2002[ Í 1 È ÑM

(1)

V

2 * × Ê Á + W N

2002[ Í 1 È ÑM

· כ ÖÜ ï ¬ £P $ ã <P

~ ç

¡q ó i ; < K 2 ç ¡

"

k] ï >6 0 È Ñz ¬ £ È ÑÜ ï 4

û B0 ² ?

2002[ Í 3 i J 3G B 4 S 2002[ Í 7 i J 8G B ¬ £+ ä 33 ¶ ; ∼78 ¶ ;

s

©\"fHÄºo #Qwn= M:ÂÒ' 6 x ~ Ãº_ 8 l, YL l, íH"f 1px`¦

^

>&hÜ¼Ð /BNÂÒôÇ. $ 0 = ∅, 1 = {0}, 2 = {0, 1} 1pxÜ¼Ð Ãº\¦ &ñ_

¦, Ãº s_ 8 l x9 YL l\¦&ñ_ôÇ. sQôÇ &ñ_\ {9y #

§¨8ZOgË:, ½+ËZOgË:, CìrZOgË: 1px`¦ 7£x"îôÇ. Ãº_ íH"fü<'aº # ©

×

æכ¹ôÇ $í|9Ér e__ Ãº|9½+ËÉrþjè°úכ`¦Hכ X<, Óüt:rÃº _

&ñ_\ {9y # s\¦ 7£x"îôÇ. &ñÃº\¦¿º Ãº_ sÐ sK l 0A

#

, Ãº_ íH"f©[þt\ &h]XôÇ 1lxu'a>\¦ ÂÒ#<ÊÜ¼Ð+ &ñÃº|9½+Ë`¦ ëßH

. ðøÍtÐ &ñÃº_ íH"f©[þt\ &h]XôÇ 1lxu'a>\¦ ÂÒ# # Ä»oÃº|9½+Ë

`

¦ ëß×¼HX<, Ä»oÃº\"fHy5px]j Ä»Ð0>. Ä»oÃº^H © Ér í

H"f^s. s ©\"fHÄ»oÃºÐÂÒ' z´Ãº\¦½¨$í H¿º t ~½ÓZO`¦è>h

HX<, H X<X<àÔ ¦îßôÇ ]Xéß`¦s6 x H כ s¦, ¢¸ Ér H ñ

ßÐØÔ ¦îßôÇ ïrÃº\P`¦s6 x Hכ s. s ¿ºt ~½ÓZOÉr 1872 ¸ °ú Ér K

\ yy µ1Ï³ð÷&%3. s ¿ºt ~½ÓZOÉrz´© °ú Érõ\¦4R¸HX<, t

} ]X\"f e__¢-aqíH"f^H½Ó© °ú 6£§`¦Ð.

2.1. ¥ o >Ê Á

|9

½+Ë A \ @/ # DhÐîr |9½+Ë A⁺\¦6£§

A⁺= A ∪ {A}

33

(2)

Ü

¼Ð &ñ_ôÇ. \V\¦ [þt#Q"f, /BN|9½+ËÂÒ' Ø¦µ1Ï # 0 = ∅ s ¿º¦, 1 = 0⁺, 2 = 1⁺, 3 = 2⁺, . . . 1pxÜ¼Ð &ñ_ 6£§

0 = ∅

1 = 0⁺= 0 ∪ {0} = ∅ ∪ {0} = {0}

2 = 1⁺= 1 ∪ {1} = {0} ∪ {1} = {0, 1}

3 = 2⁺= 2 ∪ {2} = {0, 1} ∪ {2} = {0, 1, 2}

. . .

(1)

õ

°ú s)a. 6£§¿º t $í|9

∅ ∈ A, A ∈ A =⇒ A⁺∈ A (2)

`

¦tH |9½+Ë A [þt ^_ §|9½+Ë`¦ N s æ¼¦, s |9½+Ë_ "é¶è[þt

`

¦ a:@¬£ ÂÒÉr. |9½+Ë N s $í|9 (2) \¦ tHכ Ér Ð SX)a

. "f 6£§&ño_ (), () $íwn ¦, (1) õ °ú s &ñ_)a|9

½

+Ë 0, 1, 2, 3, . . . HÃºs. ¢¸ôÇ, n⁺= n ∪ {n} Ér /BN|9½+Ës mÙ¼

Ð 6£§&ño_ () ¸ {© . 6£§&ño\ \P)a$Á t $í|9

É

rDq{¡⁽¹⁾ ðP4 Ô¦oîr.

XN

ËP 2.1.1. · כÖ N ª<:? )çHB¾ç>Ãç> ¹ÿø¶;Â6Ò.

() 0 ∈ N.

() n ∈ N =⇒ n⁺∈ N.

() »qqn ∈ N ;c 60 #l n⁺6= 0 T.

() a:@¬£¾ç>q· כÖX ⊂ N :?¨£ )çHB

0 ∈ X, n ∈ X =⇒ n⁺ ∈ X (3) Ã

ç

> ¹ÿø¶; ^@X = N T.

(1) Giuseppe Peano (1958∼1932), s»1Ïo Ãº<Æ. Turin \"f /BNÂÒ ¦Ö¸1lx

% i.

(3)

2.1. Ãº 35

() ¹ÿGBm, n ∈ N ;c 60 #l m⁺= n⁺T^@m = n T.

ë

ß{9 X ⊂ N $í|9 (2) \¦t {©y X = N s ÷&HX<, s\¦

r H כ s () s. sH¬£ÈÑ+8 Zôm(« `¦6 x½+É Ãº eHHs

. ëß{9 Ãº n ∈ N \'aôÇ "î]j P (n) s e`¦ M:, P (n) s $íwn

HÃº n ∈ N [þt_ |9½+Ë`¦ X ¿º. ëß{9 P (0) s $íwn<Ê`¦·ú¦, P (n) =⇒ P (n⁺) `¦Ðs X $í|9 (3) `¦ëß7á¤ôÇH´ús)a.

"f X = N X<, sH e__ Ãº n ∈ N \ @/ # P (n) s $íwn ô

ÇH >pws.

s

]j, &ño 2.1.1 _ () \¦ 7£x"îK Ð. Äº, n ∈ n⁺= m⁺= m ∪ {m}

s

Ù¼Ð n ∈ m s n = m s. ðøÍt ~½ÓZOÜ¼Ð m ∈ n s n = ms. "f, 6£§

n ∈ N, x ∈ n =⇒ x ⊂ n (4)

`

¦ 7£x"î n ⊂ m õ m ⊂ n s $íwn # 7£x"îs =åQèß. s]j (4) \¦ 7

£x"î l 0A #

X = {n ∈ N : x ∈ n =⇒ x ⊂ n}

s

¿º. $ 0 ∈ X eÉr ìr"î . 7£¤, x ∈ ∅ =⇒ x ⊂ ∅ Ér `Ér

"

î]js. s]j Ãº<Æ&h ) ±úZO`¦6 x l 0A # n ∈ X &ñ .

ë

ß{9 x ∈ n⁺ = n ∪ {n}s x ∈ n s x = n s. ëß{9 x = n s

x = n ⊂ n⁺s¦, x ∈ n s ) ±úZO &ñ\ _ # x ⊂ n sÙ¼Ð x ⊂ n ⊂ n⁺ $íwnôÇ. "f X = N s¦ (4) 7£x"î÷&%3.

6£§&ñoH |9½+Ë X _ "é¶è γ(0), γ(1), γ(2), . . . :£¤&ñ &hod`¦ ë

ß7á¤ ¸2¤) ±ú&hÜ¼Ð &ñ_½+É Ãº e6£§`¦´úK ÅÒHX<, Ãº_ íß

`

¦&ñ_ HX<\ ×æכ¹ôÇ %i½+É`¦ôÇ.

(4)

XN

ËP 2.1.2. · כÖX q Â6Ò4wHÐ a ∈ X Üï ÈÕ¬£ f : X → X ;c 60 #l,

:? )çHB () γ(0) = a,

() »qqn ∈ N ;c 60 #l γ(n⁺) = f (γ(n)) T )ç· Â6Ò Ã

ç

> ¹ÿø¶; ¤< ÈÕ¬£ γ : N → X ¤GB 4 +í<<Â6Ò.

