Giáo trình đại số hiện đại

✶

▼Ö❈ ▲Ö❈
❚r❛♥❣

▼ð ✣➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ữỡ ỡ ữủ ỵ tt t ❤đ♣

✺
✾

§✶✳ ❚➟♣ ❤đ♣ ✈➔ ❝→❝ ♣❤➨♣ t♦→♥ tr➯♥ t➟♣ ❤đ♣















Đ

Đ

Đ ủ tữỡ ữỡ

Đ ồ tữỡ ữỡ













t

ữỡ õ





Đ ♥❣❤➽❛ ✈➔ ✈➼ ❞ư ✈➲ ♥❤â♠

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✳ ✷✺

§✷✳ ◆❤â♠ ❝♦♥✱ ✣à♥❤ ỵ r























Đ õ t
Đ ỗ õ ✳

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§✺✳ P❤↕♠ trị ✈➔ ❤➔♠ tû

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Đ õ ỳ s
t

ữỡ trữớ tự



Đ ử

Đ ỗ

Đ ❤♦→♥ ✳

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✳ ✾✹

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§✹✳ ❱➔♥❤ ❝→❝ ♣❤➙♥ t❤ù❝
§✺✳ ❱➔♥❤ ✤❛ tự
Đ ò
t

ữỡ ổ



Đ ử

Đ ỗ

Đ ờ t trỹ t

Đ ủ t

Đ t ỡ

Đ ợ

t➟♣ ✳

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✶✺✸

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ữỡ ổ tr

Đ ổ ở
✳ ✳ ✳
§✷✳ ▼ð rë♥❣ ❝èt ②➳✉ ✈➔ ❜❛♦ ♥ë✐
§✸✳ ▼ỉ✤✉♥ ①↕ ↔♥❤ ✳ ✳ ✳ ✳
§✹✳ ▼ỉ✤✉♥ ◆♦❡t❤❡r ✳ ✳ ✳ ✳
§✺✳ ▼ỉ✤✉♥ ❆rt✐♥
✳ ✳ ✳ ✳
§✻✳ P❤➙♥ t➼❝❤ ♠æ✤✉♥ ♥ë✐ ①↕ ✳
❇➔✐ t➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳
❚r❛ ❝ù✉ tø ❦❤♦→ ✳ ✳ ✳ ✳ ✳

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①↕
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✶✺✼
✶✺✼
✶✻✺
✶✼✷
✶✽✵
✶✽✼
✶✾✺
✷✵✵
✷✵✺
✷✵✼


▼ð ✤➛✉
❈â t❤➸ ♥â✐ r➡♥❣ ♠å✐ ♥❣➔♥❤ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐ ♥❣➔② ♥❛② tr♦♥❣ q✉→ tr➻♥❤
♣❤→t tr✐➸♥ ✤➲✉ ❝➛♥ tỵ✐ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ✈➔ t➜t ♥❤✐➯♥ ❝↔ ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t
s➙✉ s➢❝ ✈➲ ❝→❝ ❝➜✉ tró❝ ♥➔②✳ ✣✐➲✉ ♥➔② ❝ơ♥❣ ❞➵ ❤✐➸✉✱ ✈➻ t❛ ❜✐➳t r➡♥❣ ❤❛✐ ✤➦❝
tr÷♥❣ ❝ì ❜↔♥ ♥❤➜t ❝õ❛ t♦→♥ ❤å❝ ❧➔ t➼♥❤ trø✉ t÷đ♥❣ ✈➔ t➼♥❤ tê♥❣ q✉→t✱ ♠➔
❤❛✐ ✤➦❝ t➼♥❤ ♥➔② ❧↕✐ ❜✐➸✉ ❤✐➺♥ ♠ët ❝→❝❤ rã r➔♥❣ ♥❤➜t tr♦♥❣ ✤↕✐ sè✳ ✣➣ ❝â
r➜t ♥❤✐➲✉ s→❝❤ ✈➲ ✤↕✐ sè ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❱✐➺t tứ t ữợ
ữủ t ❱✐➺t ◆❛♠✱ tr♦♥❣ sè ✤â ❝â ♥❤✐➲✉ q✉②➸♥ ✤➣ trð t❤➔♥❤
❦✐♥❤ ✤✐➸♥ ✈➔ ✤÷đ❝ sû ❞ư♥❣ ❧➔♠ ❣✐→♦ tr➻♥❤ ❣✐↔♥❣ ❞↕②✱ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤
✈✐➯♥ ❤å❝ t♦→♥ tr➯♥ ❦❤➢♣ t❤➳ ❣✐ỵ✐✳ ❱➻ ✈➟②✱ ✈✐➳t ♠ët ❣✐→♦ tr➻♥❤ ♠ỵ✐ ✈➲ ✤↕✐ sè ❧➔
♠ët ✈✐➺❝ ❧➔♠ r➜t ❦❤â ❦❤➠♥✱ ♥❤➜t ❧➔ ❦❤✐ t→❝ ❣✐↔ ❦❤æ♥❣ ♠✉è♥ r➟♣ ❦❤✉æ♥ ❤❛②
s❛♦ ❝❤➨♣ ❧↕✐ tø♥❣ ♣❤➛♥ ❝→❝ ❣✐→♦ tr➻♥❤ ✤➣ ❝â✳ ❈✉è♥ s→❝❤ ♥➔② ✤÷đ❝ ✈✐➳t ❞ü❛
tr➯♥ ❝→❝ ❜➔✐ ❣✐↔♥❣ ✈➲ ✤↕✐ sè ❝õ❛ t→❝ ❣✐↔ tr♦♥❣ ✈á♥❣ ✶✵ ♥➠♠ trð ❧↕✐ ✤➙② ❝❤♦
❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ t↕✐ ❱✐➺♥ ❚♦→♥ ❤å❝ ởt số trữớ
ồ tr ữợ ụ ữ ❜➔✐ ❣✐↔♥❣ tr♦♥❣ ✹ ♥➠♠ ❣➛♥ ✤➙② ❝❤♦ ❝→❝
❧ỵ♣ ❝û ♥❤➙♥ t➔✐ ♥➠♥❣ t❤✉ë❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ồ

