0

A_ ?/6 x“Ér &ño 3_ 7£x"îe”.

&

ño 3

a, b lú . ÕªQ€

(a) 6£§`¦ ëߖ7ᤠH "f–Ð ™è“ ¿º |9½+Ë A, B ”>rFôÇ:

cardA = a, cardB = b.

(b) cardA = cardA₁, cardB = cardB₁, A ∩ B = φ, A1∩ B₁ = φs€,

cardA ∪ B) = card(A₁∪ B₁).

5.4 l à º_  » ! l r

0

A_ ?/6 x“Ér &ño 3_ 7£x"îe”.

&

ño 3

a, b lú . ÕªQ€

(a) 6£§`¦ ëߖ7ᤠH "f–Ð ™è“ ¿º |9½+Ë A, B ”>rFôÇ:

cardA = a, cardB = b.

(b) cardA = cardA₁, cardB = cardB₁, A ∩ B = φ, A1∩ B₁ = φs€,

cardA ∪ B) = card(A₁∪ B₁).

8 / 13

\ V]j 2

¿

º Ä»ôÇlú 4, 3_ lú_ ½+Ë 4 + 3`¦ ½¨ #Œ.

Û

¦s

A = N4, B = {5, 6, 7}s . ÕªQ€

cardA = 4, cardB = 3, A ∩ B = φ s

. ÕªQÙ¼–Ð

4 + 3 = card(A ∪ B) = cardN7 = 7.

5.4 l à º_  » ! l r

\ V]j 2

¿

º Ä»ôÇlú 4, 3_ lú_ ½+Ë 4 + 3`¦ ½¨ #Œ.

Û

¦s

A = N4, B = {5, 6, 7}s . ÕªQ€

cardA = 4, cardB = 3, A ∩ B = φ s

. ÕªQÙ¼–Ð

4 + 3 = card(A ∪ B) = cardN7 = 7.

9 / 13

&

ño 4

x , y , z e”__ lú . ÕªQ€ (a) “§¨8ŠZOgË: x + y = y + x

(b) ½+ËZOgË: (x + y ) + z = x + (y + z) 7

£x"î

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½+Ë_ ƒíߖ ∪“Ér “§¨8ŠZOgË:õ ½+ËZOgË:s $íwn Ù¼–Ð ~1> 7

£x"î½+É Ãº e”. (7£x"î Òqt|ÄÌ)

5.4 l à º_  » ! l r

&

ño 4

x , y , z e”__ lú . ÕªQ€ (a) “§¨8ŠZOgË: x + y = y + x

(b) ½+ËZOgË: (x + y ) + z = x + (y + z) 7

£x"î

|9

½+Ë_ ƒíߖ ∪“Ér “§¨8ŠZOgË:õ ½+ËZOgË:s $íwn Ù¼–Ð ~1> 7

£x"î½+É Ãº e”. (7£x"î Òqt|ÄÌ)

10 / 13

&

ño 4

x , y , z e”__ lú . ÕªQ€ (a) “§¨8ŠZOgË: x + y = y + x

(b) ½+ËZOgË: (x + y ) + z = x + (y + z) 7

£x"î

|9

½+Ë_ ƒíߖ ∪“Ér “§¨8ŠZOgË:õ ½+ËZOgË:s $íwn Ù¼–Ð ~1> 7

£x"î½+É Ãº e”. (7£x"î Òqt|ÄÌ)

5.4 l à º_  » ! l r

&

ño 4

x , y , z e”__ lú . ÕªQ€ (a) “§¨8ŠZOgË: x + y = y + x

(b) ½+ËZOgË: (x + y ) + z = x + (y + z) 7

£x"î

|9

½+Ë_ ƒíߖ ∪“Ér “§¨8ŠZOgË:õ ½+ËZOgË:s $íwn Ù¼–Ð ~1> 7

£x"î½+É Ãº e”. (7£x"î Òqt|ÄÌ)

10 / 13

cardN = ℵ0 (·ú˜YUáÔ ]j–Ð) cardR = c

\ V]j 3 l

ú_ ½+Ë ℵ₀+ ℵ0`¦ ½¨ #Œ.

Û

¦s

Ne = {2n | n ∈ N}, No = {2n − 1 | n ∈ N} .

Õ

ªQ€ Ne∩ No = φs“¦ Ne∪ No = Ns. ÕªQÙ¼–Ð ℵ₀+ ℵ0 = cardNe+ cardNo = card(Ne∪ No) = cardN = ℵ0.

5.4 l à º_  » ! l r

cardN = ℵ0 (·ú˜YUáÔ ]j–Ð) cardR = c

\ V]j 3 l

ú_ ½+Ë ℵ₀+ ℵ0`¦ ½¨ #Œ.

Û

¦s

Ne = {2n | n ∈ N}, No = {2n − 1 | n ∈ N} .

Õ

ªQ€ Ne∩ No = φs“¦ Ne∪ No = Ns. ÕªQÙ¼–Ð ℵ₀+ ℵ0 = cardNe+ cardNo = card(Ne∪ No) = cardN = ℵ0.

11 / 13

\ V]j 4 l

ú_ ½+Ë ℵ0+ c`¦ ½¨ #Œ.

Û

¦s

(0, 1) ∼ RsÙ¼–Ð, card(0, 1) = cardR = c s.

S = N ∪ (0, 1)s Z~. N ∩ (0, 1) = φsl M:ëH\, cardS = ℵ0+ c.

ô

Ǽ#, R ∼ (0, 1) ⊂ Ss“¦ S ∼ S ⊂ Rsl M:ëH\, Cantor-Berstein Theorem\ _K R ∼ Ss. ÕªQÙ¼–Ð

ℵ₀+ c = cardS = cardR = c s

.

5.4 l à º_  » ! l r

\ V]j 4 l

ú_ ½+Ë ℵ0+ c`¦ ½¨ #Œ.

Û

¦s

(0, 1) ∼ RsÙ¼–Ð, card(0, 1) = cardR = c s.

S = N ∪ (0, 1)s Z~. N ∩ (0, 1) = φsl M:ëH\, cardS = ℵ0+ c.

ô

Ǽ#, R ∼ (0, 1) ⊂ Ss“¦ S ∼ S ⊂ Rsl M:ëH\, Cantor-Berstein Theorem\ _K R ∼ Ss. ÕªQÙ¼–Ð

ℵ₀+ c = cardS = cardR = c s

.

12 / 13

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