7

£x"î: $, () ü< () \¦ 1lxr\ ëß7á¤ H<ÊÃº γ¹õ γ² e¦

&ñ ¦,

X = {n ∈ N : γ¹(n) = γ2(n)}

s

¿º. $ () \ _ # 0 ∈ X s. ëß{9 n ∈ X s γ₁(n⁺) = f (γ₁(n)) = f (γ₂(n)) = γ₂(n⁺) s

Ù¼Ð n⁺∈ Xs. "f, Ãº<Æ&h ) ±úZO\ _ # X = N s¦, e _

_ n ∈ N \ @/ # γ¹(n) = γ2(n) e`¦·ú Ãº e.

s

]j, >rF$í`¦Ðsl 0A # 6£§$í|9

(0, a) ∈ A, (n, x) ∈ A =⇒ (n⁺, f (x)) ∈ A (5)

`

¦ëß7á¤ H A ⊂ N × X ^_ |9½+Ë`¦A ¿º N × X ∈ A sÙ¼Ð A Hq#Q et ·ú§. s]j, γ =T A ¿º. ëß{9 γ ⊂ N × X <ÊÃº s

6£§

γ(n) = x =⇒ (n, x) ∈ γ

=⇒ (n⁺, f (x)) ∈ γ

=⇒ γ(n⁺) = f (x) = f (γ(n)) s

$íwn Ù¼Ð, γ ⊂ N × X <ÊÃºe`¦Ðs 7£x"îs =åQèß.

s

\¦0A # r

X = {n ∈ N : (n, x) ∈ γ \¦ëß7á¤ H x ∈ X Ä»{9 > >rFôÇ}

(5)

2.1. Ãº 37

¿º¦ ) ±úZO`¦ 6 xôÇ. ëß{9 0 /∈ X (0, b) ∈ γ, b 6= a b ∈ X >rFôÇ. ÕªX< b 6= a sÙ¼Ð, &ño 2.1.1 () \ _ # γ \ {(0, b)} ⊂ N × X %ir $í|9 (5) \¦ëß7á¤ôÇ. sH γ =T A \ ¸íH s

. "f, 0 ∈ X s. s]j n ∈ X stëß n⁺ ∈ X/ &ñ . Õª

Q (n⁺, f (x)) ∈ γsÙ¼Ð (n⁺, y) ∈ γ, y 6= f (x) y ∈ X >rFôÇ

. s\¸ %ir γ \ {(n⁺, y)} ∈ A e`¦Ðs n ∈ X =⇒ n⁺ ∈ X

7£x"î÷&¦, "f ¸H 7£x"îs =åQèß.

Ä

º n⁺ 6= 0sÙ¼Ð (0, a) ∈ γ \ {(n⁺, y)}s. s]j (m, z) ∈ γ \ {(n⁺, y)} &ñ . ëß{9 m = n s z = x s¦ ¢¸ôÇ y 6= f (x) sÙ¼

Ð (m⁺, f (z)) = (n⁺, f (x)) ∈ γ \{(n⁺, y)} e`¦·ú Ãº e. ëß{9 m 6= n s

&ño 2.1.1 () \ _ # m⁺ 6= n⁺s¦, "f (m⁺, f (z)) ∈ γ \ {(n⁺, y)}s.

&

ño 2.1.2 \¦&h6 x , y Ãº m ∈ N \ @/ # γm(0) = m, n ∈ N =⇒ γ^m(n⁺) = [γm(n)]⁺

\

¦ëß7á¤ H<ÊÃº γ^m: N → N s Ä»{9 > >rF<Ê`¦·ú Ãº e. s]j,

¿

º Ãº_ G M\¦

m + n = γm(n), m, n ∈ N s

&ñ_

m + 0 = m, m + n⁺= (m + n)⁺, m, n ∈ N s

$íwnôÇ. ½+ËZOgË:, §¨8ZOgË:, ½Ó1px"é¶\'aôÇ כ [þt`¦ 7£x"î l ·ú¡"f n⁺= 1 + n, n ∈ N

e

`¦$ Ðs. $ 0⁺= 1 = 1 + 0s. s 1pxds Ãº n \ @/

# $íwnôÇ &ñ (n⁺)⁺= (1 + n)⁺= 1 + n⁺s $íwn Ù¼Ð, Ãº

(6)

<Æ&h ) ±úZO`¦&h6 x½+É Ãº e. YL l\¦&ñ_ Hכ ¸ &ño 2.1.2 \¦ s

6 xôÇ. $, y Ãº m ∈ N \ @/ #

δ_m(0) = 0, n ∈ N =⇒ δm(n⁺) = δ_m(n) + m s

$íwn H<ÊÃº δm: N → N `¦¸úÉrÊê\,

mn = δ_m(n), m, n ∈ N s

&ñ_ôÇ. ÕªQ

m0 = 0, mn⁺= mn + m, m, n ∈ N s

$íwnôÇ.

XN

ËP 2.1.3. »qqm, n, k ∈ N ;c 60 #l :?T )ç· Â6Ò.

() 0 + n = n, (m + n) + k = m + (n + k)Dm + n = n + m T.

() 0n = 0, 1n = n, (mn)k = m(nk)Dmn = nm T.

() m(n + k) = mn + mkD(n + k)m = nm + km T.

¸H 7£x"îÉr) ±úZO`¦6 xôÇ. $ 0 + 0 = 0 Érsp ·ú¦ e¦, 0 + n = ns 0 + n⁺ = (0 + n)⁺ = n⁺sÙ¼Ð, () _ 'Í P: "î]j

7

£x"î)a. ÑütP: "î]jH m, n `¦¦&ñ ¦ k \'aôÇ ) ±úZO`¦6 x

)

a. [jP: "î]j %ir m `¦¦&ñ ¦ n \'aôÇ ) ±úZO`¦6 x )a

. "î]j (), () %ir Ãº<Æ&h ) ±úZO`¦6 x # 7£x"îôÇ.

' Ö

<<K 2.1.1. &ño 2.1.3 `¦ 7£x"î #.

' Ö

<<K 2.1.2. &ño 2.1.2 \¦s6 x # mⁿ`¦&ñ_ ¦, 6£§íß ZOgË:[þt m^n+k= mⁿm^k, (mn)^k= m^kn^k (mⁿ)^k= m^nk

`

¦ 7£x"î #.

(7)

2.1. Ãº 39

s

]j |9½+Ë N \ íH"f\¦ &ñ_ ¦ s ]X`¦ ëBlÐ ôÇ. ¿º Ãº m, n ∈ N \ @/ #

m ≤ n ⇐⇒ m ∈ n <ÊÉr m = n s

&ñ_ôÇ. $ m ≤ m eÉr{© . ¢¸ôÇ m ≤ n õ n ≤ m s 1

l

xr\ $íwnôÇ &ñ . ëß{9 m 6= n s m ∈ n õ n ∈ m s¦, (4)

\

_ # m ⊂ n, n ⊂ m s $íwnÙ¼Ð m = n s)a. s]j, m ≤ n, n ≤ k &ñ . ÕªQ 6£§

m ∈ n, n ∈ k m ∈ n, n = k m = n, n ∈ k m = n, n = k W

1 t âÄº Òqt|. %6£§ [j âÄº\H m ∈ k $íwn ¦, Qt

â

Äº\H m = k $íwn Ù¼Ð m ≤ k e`¦ ·ú Ãº e. "f, ≤ H í

H"f'a>)a. 6£§ &ñoHÃº|9½+Ë\ &ñ_)a íH"f'a>_ Ùþd

&

h $í|9s. ¿º Ãº m, n ∈ N s m ≤ n s"f m 6= n {9 M:, m < ns H. "f, m < n ⇐⇒ m ∈ n s. :£¤y, m ∈ m⁺sÙ¼

Ð m < m⁺s.

XN

ËP 2.1.4. R#eU óm¬ª<a:@¬£¾ç>q· כÖª< L|Ð 4wHÐ¿ì>.> .

7

£x"î: q#Q et ·ú§Ér Ãº[þt_ |9½+Ë A ⊂ N þjè "é¶è\¦tt

· ú

§H¦ &ñ ¦,

X = {n ∈ N : m ∈ A =⇒ n ≤ m}

s

¿º. ëß{9 k ∈ X ∩A s k H A_ þjè "é¶è ÷&Ù¼Ð X ∩A = ∅ s

#Q ôÇ. s]j ) ±úZO`¦s6 x # X = N e`¦Ðs A = ∅ s ÷&

#

Q"f ¸íH`¦%3H.

s

\¦0A # $ 0 ∈ X, 7£¤ e__ m ∈ N \ @/ # 0 ≤ m e`¦

ÐsHX<, s ¢¸ôÇ ) ±úZO`¦6 xôÇ. $ 0 ≤ 0 Ér "î . ëß{9

(8)

0 ≤ ms, m ∈ m⁺\"f m ≤ m⁺sÙ¼Ð 0 ≤ m⁺\¦%3H. 6£§Ü¼

Ð n ∈ X =⇒ n⁺∈ X `¦Ðsl 0A # n ∈ X \¦&ñ . 7£¤, m ∈ A =⇒ n ≤ m

`

¦ &ñ . ëß{9 n ∈ A s n Ér A_ þjè "é¶ès¦, sH&ñ\

#

QFMèß. "f n /∈ As¦, 6£§

m ∈ A =⇒ n < m s

$íwnôÇ. s]j 6£§

n < m =⇒ n⁺≤ m (6)

`

¦Ðs ¸H 7£x"îs =åQèß. s %ir n `¦¦&ñ ¦ m \ @/ôÇ ) ±ú ZO

`¦6 x l 0A #

Y = {m ∈ N : n < m =⇒ n⁺≤ m}

s

¿º. $ n < 0 s n ∈ ∅ X< sQôÇ "é¶è n s \OÜ¼Ù¼Ð 0 ∈ Y eÉr "î . s]j, m ∈ Y &ñ . ëß{9 n < m⁺s n ∈ m⁺ = m ∪ {m}s¦, "f n ∈ m s n = m s. $ n ∈ ms n < m sÙ¼Ð ) ±úZO &ñ\ _ # n⁺ ≤ m < m⁺s

¦, n = m s n⁺= m⁺sÙ¼Ð, #QÖ¼ âÄº n⁺≤ m⁺s. "f, n < m⁺=⇒ n⁺≤ m⁺ $íwn ¦ m⁺∈ Y e`¦·ú Ãº e. '

Ö

<<K 2.1.3. m ≤ n ⇐⇒ m⁺≤ n⁺e`¦Ð#.