ố ở õ ữủ t ữợ tợ ử t
ử t t ố ữ ồ ❣✐→♦ tr➻♥❤ ✈➲ ✤↕✐ sè✱ ❧➔ ♥❤➡♠ ❝✉♥❣
❝➜♣ ❝→❝ ❝➜✉ tró❝ ✤↕✐ sè ❝ì ❜↔♥ ♥❤➜t ♠➔ ❦❤ỉ♥❣ ✤á✐ ❤ä✐ ♥❣÷í✐ ✤å❝ ♣❤↔✐ ❝â
❜➜t ❝ù ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ số trữợ õ trứ ởt út
t t♦→♥ ❤å❝✳
▼ö❝ t✐➯✉ t❤ù ❤❛✐ ❝õ❛ ❝✉è♥ s→❝❤ ❧➔ tr➻♥❤ trú
số ữợ ởt ổ ♥❣ú tê♥❣ q✉→t✱ t❤è♥❣ ♥❤➜t ✈ỵ✐ sü ❝❤ó trå♥❣ ♥❤✐➲✉ ❤ì♥
❝→❝ t➼♥❤ ♣❤ê ❞ư♥❣ ❝õ❛ ❝→❝ ❦❤→✐ ♥✐➺♠✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ t→❝ ❣✐↔ ♠✉è♥ ♥❣÷í✐
✤å❝ ♥❤➟♥ t❤➜② ❝→❝ ♠è✐ q✉❛♥ ❤➺ q✉❛ ❧↕✐ ❣✐ú❛ ❝→❝ ❦❤→✐ ♥✐➺♠✱ ❝➜✉ tró❝ ✤↕✐ sè
✸


✹

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

❦❤→❝ ♥❤❛✉ ✈➔ ❦❤✉②➳♥ ❦❤➼❝❤ ❝❤♦ ♥❤ú♥❣ t÷ ❞✉② tê♥❣ q✉→t✱ trø✉ t÷đ♥❣ ❤ì♥
♥ú❛✳
❉♦ ✤â✱ ❣✐→♦ tr➻♥❤ ♥➔② ✤÷đ❝ ✈✐➳t t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ✤✐ tø trứ tữủ
ử t ởt tr ợ t ố s số trữợ ũ
♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❝❤♦ ♣❤➨♣ t❛ ❝â ♠ët ❝→❝❤ ♥❤➻♥ tê♥❣ t❤➸ ❤ì♥✱ rót ♥❣➢♥
✤→♥❣ ❦➸ ❝→❝❤ tr➻♥❤ ❜➔② ✈➻ ❞➵ ❞➔♥❣ ✤÷❛ ❝→❝ ❝➜✉ tró❝ ❦❤→❝ ♥❤❛✉ ✈➔♦ tr♦♥❣
♠ët ú ữớ ồ q ợ ữỡ ♣❤→♣ t÷ ❞✉② ❤➻♥❤
t❤ù❝ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ q✉❛♥ trå♥❣ ♥❤➜t tr♦♥❣ ✤↕✐ sè✳ ❚✉② ♥❤✐➯♥ ✤➸ ❣✐↔♠ ❜ỵt
t➼♥❤ ❤➻♥❤ t❤ù❝✱ s❛✉ ♠é✐ ❦❤→✐ ♥✐➺♠ trø✉ t÷đ♥❣ ❝❤ó♥❣ tỉ✐ ❝è ❣➢♥❣ ✤÷❛ r❛
♥❤✐➲✉ ✈➼ ❞ư ❦❤→❝ ♥❤❛✉ ♥❤➡♠ ❣✐ó♣ ❝❤♦ ♥❣÷í✐ ✤å❝ ❞➵ ❤➻♥❤ ❞✉♥❣ ✈➔ t✐➳♣ ♥❤➟♥
✤÷đ❝ ❦❤→✐ ♥✐➺♠ ♥➔②✳
❙→❝❤ ỗ ữỡ ữỡ tr tt ỵ tt t
ủ q tố t ỵ t ữỡ
t t r ữỡ ỵ tt õ ú tổ ọ q✉❛ ♥❤ú♥❣ ❝➜✉