¿

º Ãº m, n ∈ N \ @/ # {m, n} ⊂ N Hþjè "é¶è\¦tH X

<, þjè "é¶è m s m ≤ n s¦, þjè "é¶è n s n ≤ m s.

¨XNËP 2.1.5. »qq ¨£ a:@¬£ m, n ∈ N ;c 60 #l m ≤ n ýnª<

n ≤ m T )ç· Â6Ò.

(9)

2.1. Ãº 41

6£§$í|9

n ∈ N =⇒ n = 0 <ÊÉr n = m⁺ m ∈ N s >rFôÇ (7)

É

r2£§&ño 2.1.6 õ &ño 2.1.7 _ 7£x"îõ&ñ\"f Ä»6 x > æ¼.

' Ö

<<K 2.1.4. ₎_±_ú_ZO`¦ s6 x # "î]j (7) `¦ 7£x"î #.

¨XNËP 2.1.6. a}¹ ¤4T f R#eU óm¬ª<a:@¬£¾ç>q· כÖA ⊂ N ¤< L|60 4wHÐ¿ì>.> .

7

£x"î: |9½+Ë A _ ©> ^_ |9½+Ë`¦ B ¿º, &ñ\ _ # B 6= ∅ s

¦ &ño 2.1.4 \ _ # þjè "é¶è k ∈ N \¦ . e__ n ∈ A

\

@/ # n ≤ k sÙ¼Ð k ∈ A e`¦ Ðs k A _ þj@/ "é¶èes 7

£x"î)a. s]j, k /∈ As &ñ . ÕªQ, n ∈ A `¦ ¸ú¤`¦ M

: k > n ≥ 0 s¦, "f (7) \ _ # k = s⁺s. ëß{9 s /∈ Bs s < t t ∈ A >rF ¦, s⁺≤ t ≤ k = s⁺sÙ¼Ð k = t ∈ A ÷&#Q

"

f ¸íHs. "f s ∈ B X<, s < k sÙ¼Ð sH k B _ þjè "é¶

èHX<\ ¸íHs. ÕªQÙ¼Ð, k ∈ A e`¦·ú Ãº e.

6£§&ñoH&ñÃº\¦&ñ_ HX<\ ×æכ¹ôÇ %i½+É`¦ HX<, HÃº\

"

f ÉrÃº\¦õü Ãº e6£§`¦´úK ïr.

XN

ËP 2.1.7. ¹ÿGB n = m + k T^@ n ≥ m T. *9Ä}¹, ¨£ a:@¬£ m, n ∈ N ;c 60 #l n ≥ m T^@, n = m + k Ãç> ¹ÿø¶; ¤<a:@¬£ k ∈ N T

¤GB 4 +í<<Â6Ò.

&

ño_ Ä»{9$í`¦"îSX > æ¼ 6£§

m + k = m + ` =⇒ k = ` (8) õ

°ú s)a.

(10)

7

£x"î: 'Í P: "î]jH m + k ≥ m `¦ Ðs ÷&HX<, k \ 'aôÇ ) ±úZO

`

¦ 6 xôÇ. $, k = 0 s {© ¦, m + k ≥ m s &ñ , m + k⁺= (m + k)⁺> m + k ≥ ms)a.

Ñ ü

tP: "î]j_ >rF$í`¦Ðsl 0A # X = {` ∈ N : m + ` ≥ n}

¿º, n ∈ X sÙ¼Ð X 6= ∅ s. "f, &ño 2.1.4 \ _ # X H þ

jè "é¶è k ∈ X \¦tHX<, n = m + k e`¦ 7£x"î )a. s\¦0A

# m + k > n &ñ . ëß{9 k = 0 s m > n s &ñ\ #QFM

Ù¼Ð k 6= 0 s¦, (7) \ _ # k = s⁺(éß, s ∈ N) Ü¼Ð jþt Ãº e.

Õ

ªQ (m + s)⁺ = m + s⁺ = m + k > n\"f m + s ≥ n s¦,

"

f s ∈ X X< s < s⁺ = ksÙ¼Ð k X _ þjè "é¶èHX<\ ¸íH s

.

s

]j (8) `¦Ðs 7£x"îs =åQHX<, k \'aôÇ ) ±úZO`¦6 xôÇ.

$

m + 0 = m + ` =⇒ 0 = `

`

¦Ðs. ëß{9 m = m + ` X< ` 6= 0 s ` = s⁺s¦, m = m + ` = m + s⁺= (m + s)⁺≥ m⁺> m s

÷&#Q"f ¸íHs. s]j (8) s $íwnôÇ¦ &ñ ¦ m + k⁺= m + ` s

. ëß{9 ` = 0 s ~½ÓFK 7£x"îôÇ \ _ # k⁺ = 0sÙ¼Ð,

` 6= 0s¦ ` = t⁺s. ÕªQ

(m + k)⁺= m + k⁺= m + ` = m + t⁺= (m + t)⁺

\

"f m+k = m+t \¦%3¦, ) ±úZO &ñ\ _ # k = t x9 k⁺= t⁺= `

`

¦%3H. '

Ö

<<K 2.1.5. m + k ≤ m + ` ⇐⇒ k ≤ ` \¦ 7£x"î #.

(11)

2.2. &ñÃº 43

2.2. Ç a Ê Á

Ãº m, n ∈ N s n ≤ m s &ño 2.1.7 \ _ # m = n + k

Ãº k ∈ N s Ä»{9 > >rF HX<, sQôÇ k \¦ m − nÜ¼Ð æ¼.

s

]j, m < n âÄº\¸ m − n s >pw`¦t>FKÃº_ #30A\¦V,y9

¦ ôÇ. sQôÇ ³ð&³ m − n Ér ¿º Ãº m, n _ íH"f\ _>r Ù¼Ð, m − n @/\ íH"f© (m, n) `¦ Òqty ÷& m ≥ n sêøÍ ]jôÇ`¦ \OE¦

|9

½+Ë

N × N = {(m, n) : m, n ∈ N}

\

"f r lÐ . |9½+Ë N × N \ 6£§

(m, n) ∼ (m⁰, n⁰) ⇐⇒ m + n⁰= n + m⁰ õ

°ú s 'a>\¦ &ñ_ , ∼ Ér 1.3 ]X_ Ðl 2 \"f ÐH ü< °ú s N × N _ 1lxu'a>s¦

m ≥ k, n ≥ k =⇒ (m, n) ∼ (m − k, n − k) (9)

$íwn<Ê`¦ FK~½Ó ·ú Ãº e.

' Ö

<<K 2.2.1. "î]j (9) $íwn<Ê`¦Ð#.

s

]j (m, n) ∈ N × N `¦ "é¶èÐ tH N × N _ 1lxuÀÓ [(m, n)] `¦ ç

ßéßy [m, n] Ü¼Ð ³ðrôÇ. |9½+Ë N × N/∼ `¦ Z æ¼¦ Z _ "é¶è\¦ +ä

¬£ ÂÒÉr. 0A_'a> (9) \¦ &h6 x , |9½+Ë Z _ ¸H "é¶è\¦ &h {

©ôÇ Ãº n = 1, 2, . . . \ @/ # 6£§

[n, 0], [0, 0], [0, n] (10)

×

æ_ Ð ³ð&³½+É Ãº e6£§`¦ ~1> ·ú Ãº e. s]j |9½+Ë Z \ 6£§ [m, n] ≥ [k, `] ⇐⇒ m + ` ≥ n + k (11)

(12)

õ

°ú s'a>\¦ &ñ_ . ëß{9 (m, n) ∼ (m⁰, n⁰)s¦ (k, `) ∼ (k⁰, `⁰) s

&ño 2.1.7 \ _ #

m + ` ≥ n + k ⇐⇒ m⁰+ `⁰≥ n⁰+ k⁰ e

`¦ ·ú Ãº e. "f, &ñ_ (11) s ¸ú &ñ_÷&#Q e¦ sH íH"f'a

>

)a.

' Ö

<<K 2.2.2. e__ &ñÃºH (10)\ \P)a כ ×æ\ e`¦Ð#. ¢¸ôÇ,

&

ñ_ (11) s ¸ú &ñ_)a íH"f'a>e`¦Ð#.

XN

ËP 2.2.1. +ä¬£· כÖ Z ;c 60 #l :?T )ç· Â6Ò.

() »qq4wHÐ a, b ∈ Z ;c 60 #l a ≥ b T b ≥ a T.

() R#eU óm¬ª< Z q «»(Ûo· כÖS a}¹ ¤4T^@S ¤< L|604wHÐ

¿ ì

>.> . ¾6ÒU}¹, R#eU óm¬ª< Z q «»(Ûo· כÖS 7 }

¹ ¤4T^@S ¤< L|Ð4wHÐ¿ì>.> .

7

£x"î: ëß{9 m ≥ n s

[m, 0] ≥ [n, 0] ≥ [0, 0] ≥ [0, n] ≥ [0, m]

s

$íwn Ù¼Ð () 7£x"î)a.

s

]j, () \¦ 7£x"î l 0A # |9½+Ë A 0AÐ Ä»> &ñ ¦, ¸

H&ñÃº\¦ (10)\ eH+þAIÐ ³ðr # B = {k ∈ N : [k, 0] ∈ A}

¿º. ëß{9 B 6= ∅ s B 0AÐ Ä»>sÙ¼Ð 2£§&ño 2.1.6 \ _

# þj@/ "é¶è n `¦tHX<, [n, 0] s A _ þj@/ "é¶èe`¦ÐSX

½

+É Ãº e. ëß{9 B = ∅ s A _ ¸H "é¶è[þtÉr [0, k]_ g1JÐ ³ðr)a

. s M:, C = {k ∈ N : [0, k] ∈ A} _ þjè "é¶è\¦ ms ¿º [0, m]

s

A _ þj@/ "é¶èe`¦·ú Ãº e. |9½+Ë A AÐ Ä»> âÄº¸ ð

øÍts.

(13)

2.2. &ñÃº 45

' Ö

<<K 2.2.3. &ño 2.2.1 _ 7£x"î`¦Áºo #.

s

]j, &ñÃº s_ 8 l\¦6£§

[m, n] + [k, `] = [m + k, n + `] (12)

õ

°ú s &ñ_ . ÕªQ s íßs ¸ú &ñ_÷&#Q e¦ §¨8ZOgË:õ

½

+ËZOgË:`¦ëß7á¤ôÇ. :£¤y Z _ "é¶è [0, 0] Ér½Ó1px"é¶_ %i½+É`¦ôÇ. ¢¸ ô

Ç, &ñÃº [n, m] s [m, n] _ %i"é¶sHd`¦·ú Ãº e.

' Ö

<<K 2.2.4. |9½+Ë Z \ &ñ_)aíß (12) ¸ú &ñ_÷&#Q e¦, 2.3 ]X\

¸H (^1), (^2), (^3), (^4) $íwn<Ê`¦Ð#.

s

]j &ñÃº_ YL l\¦&ñ_ . |9½+Ë N × N _ "é¶è [m, n] õ [k, `]

\

@/ #

[m, n] · [k, `] = [mk + n`, m` + nk] (13) s

&ñ_ , 8 l_ âÄºü< ðøÍtÐ ]j@/Ð &ñ_÷&#Q e¦

½

+ËZOgË:õ §¨8ZOgË:s ëß7á¤Hd`¦·ú Ãº e. :£¤y, [1, 0] Ér YL l_ ½Ó 1

px"é¶s.

' Ö

<<K 2.2.5. íß (13) s ¸ú &ñ_÷&#Q e6£§`¦ Ð#. ¢¸ôÇ, |9½+Ë Z \ &ñ _

)a íß (12) ü< (13) \ @/ # 2.3 ]X\ ¸H (^5), (^6), (^8), (^9)

$í

wn<Ê`¦Ð#.

' Ö

<<K 2.2.6. ëß{9 [m, n]·[k, `] = [0, 0] s [m, n] = [0, 0] s [k, `] = [0, 0]

s

$íwn<Ê`¦Ð#.

'

Ö<<K 2.2.7. &ñÃº |9½+Ë Z \ YL l\'aôÇ %i"é¶s >rF t ·ú§6£§`¦Ð#.

' Ö

<<K 2.2.8. _ëß{9 P = {[n, 0] ∈ Z : n ∈ N, n 6= 0} s ¿º 2.3 ]X\ ¸

H (íH1)s $íwn<Ê`¦Ð#.

<ÊÃº f : N → Z \¦

f : n 7→ [n, 0] (14)

(14)

Ð &ñ_ f : N → Z éß<ÊÃºs¦ y Ãº m, n ∈ N \ @/

#

6£§

f (m + n) = f (n) + f (m) f (mn) = f (m)f (n) m ≥ n ⇐⇒ f (m) ≥ f (n) s

$íwn Ù¼Ð, 8 l, YL l x9 íH"f\'aôÇ ôÇ N Ér Z _ ÂÒìr|9½+Ë Ü

¼Ð Òqty½+É Ãº e. s]jÂÒ', (10) \ _ # ³ðr÷&H&ñÃº\¦yy n, 0, −nÜ¼Ð æ¼lÐ ôÇ.

2.3. Ë Âh Ê Á

s

]j, &ñÃº ^_ |9½+Ë Z `¦ y5px]j Ä»\v> ÷&¸2¤ 9¦

HX<, Õª \ y5px]j_ _p\¦ ìr"î > |9¦ Å#QH כ s ¼#o

. |9½+Ë F \ ¿º s½Óíß

(x, y) 7→ x + y, (x, y) 7→ x · y s

ÅÒ#Q4R"f 6£§\ \PôÇ $í|9 (^1) ∼ (^9) \¦ëß7á¤ s\¦=i

¦ ôÇ. ·ú¡Ü¼Ð x · y HÕªzª xy Ð æ¼l¸ ôÇ.

(^1) e__ a, b, c ∈ F \ @/ # a + (b + c) = (a + b) + c s.

(^2) 6£§$í|9

e

__ a ∈ F \ @/ # a + e = e + a = a

`

¦ëß7á¤ H "é¶è e ∈ F >rFôÇ.

0

A $í|9`¦ ëß7á¤ H "é¶è e⁰ ∈ F ¢¸ e e = e + e⁰ = e⁰s

. "f sQôÇ $í|9`¦ ëß7á¤ H "é¶èH µ1Ú\ \OHX<, s\¦·ú¡ Ü

¼Ð 0 s æ¼¦ 8 l_ ØûB4wHs ôÇ.

(15)

2.3. Ä»oÃº 47

(^3) y a ∈ F \ @/ # 6£§$í|9

a + x = x + a = 0

`

¦ëß7á¤ H "é¶è x ∈ F e.

ë

ß{9 0A $í|9`¦tH "é¶è y ∈ F ¢¸ e x = x + 0 = x + (a + y) = (x + a) + y = 0 + y = y

÷&#Q"f s $í|9`¦ tH "é¶ÉrÄ»{9 . "é¶è a ∈ F \ _ #

&

ñ÷&Hs "é¶è\¦−a æ¼¦, s\¦8 l\'aôÇ a _ *94wHs ôÇ.

¢

¸ôÇ b + (−a) Hçßéßy b − a Ð H.

(^4) e__ a, b ∈ F \ @/ # a + b = b + a s.

(^5) e__ a, b, c ∈ F \ @/ # a(bc) = (ab)c s.

(^6) 6£§$í|9 e

__ a ∈ F \ @/ # a · 1 = 1 · a = a

`

¦ëß7á¤ H 0 "é¶è 1 ∈ F s >rFôÇ.

(^7) y a ∈ F \ {0} \ @/ # 6£§$í|9 ax = xa = 1 `¦ëß7á¤ H "é¶

è x ∈ F >rFôÇ.

Ó ü

t:r (^6) _ "é¶è 1 ¸ Ä»{9 9, s :£¤&ñ "é¶è\¦ 1Ð æ¼¦ YL l_

ØûB4wHs ôÇ. (^7) _ "é¶è x ¸ (^3) _ âÄºü< ðøÍtÐ a \ _

# &ñ÷&HX<, s\¦ a⁻¹ <ÊÉr 1

as æ¼¦ YL l\'aôÇ a _ *9 4

w

Hs ôÇ.

(^8) e__ a, b ∈ F \ @/ # ab = ba s.

(^9) e__ a, b, c ∈ F \ @/ # a(b + c) = ab + ac s.

0

A \PôÇ $í|9 îrX< (^1) ÂÒ' (^4) tH 8 l\ 'aôÇ $í

|9

[þts¦, (^5) ÂÒ' (^8) tH YL l\'aôÇ $í|9[þte`¦·ú Ãº e

(16)

. /BNo (^9) H Óüt:r8 lü< YL l #QbG>'aº÷&#Q eH

H&h`¦·p. #Q" |9½+Ës ^ <ÊÉrçßéßy ´ú # 8 lü< YL

l &ñ_÷&¦ y5px]j Ä»\vH >pws.

|

ºM 1. &ñÃº ^_ |9½+Ë Z \ 6£§

a ∼_nb ⇐⇒ a − b H n_ CÃºs õ

°ú s &ñ_ sH 1lxu'a>)a. #l"f, n = 2, 3, 4, . . . s.

]

|9½+Ë Z/∼ⁿ`¦ Zⁿ= {[0], [1], . . . , [n − 1]}s æ¼¦, 6£§ [i] + [j] = [i + j], [i][j] = [ij]

õ °ú s íß`¦&ñ_ôÇ. \V[þt [þt#Q, Z⁵\"fH

[1] + [3] = [4], [3] + [4] = [7] = [2], [2][4] = [8] = [3]

õ

°ú s )a. s íßÉr (^7) `¦ ]jü@ôÇ ¸H ^_ $í|9[þt`¦ ëß7á¤ôÇ

. s íßs ]j (^7) `¦ëß7á¤ Ht y 4R Ðl êøÍ. '

Ö

<<K 2.3.1. _Ðl_{1 \}_¸H 'a> 1lxu'a>e`¦ Ðs¦, ¿º t íß s

¸ú &ñ_÷&#Q eÜ¼ 9, (^7) `¦]jü@ôÇ ¸H^_ $í|9[þt`¦ëß7á¤<Ê`¦Ð#.

s

íßs (^7) `¦ëß7á¤½+É n _ 9כ¹Øæìr¸| `¦¹1Ô.

s

]j ^ 0A\ íH"f\¦Òqty 9 HX<, s\¦[O"î l 0A # 6£§ õ

°ú s ªÃºH >h¥Æ`¦¸{9ôÇ. ^ F _ ÂÒìr|9½+Ë S \ @/ # |9

½ +Ë −S \¦

−S = {−a : a ∈ S}

Ð &ñ_ .

^

F \ q#Q et ·ú§ÉrÂÒìr|9½+Ë P >rF # 6£§ (íH1) a, b ∈ P =⇒ a + b, ab ∈ P ,

(íH2) F = P ∪ {0} ∪ (−P ),

(íH3) |9½+Ë P , {0} x9 −P H"fÐès

(17)

2.3. Ä»oÃº 49

õ

°ú Ér $í|9`¦ t s\¦ )Ö<"k=i ¦ P _ "é¶è\¦ è¡¬£ ôÇ.

í

H"f^ F _ ¿º "é¶è a, b ∈ F \ @/ #, a − b ∈ P s a b Ð Ç

¦ ´ú ¦ s\¦ a > b¢¸ H b < aÐ H. y &ñÃº a, b, c ∈ Z \ @/

# (^1) – (^6) õ (^8) – (^9) $íwn ¦, 0 `¦]jü@ôÇ Ãº

^

_ |9½+Ë`¦ P_Z⊆ Z ¿º (íH1) – (íH3)s $íwnôÇ.

' Ö

<<K 2.3.2. _Ä_»ô_Ç|9½+Ë⁽²⁾Ér íH"f^|¨cÃº \O6£§`¦Ð#.

' Ö

<<K 2.3.3. íH"f^\ &ñ_)aªÃº |9½+Ë P ÐÂÒ' 6£§

a ≤ b ⇐⇒ b − a ∈ P <ÊÉr a = b (15) õ

°ú s &ñ_ , sH íH"f'a>Hd`¦Ð#.

' Ö

<<K 2.3.4. íH"f^ F _ e__ "é¶è a, b, c ∈ F \ @/ # 6£§`¦Ð#.

() a ≥ b, a ≤ b =⇒ a = b.

() a ≤ b, b ≤ c =⇒ a ≤ c.

() a + b < a + c ⇐⇒ b < c.

() a > 0, b < c =⇒ ab < ac.

() a < 0, b < c =⇒ ab > ac.

() a²≥ 0, :£¤y 1 > 0.

() 0 < a < b =⇒ 0 < 1 b < 1

a. () a, b > 0 s a²< b²⇐⇒ a < b.

í

H"f^ F _ "é¶è a ∈ F _ }B60ìm´|a| \¦6£§

|a| =

(a, a ≥ 0,

−a, a < 0.

õ °ú s &ñ_ôÇ. sH ïrÃº\P`¦s6 x # z´Ãº\¦½¨$í½+É M:\ ×æ כ

¹ôÇ %i½+É`¦ôÇ.

XN

ËP 2.3.1. )Ö<"k=i F q»qq4wHÐ a, b, c ∈ F ;c 60 #l :?T )ç

· Â6Ò.

(2) |9½+Ë X ü< &h]XôÇ Ãº n = {0, 1, 2, . . . , n − 1} s\ éß<ÊÃº eÜ¼ X

\

¦Ä»ôÇ|9½+Ës ôÇ. Ä»ôÇ|9½+Ëõ ÁºôÇ|9½+Ë\ @/K"fH 3.4 ]X\"f [jy êr.

(18)

() |a| ≥ 0 T. ¡Â6Ò, |a| = 0 ⇐⇒ a = 0.

() |ab| = |a| |b|.

() b ≥ 0 T^@|a| ≤ b ⇐⇒ −b ≤ a ≤ b.

() ||a| − |b|| ≤ |a ± b| ≤ |a| + |b|.

() |a − c| ≤ |a − b| + |b − c|.

7

£x"î: (), () x9 () H0pxôÇ ¸HâÄº\¦ÐÐ 4R 4§Ü¼Ð +

~1> µ1ßn= Ãº e. s]j, () _ :£¤ÃºôÇ âÄºÐ"f −|a| ≤ a ≤ |a| e

`

¦·ú Ãº e¦ b −b \ @/ #¸ ðøÍtsÙ¼Ð

−(|a| + |b|) ≤ a ± b ≤ |a| + |b|

\

¦%3¦, () \¦r &h6 x

|a ± b| ≤ |a| + |b| (16) e

`¦·ú Ãº e. ÂÒ1pxd ||a| − |b|| ≤ |a ± b| H~½ÓFK 7£x"îôÇ (16) Ü¼ÐÂÒ '

~1> Ä»¸÷&¦, () H () \"f Ð :r. '

Ö<<K 2.3.5. &ño 2.3.1 _ (), (), () x9 () _ 'Í P: ÂÒ1pxd`¦ 7£x"î

#

.

s

]j, (^7) t $íwn ¸2¤ ‘Ãº’_ #30A\¦V,y¦ HX<, sH N

Sl\¦ l 0A # Ãº\¦&ñÃºÐSX© Hõ&ñõ q5pw . |9½+Ë Z × (Z \ {0}) \ 6£§

(a, b) ∼ (c, d) ⇐⇒ ad = cb (17) õ

°ú s'a>\¦&ñ_ , sH Z × (Z \ {0}) _ 1lxu'a>)a. |9½+Ë Z × (Z \ {0})/∼ `¦ Q æ¼¦, Q _ y "é¶è\¦ ¤P¬£ ÂÒÉr. #l

"

f¸, y Ä»oÃº\¦?/H 1lxuÀÓ [(a, b)] \¦Õªzª [a, b] Ð H. s]j,

8 lü< YL l\¦

[a, b] + [c, d] = [ad + cb, bd], [a, b] · [c, d] = [ac, bd] (18)

(19)

2.3. Ä»oÃº 51

&ñ_ôÇ. ëß{9 y a ∈ Z \ @/ # a^∗= [a, 1]s æ¼ 0^∗õ 1^∗Ér y

y 8 lü< YL l\ @/ôÇ ½Ó1px"é¶s)a.

' Ö

<<K 2.3.6. 'a> (17) s 1lxu'a>e`¦Ð#. &ñ_ (18) s ¸ú &ñ_÷&#Q e

6£§`¦Ð#. Ä»oÃº ^_ |9½+Ë Q ^e`¦Ð#.

|9

½+Ë P_Z (íH2), (íH3) `¦ ëß7á¤ Ù¼Ð, |9½+Ë Z × (Z \ {0}) _ ¸H

"

é

¶èH

{0} × (Z \ {0}), PZ× P_Z, P_Z× (−P_Z), (−P_Z) × P_Z, (−P_Z) × (−P_Z)

Ð ìr½+É)a. ÕªX<, e__ (a, b) ∈ Z × (Z \ {0}) \ @/ # (a, b) ∼ (−a, −b)sÙ¼Ð, e__ Ä»oÃºH6£§[j |9½+Ë

{0} × P_Z, P_Z× P_Z, (−P_Z) × P_Z

\

5Åq H "é¶è[þt`¦@/³ð"é¶Ü¼Ð H 1lxuÀÓ\ _ # &ñ)a. s]j

P_Q= {[a, b] : (a, b) ∈ P_Z× P_Z}

&ñ_ , (íH2), (íH3) `¦ ëß7á¤ôÇ. ¢¸ôÇ, P_Z (íH1) `¦ ëß7á¤ Ù¼

Ð, &ñ_ (18) \ _ # {©y PQ¸ (íH1) `¦ëß7á¤ôÇ. "f, Q H í

H"f^e`¦·ú Ãº e¦, (15) \ _ # íH"f'a> ÅÒ#Q.

'

Ö<<K 2.3.7. _e_{_}_{_} [a, b], [c, d] ∈ Q \ @/ # [a, b] ≥ [c, d] ⇐⇒ abd²≥ cdb² s

$íwn<Ê`¦Ð#.

<ÊÃº a 7→ a^∗= [a, 1] : Z → Q éß<ÊÃºeÉr "î . ¢¸ôÇ, 6

£

§$í|9[þt

(a + b)^∗= a^∗+ b^∗ (ab)^∗= a^∗b^∗ a ≥ b ⇐⇒ a^∗≥ b^∗ s

$íwn Ù¼Ð, 8 l, YL l x9 íH"f\'aôÇ ôÇ Z Ér Q _ ÂÒìr|9½+Ë Ü

¼Ð Òqty½+É Ãº e. s]jÂÒ', Ä»oÃº [a, b] \¦Õªzª a

b H.

(20)

e

__ íH"f^ F H8 lü< YL l\'aôÇ ½Ó1px"é¶0õ 1 `¦

. &ño 2.1.2 \¦&h6 x 6£§$í|9 γ(0) = 0

γ(n + 1) = γ(n⁺) = γ(n) + 1, n ∈ N

`

¦ëß7á¤ H<ÊÃº γ : N → F Ä»{9 > >rFôÇ. #l"f ýa_ 8

lH N \"f &ñ_)a8 ls¦, Äº_ 8 lH íH"f^ F _ 8 l

s. :£¤y, Äº_ 0 õ 1 Ér íH"f^ F _ 8 lü< YL l\'aôÇ ½Ó 1

px"é¶s. s]j,

γ(n + m) = γ(n) + γ(m), γ(nm) = γ(n)γ(m), m, n ∈ N (19) s

$íwn<Ê`¦ Ðs. $ m = 0 s {© . ëß{9 m ∈ N \ @/

#

$íwnôÇ

γ(n + m⁺) = γ((n + m)⁺) = γ(n + m) + 1

= γ(n) + γ(m) + 1 = γ(n) + γ(m⁺) s

Ù¼Ð, e__ m, n ∈ N \ @/ # (19) _ 'Í P: ds $íwnôÇ. ÑütP: d

%ir m = 0 s {© ¦, ) ±úZO &ñ\ _ # γ(nm⁺) = γ(nm + n) = γ(nm) + γ(n)

= γ(n)γ(m) + γ(n) = γ(n)[γ(m) + 1] = γ(n)γ(m⁺)

÷&#Q 7£x"î)a. ¢¸ôÇ, e__ íH"f^\"f 0 < 1 sÙ¼Ð 1 ∈ P^Fs.

ë

ß{9 γ(n) ∈ P_Fs γ(n⁺) = γ(n) + 1 ∈ P_F ÷&#Q 6£§

γ(n) ∈ PF, n = 1, 2, . . . (20) s

$íwn<Ê`¦·ú Ãº e. =åQÜ¼Ð γ : N → F éß<ÊÃºe`¦ Ðs.

s

\¦0A # n > m, γ(n) = γ(m) s &ñ . ÕªQ n = m + k k ∈ N \ {0} \¦¸ú`¦Ãº e¦,

γ(k) = [γ(m) + γ(k)] − γ(m) = γ(m + k) − γ(m) = γ(n) − γ(m) = 0

(21)

2.3. Ä»oÃº 53

X<, sH (20)\ ¸íHs. "f, γ : N → F H (19)ü< (20) `¦ ëß 7

á

¤ Héß<ÊÃºs.

s

]j, N ⊂ Z ⊂ Q e`¦%i¿º\ ¿º¦, <ÊÃº γ : N → F _ &ñ_%i`¦ Q

\

SX© . $, <ÊÃº γ : Z → F \¦

γ(n) = γ(n), γ(−n) = −γ(n), n ∈ N s

&ñ_ôÇ. ¿º <ÊÃº γ : N → F ü< γ : Z → F HÕª <ÊÃº°úכs N 0A\

"

f {9u Ù¼Ð, °ú Érl ñ\¦6 xK¸ Áº~½Ó . ðøÍtÐ γ : Q → F

\

¦6£§

γa b

=γ(a)

γ(b), (a, b) ∈ Z × (Z \ {0}) õ

°ú s &ñ_ôÇ. sXO> &ñ_)a <ÊÃº γ : Q → F ]X &ñ_÷&#Q e

¦, (19) ü< (20) `¦ëß7á¤ Héß<ÊÃºHdÉry ¶ú(R Ðl êøÍ.

XN

ËP 2.3.2. »qq )Ö<"k=i F ;c 60 #l :? )çHB () »qqr, s ∈ Q ;c 60 #l

γ(r + s) = γ(r) + γ(s), γ(rs) = γ(r)γ(s)

)ç· Â6Ò,

() γ(P_Q) = γ(Q) ∩ P^FT Ã

ç

> ¹ÿø¶; ¤< ·ÿÈÕ¬£ γ : Q → F ¤GB 4 +í<<Â6Ò.

'

Ö<<K 2.3.8. &ño 2.3.2 _ 7£x"î`¦Áºo #. ¢¸ôÇ F _ YL l\ 'aôÇ

½Ó1px"é¶`¦ 1Fs ¿º, e__ r ∈ Q \ @/ # γ(r) = r · 1^Fe`¦Ð#.

&

ño 2.3.2 H e__ íH"f^ F Ä»oÃº^ Q \¦í<Ê½+É ÷rm

¿

º íH"f^_ íßõ íH"f\¦ ½¨Z>½+É 9כ¹ \O6£§`¦´úK ïr. ·ú¡Ü¼Ð í

H"f^ F \ @/ # 7H_ HâÄº Õª îß\ Ä»oÃº^ eH כ Ü¼Ð, 7

£¤, Q ⊂ F כ Ü¼Ð çßÅÒôÇ.

(22)

XN

ËP 2.3.3. )Ö<"k=i F ;c 60 #l :?ª<ò6BVT.

() x > 0 T^@x > 1

n-> a:@¬£ n = 1, 2, . . . T +í<<Â6Ò.

() y > 0 T^@y < n-> a:@¬£ n = 1, 2, . . . T +í<<Â6Ò.

() · כÖ N (⊂ F ) ª< a}¹ ¤4 N.

() »qq x, y > 0 ;c 60 #l y < nx ¿ì> ¹ÿø¶; ¤< a:@¬£ n ∈ N T +

í

<<Â6Ò.

í

H"f^ F 0A "î]j_ 1lxu ¸| [þt`¦ëß7á¤ ÀW8K62Ã⁽³⁾)ç H

B`¦ëß7á¤ôÇ¦ ´úôÇ.

' Ö

<<K 2.3.9. &ño 2.3.3 `¦ 7£x"î #.

XN

ËP 2.3.4. ¤P¬£=i¤<ÀW8K62Ã )çHBÃç> ¹ÿø¶;Â6Ò.

7

£x"î: ëß{9 m, n ∈ P_Zs n

m < 2ns.

2.4. P LP L í >ç Ãâ í 5 ø Ê Á

s

]j, Ä»oÃºÐÂÒ' z´Ãº\¦½¨$í # Ð. Ä»oÃº[þt_ |9½+Ë α ⊂ Q

6£§$í|9[þt

(]X1) α 6= ∅, α 6= Q s,

(]X2) ëß{9 p ∈ α, q ∈ Q, q < p s q ∈ α s, (]X3) ëß{9 p ∈ α s p < r r ∈ α >rFôÇ

`

¦ëß7á¤ , s\¦62620> È⁽⁴⁾}B·ÿ <ÊÉrÕªzª }B·ÿs ôÇ. X<X< à

Ô]Xéß ^_ |9½+Ë`¦ R s æ¼¦, s |9½+Ë_ "é¶è, 7£¤ X<X<àÔ]Xéß

(3) Archimedes (287∼212 B.C.).ØÔvBjX<Û¼ $í|9`¦ëß7á¤ t ·ú§H íH"f^_ \V\¦

· ú

¦ z·ÉrsHÃÐ¦ëH³ [27], 2.5 ]X_ _þvëH]j 8∼10 <ÊÉr [15], 1©`¦ÃÐ¸

.

(4) Julius Wihelm Richard Dedekind (1831∼1916), 1lq{9 Ãº<Æ. G¨ottingen \"f <Æ 0

A\¦~ÃÎÉrÊê, 2[oy x9 1lq{9 ×æÂÒ_ Braunschweig 1px\"fÖ¸1lx %i.

(23)

2.4. X<X<àÔ]Xéßõ z´Ãº 55

`

¦FB¬£ ÂÒÉr. e__ r ∈ Q \ @/ # r^∗= {p ∈ Q : p < r}

É

r]Xéßs.

' Ö

<<K 2.4.1. e__ r ∈ Q \ @/ # r^∗ ]Xéße`¦Ð#.

|

ºM 1. |9½+Ë

α = {p ∈ Q : p ≤ 0} ∪ {p ∈ Q : 0 < p, p²< 2}

É

q²< 2 e`¦ÐSX½+É Ãº eÜ¼Ù¼Ð q ∈ α s. s]j (]X3) `¦Ð s

l 0A # p ∈ α \¦×þ . ëß{9 p ≤ 0 s p < 1 ∈ α sÙ¼Ð 0 < p, p²< 2 &ñ ¦, &ño 2.3.4 \¦ &h6 x # 1

n(2p + 1) < 2 − p²

Ãº n `¦¸ú. ÕªQ

p + 1

n

²

≤ p²+ 2 np + 1

n < 2

÷&#Q"f p + 1

n ∈ αs¦, α H]Xéße`¦·ú Ãº e.

s

]j, #Q" r ∈ Q \ @/ "f¸ α 6= r^∗e`¦ Ðs. s\¦ 0A # α = r^∗ Ä»oÃº r ∈ Q e¦ &ñ . ëß{9 r² < 2s 0A\"f

¶ ú

(R:rü< °ú s

r + 1

n

²

< 2 Ãº n ∈ M `¦¸ú`¦Ãº e. Õª

Q r +1

n ∈ αstëß r +1

n ∈ r/ ^∗s#Q"f ¸íHs. s\H r²> 2

&ñ . ÕªQ ðøÍt ~½ÓZOÜ¼Ð

r − 1

m

²

> 2 Ãº m ∈ N

`

¦¸ú`¦Ãº e. ÕªQ r − 1

m ∈ α/ stëß r − 1

m ∈ r^∗ ÷&#Q"f r

¸íHs. "f r²= 2 X<, sQôÇ Ä»oÃº >rF t ·ú§HHכ

`

¦·ú¦ e.

(24)

' Ö

<<K 2.4.2. ^ª_{_}_Ä_»o_Ã_{º r ∈ P}_Q_r² > 2s

r − 1

m

2

> 2 Ãº m ∈ N s >rF<Ê`¦Ð#.

]X

éß α \ @/ # α^c= Q \ α ¿º, 6£§

p ∈ α, q ∈ α^c=⇒ p < q, r ∈ α^c, r < s =⇒ s ∈ α^c s

$íwnôÇ. "f, ¿º |9½+Ë α, α^cH Q \¦Ãºf 0A_ &h[þtÐ Òqty

‘¢,aAá¤’õ ‘¸ÉrAá¤’Ü¼Ð ìr½+ÉôÇ. s M:, ¢,aAá¤ |9½+ËÉr (]X3) \ _

# þj@/°úכ`¦tt ·ú§Hכ Ü¼Ð çßÅÒôÇ.

s

]j, ¿º ]Xéß α, β \ @/ #

α ≤ β ⇐⇒ α ⊂ β Ü

¼Ð &ñ_ íH"f'a>\¦%3H. Óüt:r,#l"f¸ α ≤ β s"f α 6= β s

α < β H. ëß{9 α ⊂ β s m, p /∈ β p ∈ α >rFôÇ

. ëß{9 q ∈ β s p /∈ βÐÂÒ' q < p e`¦·ú Ãº e¦, "f q ∈ α s

. 7£¤, α ⊂ βs m β ⊂ α ÷&Ù¼Ð, ¿º ]Xéß α, β ÅÒ#Qt α ≤ β <ÊÉr α ≥ β $íwn<Ê`¦·ú Ãº e. "f, 6£§"î]j 7£x"î

÷

&%3.

XN

ËP 2.4.1. »qqFB¬£ α, β ∈ R ;c 60 #l :?

α > β, α = β, α < β

^ ï

B Â6Ò'å<K )ç· z, ¡Â6Ò ¨£ 'å<Kò6BS;c )ç· U óm¬¤<.

· ú

¡Ü¼Ð,

P_R= {α ∈ R : α > 0^∗} s

éH. &ño 2.4.1 Ér (íH2) x9 (íH3)s $íwn<Ê`¦´úK ïr.

(25)

XN

ËP 2.4.2. R#eU óm¬ª< · כÖA ⊂ R a}¹ ¤4T^@A ¤< ç¡Â6Ò Ã

ç

>.> .

7

£x"î: q#Q et ·ú§Ér |9½+Ë A ⊂ R 0AÐ Ä»>{9 M:, α =S{β ∈ A}

α ⊂ γ ( Q s. ëß{9 p ∈ α s p ∈ β β ∈ A >rFôÇ. ëß{9 q < ps q ∈ β ⊂ α sÙ¼Ð (]X2) 7£x"î)a. ëß{9 p < q q ∈ β \¦

¸ ú

Ü¼ q ∈ α sÙ¼Ð (]X3) %ir 7£x"î÷&¦, "f α ∈ R s.

s

Õ

ªQ r ∈ β s¦ r /∈ δsÙ¼Ð β δ s¦, &ño 2.4.1 \ _ # δ < β

©>e`¦´úK ïr. s

]j, íH"f|9½+Ë R \ íß`¦&ñ_½+É YVs. $ α, β ∈ R \ @/

#

α + β = {s + t ∈ Q : s ∈ α, t ∈ β} (21) s

&ñ_ôÇ. $, α + β ∈ R e`¦ÐsHX<, α + β 6= ∅ eÉr"î

. ëß{9 u /∈ α, v /∈ βs e__ s ∈ α, t ∈ β \ @/ # u + v > s + t

X<, sH e__ r ∈ α + β \ @/ # u + v > r êøÍ ´ús. "f u + v /∈ α + βs¦, α+β ( Q s. s]j, (]X2) ü< (]X3) `¦Ðsl 0A

#

p ∈ α+β . ÕªQ p = s+t (éß, s ∈ α, t ∈ β) s. ëß{9 q < p s

q − t < s, s ∈ α \"f q − t ∈ α $íwn ¦, q = (q − t) + t ∈ α + β s

Ù¼Ð (]X2) 7£x"î)a. ëß{9 s < r r ∈ α `¦¸úÜ¼ p < r + t s

¦, r + t ∈ α + β sÙ¼Ð (]X3) s 7£x"î÷&%3.

' Ö

<<K 2.4.3. íß (21) \ @/ # ½+ËZOgË: (^1) õ §¨8ZOgË: (^4) $íwn

<Ê`¦Ð#.

(26)

s

]j, 0^∗ ∈ R 8 l\ 'aôÇ ½Ó1px"é¶s Hd`¦ Ðs. s\¦ 0A

#

e__ α ∈ R \ @/ # α + 0^∗= α e`¦ Ðs )a. $ r ∈ α, s ∈ 0^∗s r + s < r sÙ¼Ð r + s ∈ α s¦, "f α + 0^∗ ⊂ α e`¦

· ú

Ãº e. ëß{9 p ∈ α s p < r r ∈ α \¦ ×þ½+É Ãº e. ÕªQ p − r ∈ 0^∗sÙ¼Ð p = r + (p − r) ∈ α + 0^∗s ÷&#Q, α ⊂ α + 0^∗e`¦·ú Ã

º e.

ë

ß{9 α > 0^∗s p ∈ α \ 0^∗`¦×þ½+É Ãº e. sH 0 ≤ p, p ∈ α \¦

>

p

w Ù¼Ð 0 ∈ α e`¦ ·ú Ãº e. %iÜ¼Ð, 0 ∈ α s p < 0 =⇒ p ∈ α

X<, sH 0^∗< α e`¦ >pwôÇ. "f, 6£§ 0^∗< α ⇐⇒ 0 ∈ α

`

¦%3H. sÐÂÒ', (íH1)_ 'Í P:"î]jHÐ :r.

ô

Ç¼#, 8 l\'aôÇ %i"é¶s e6£§`¦Ðsl 0A #, y α ∈ R \ @/

#

β = {p ∈ Q : r > p, −r /∈ α r ∈ Q s >rFôÇ}

¿º¦, β ∈ R õ α + β = 0^∗e`¦Ðs. $ s ∈ Q \ α \¦×þ ¦ p < −s p ∈ Q \¦×þ p ∈ β s. ¢¸ôÇ, q ∈ α \¦×þ

r > −q =⇒ −r < q =⇒ −r ∈ α s

Ù¼Ð −q /∈ βs. s]j, p ∈ β ¦ r > p, −r /∈ α r ∈ Q \¦¸ú

. ëß{9 q < p s r > q, −r /∈ αsÙ¼Ð q ∈ β s. ëß{9 s = p + r 2

¿º r > s, −r /∈ αsÙ¼Ð s ∈ β s¦, p < s sÙ¼Ð β ]Xéßes 7

£x"î÷&%3.

ë

ß{9 q ∈ α, p ∈ β s r > p, −r /∈ α r ∈ Q \¦ ¸ú`¦Ãº e.

Õ

ªQ q ∈ α, −r /∈ αÐÂÒ' q < −r `¦ %3H. "f, −(q + p) = (r − p) + (−r − q) > 0s¦ q + p < 0 `¦%3#Q"f, α + β ⊂ 0^∗es 7£x"î

÷

&%3. =åQÜ¼Ð 0^∗⊂ α + β e`¦Ðs. s\¦0A # s ∈ 0^∗\ @/ #

(27)

t = −s

2 > 0s ¦

A = {n ∈ Z : nt ∈ α}

¿º. ëß{9 A 0AÐ Ä»> m &ñ . ÕªQ &ño 2.3.4

\

¦ &h6 x # e__ q ∈ Q \ @/ # q < mt m ∈ N `¦ ¸ú`¦Ãº e

²DG, q < nts¦, nt ∈ α sÙ¼Ð q ∈ α s¦, "f α = Q ÷&#Q

"

f ¸íHs. ÕªQÙ¼Ð A H 0AÐ Ä»>s 9, &ño 2.1.4 \ _ # þj

@

/ "é¶è n0 ∈ A \¦ . 7£¤, n0t ∈ αs¦ (n0+ 1)t /∈ αs. s]j r = s−n0t = −(n0+2)t ¿º. ÕªQ −(n₀+1)t > rõ (n⁰+1)t /∈ α

ÐÂÒ' r ∈ β e`¦·ú>)a. "f, s = n0t + r ∈ α + βs. ²DG, α + β = 0^∗es 7£x"î÷&%3Ü¼Ù¼Ð β H α_ %i"é¶s¦, ·ú¡Ü¼ÐH−αÐ æ

¼lÐ ôÇ.

s

]j, z´Ãº_ YL l\¦&ñ_ HX<, $ ªÃº α, β ∈ PR\ @/ #

&

ñ_ . e__ α, β ∈ PR\ @/ #

αβ = {p ∈ Q : p ≤ rs r ∈ α ∩ P_Q, s ∈ β ∩ P_Q >rFôÇ}

Ð &ñ_ôÇ.

' Ö

<<K 2.4.4. e__ α, β ∈ PR\ @/ #

αβ = 0^∗∪ {rs : 0 ≤ r ∈ α, 0 ≤ s ∈ β}

e

`¦Ð#.

$ αβ ]Xéße`¦ Ðs. Äº 0 ∈ α, 0 ∈ β sÙ¼Ð 0 ∈ αβ s

. ¢¸ôÇ, u /∈ α, v /∈ βs uv /∈ αβs. z´]jÐ e__ s ∈ α, t ∈ β

\

@/ # s < u, t < v sÙ¼Ð st < uv s¦, "f (]X1) s 7£x"î)a.

(]X2) ü< (]X3) ¢¸ôÇ &ñ_\ _ # "î . "f αβ ∈ R s¦, :£¤ y

(íH1)_ ÑütP: "î]j 7£x"î÷&%3. s]j, y α, β ∈ R \ @/ # 6£§

(28)

αβ =











0^∗, α = 0^∗ <ÊÉr β = 0^∗

−(−α)β, α ∈ −P_R, β ∈ P_R

−α(−β), α ∈ P_R, β ∈ −P_R (−α)(−β), α ∈ −P_R, β ∈ −P_R õ

°ú s &ñ_ôÇ.

' Ö

<<K 2.4.5. |9½+Ë R \ &ñ_)a8 lü< YL l\ @/ # (^5), (^8), (^9)

$íwn<Ê`¦Ð#. [¸¹¡§´ú: $ PR_ "é¶è[þt\ @/ # 7£x"îôÇ.]

'

Ö<<K 2.4.6. _y_z´Ãº α ∈ R \ @/ # α1^∗= 1^∗α = α e`¦Ð#. ¢¸ôÇ, 0^∗< 1^∗e`¦Ð#.

s

]j, YL l\ 'aôÇ %i"é¶_ >rF\¦ Ðs R s íH"f^e`¦ 7£x"î

Hõ&ñs ¸¿º =åQèß. y α ∈ PR\ @/ # γ = 0^∗∪ {0} ∪ {q ∈ P_Q: r > q, 1

r ∈ α / r ∈ P_Q s >rFôÇ} (22)

&ñ_ôÇ.

' Ö

<<K 2.4.7. (22)\"f &ñ_)aγ ]Xéße`¦Ðs¦, αγ = 1^∗e`¦Ð#.

s

]j, y α ∈ −PR\ @/ # α

− 1

−α

= −α

1

−α

= (−α)

1

−α

= 1 s

Ù¼Ð, α 6= 0 α ∈ R H YL l\'aôÇ %i"é¶`¦.

:r&hÜ¼Ð, s ]X\"f ½¨$íôÇ |9½+Ë R Ér&ño 2.4.2 \¦ëß7á¤ H íH

"

f^e`¦ 7£x"î %i. t}Ü¼Ð, e__ r, s ∈ Q \ @/ # 6£§$í

|9

[þt

r = s ⇐⇒ r^∗= s^∗ (r + s)^∗= r^∗+ s^∗

(rs)^∗= r^∗s^∗ r ∈ P_Q⇐⇒ r^∗∈ P_R

(23)

s

$íwn<Ê`¦ÐsHX<, sH

r 7→ r^∗: Q → R

(29)

2.5. ïrÃº\Põ z´Ãº 61

s

íßõ íH"f\¦Ð>r Héß<ÊÃºe`¦´úôÇ.

2.5. ók Ê Áá ~ ø Ê Á

Ãº|9½+Ë N \"f |9½+Ë X Ð H<ÊÃº x : N → X \¦ X_ ¬£~xs

ÂÒÉr. íH"f^ F _ Ãº\P x : N → F ü< a ∈ F ÅÒ#Q4R e`¦M:, e

__ e ∈ PF\ @/ # 6£§$í|9

i ≥ N =⇒ |x(i) − a| < e

`

¦ëß7á¤ HÃº N ∈ N s >rF , x a ∈ F Ð ¬£&9ôÇ¦ ´ú ô

Ç. ¢¸ôÇ, e__ e ∈ PF\ @/ # 6£§$í|9 i, j ≥ N =⇒ |x(i) − x(j)| < e

`

¦ ëß7á¤ H Ãº N s >rF , x \¦ØS⁽⁵⁾¬£~xs ÂÒÉr. :£¤ y

, Ä»oÃº_ ïrÃº\P`¦ M(è<~xs ÂÒÉr. e__ Ä»oÃº r ∈ Q \

@ / #

r^∗(i) = r, i ∈ N s

\

@/ # 6£§$í|9

|x(i)| ≤ M, i ∈ N

`

¦ëß7á¤ H M ∈ F eÜ¼, sH¤4¬£~xs ôÇ.

' Ö

<<K 2.5.1. ^íH"f^ F _ Ãº\P x : N → F Ä»>{9 9כ¹Øæìr¸| Ér |9½+Ë {x(i) ∈ X : i ∈ N} s 0AÐ Ä»>s¦ 1lxr\ AÐ Ä»>e`¦Ð#.

(5) Augustin Louis Cauchy (1789∼1857), áÔ|½ÓÛ¼ Ãº<Æ. "é¶A, Ð3lq/BN<Æ`¦/BN

9 %iÜ¼, ´Ecole Polytechnique_ e¦Û¼ 1pxs Ý¶ # Ãº<Æ`¦ > ÷&%3.

1816 ¸ÂÒ' ´Ecole Polytechnique_ §ÃºÐ eÜ¼"f Ãº´ú§Ér 7HëHõ $`¦z Ü

%i. Õª\'aôÇ lÐ [9] 1pxs e.

(30)

XN

ËP 2.5.1. )Ö<"k=i F q ¬£~xx : N → F ¬£&9 ^@ ØS¬£~xT.

¡Â6Ò»qq ØS¬£~xª<¤4T.

7

£x"î: x : N → F a ∈ F Ð Ãº§4 ¦,

i ≥ N =⇒ |x(i) − a| < e 2

Ãº N `¦¸ú. ÕªQ, e__ i, j ≥ N \ @/ #

|x(i) − x(j)| ≤ |x(i) − a| + |x(j) − a| < e 2 +e

2 = e

÷&#Q, x HïrÃº\Ps. s]j x ïrÃº\Ps &ñ ¦, i, j ≥ N =⇒ |x(i) − x(j)| < 1

Ãº N ∈ N `¦ ¸ú. ÕªQ, e__ Ãº i ≥ N \ @/ #

|x(i) − x(N )| < 1sÙ¼Ð, |x(i)| ≤ |x(N)| + 1 e`¦·ú Ãº e. s]j, M = sup{|x(0)|, |x(1)|, . . . , |x(N − 1)|, |x(N )| + 1}

s

¿º, e__ i ∈ N \ @/ # |x(i)| ≤ M s.

¿

º l:r\P α, β : N → Q ÅÒ#Q4R e¦ . e__ Ä»oÃº e > 0\ @/ #

i ≥ N =⇒ |α(i) − β(i)| < e s

i ≥ N1=⇒ |α(i) − β(i)| < e

2, i ≥ N2=⇒ |β(i) − γ(i)| < e 2

Ãº N1, N₂\¦ ¸ú`¦ Ãº e. "f N = sup{N1, N₂} ¿º, e

__ i ≥ N \ @/ #

|α(i) − γ(i)| ≤ |α(i) − β(i)| + |β(i) − γ(i)| < e 2 +e

2 = e

(31)

2.5. ïrÃº\Põ z´Ãº 63

s

Ù¼Ð α ∼ γ e`¦·ú Ãº e. "f, ~½ÓFK&ñ_ôÇ'a> ∼ Hl:r\P

^_ |9½+Ë F _ 1lxu'a> )a. s]j, F/∼ `¦ R Ð ³ðr ¦, s ]

|9½+Ë_ "é¶è[þt`¦ z´Ãº ÂÒÉr.

¿

º z´Ãº [α], [β] ∈ R \ @/ #, 6£§$í|9 i ≥ N =⇒ α(i) − β(i) > d

`

¦ëß7á¤ HÄ»oÃº d > 0 ü< Ãº N s e`¦M:,

[α] > [β] (24) s

&ñ_ .

' Ö

<<K 2.5.2. &ñ_ (24) ¸ú &ñ_÷&#Q e6£§`¦Ð#.

XN

ËP 2.5.2. »qqFB¬£ [α], [β] ∈ R ;c 60 #l :? [α] > [β], [α] = [β], [α] < [β]

^ ï

B Â6Ò'å<K )ç· z, ¡Â6Ò ¨£ 'å<Kò6BS;c )ç· U óm¬¤<.

7

£x"î: e__ Ä»oÃº e > 0 \ @/ # 6£§$í|9

i ≥ Ne =⇒ |α(i) − α(Ne)| < e, |β(i) − β(Ne)| < e s

$íwn H þjè Ãº Ne\¦ ¸ú. s Ãº NeH Óüt:r Ä»oÃº e > 0\ _ # ÅÒ#Q. ÕªQ y i ≥ N^e\ @/ #

α(N_e) − β(N_e) − 2e < α(i) − β(i) < α(N_e) − β(N_e) + 2e (25)

$íwnôÇ. s]j,

de= α(Ne) − β(Ne) − 2e, d⁰_e= α(Ne) − β(Ne) + 2e

¿º.

2002[ Í 1 È ÑM

V

 2 *  × Ê Á  + W N 

2002[ Í 1 È ÑM 



·

 כ ÖÜ ï ¬ £P $ ã <P 

~ ç

¡q  ó i ; < K 2  ç ¡

"

k] ï >6 0 È Ñz  ¬ £ È ÑÜ ï 4

 û B0 ² ?

2002[ Í 3 i J 3G  B 4 S  2002[ Í 7 i J 8G  B ¬ £+ ä  33 ¶ ; ∼78 ¶ ;

2.1.  ¥ o >Ê Á

2.2. Ç a Ê Á

2.3. Ë Âh Ê Á

2.4. P LP L í  >ç Ãâ  í 5 ø   Ê Á

2.5.  ók Ê Áá ~ ø   Ê Á

2002[ Í 1 È ÑM

2 * × Ê Á + W N

2002[ Í 1 È ÑM

כ ÖÜ ï ¬ £P $ ã <P

¡q ó i ; < K 2 ç ¡

k] ï >6 0 È Ñz ¬ £ È ÑÜ ï 4

û B0 ² ?

2002[ Í 3 i J 3G B 4 S 2002[ Í 7 i J 8G B ¬ £+ ä 33 ¶ ; ∼78 ¶ ;

2.1. ¥ o >Ê Á

2.2. Ç a Ê Á

2.3. Ë Âh Ê Á

2.4. P LP L í >ç Ãâ í 5 ø Ê Á

2.5. ók Ê Áá ~ ø Ê Á