tró❝ ♥û❛ ♥❤â♠✱ t✐➲♥ ♥❤â♠ ♠➔ ✤✐ ♥❣❛② õ ú tổ
ụ ọ q ỵ t❤✉②➳t ♥❤â♠ ❤ú✉ ❤↕♥ ♠➔ ❞➔♥❤ tr➻♥❤ ❜➔② ❦ÿ ❤ì♥ ✈➲
❝➜✉ tró❝ ♥❤â♠ ❆❜❡❧ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤→✐ ♥✐➺♠ ♣❤↕♠ trị ✈➔ ❤➔♠ tû ❝ơ♥❣
✤÷đ❝ ✤÷❛ ✈➔♦ ❝❤÷ì♥❣ ♥➔② ♥❤➡♠ ♣❤ö❝ ✈ö ♥❣❛② ❝❤♦ ✈✐➺❝ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ❦❤→✐
♥✐➺♠ q✉❛♥ trå♥❣ ♠❛♥❣ t➼♥❤ ♣❤ê ❞ö♥❣ ❝õ❛ ✤↕✐ sè tr♦♥❣ s✉èt tr ởt
t q r ữỡ ỵ tt õ ởt ú ỵ tr
ởt t ỏ ọ sỹ tỗ t tỷ ỡ ✈à✱ ✤➙② ❝ô♥❣ ❧➔ ✤✐➲✉
♠➔ ♥❤✐➲✉ ❣✐→♦ tr➻♥❤ ✤↕✐ sè ổ ỏ ọ ỵ t
❧➔ ✈➻ ❣✐→♦ tr➻♥❤ ✤÷đ❝ ✈✐➳t t❤✐➯♥ ♥❤✐➲✉ ❤ì♥ ✈➲ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✳ ❈❤÷ì♥❣ ✹
tr➻♥❤ ❜➔② ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➔ ỡ ừ ỵ tt ổ
trú q✉❛♥ trå♥❣ ♥❤➜t ❝õ❛ ✤↕✐ sè✳ ❍❛✐ ❤➔♠ tû q✉❛♥ trồ t ừ ỵ tt
ổ tỷ t❡♥ ①ì ❝ơ♥❣ ♥❤÷ t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ✤➛✉ t✐➯♥ ❝õ❛
❝❤ó♥❣ ❝ơ♥❣ ✤÷đ❝ ①➨t ✤➳♥ tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳ ❈❤÷ì♥❣ ❝✉è✐ ❝ị♥❣ ❞➔♥❤ ❝❤♦
✈✐➺❝ tr➻♥❤ ❜➔② ❝➜✉ tró❝ ♠ët sè ❧ỵ♣ ♠ỉ✤✉♥ ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ ♥❤÷ ♠ỉ✤✉♥
♥ë✐ ①↕✱ ♠ỉ✤✉♥ ①↕ ↔♥❤✱ ♠ỉ✤✉♥ ◆♦❡t❤❡r ✈➔ ❆rt✐♥ tr➯♥ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✳ ◆❤÷
✈➟②✱ ❤❛✐ ❝❤÷ì♥❣ ❝✉è✐ ❝õ❛ ❣✐→♦ tr➻♥❤ ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ ♠ët sü ❝❤✉➞♥ ❜à ❦✐➳♥
t❤ù❝ ❦❤ð✐ ✤➛✉ ❝❤♦ ♥❤ú♥❣ ồ õ ỵ t tử s ự
q trồ ừ số ữ ỵ tt ổ tr t ủ
số ỗ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✳


▼ð ✤➛✉

✺

❈✉è✐ ♠é✐ ❝❤÷ì♥❣ ❝õ❛ ❝✉è♥ s→❝❤ ✤➲✉ ❝â ♣❤➛♥ ❜➔✐ t➟♣ ✤÷đ❝ ❝❤å♥ ❧å❝✳
❈→❝ ❜➔✐ t➟♣ ♥➔② ❦❤ỉ♥❣ ❝❤➾ ✤➸ ♥❣÷í✐ ✤å❝ ❣✐↔✐ ♥❤➡♠ tü ❦✐➸♠ tr❛ sü t✐➳♣ t❤✉
♥❤ú♥❣ ✤✐➲✉ ✤➣ ❤å❝✱ ♠➔ ♥❤✐➲✉ ❜➔✐ t➟♣ ❧➔ ♥❤ú♥❣ ❜ê s✉♥❣ ❤❛② ♠ð rë♥❣ ❦✐➳♥
t❤ù❝ ❝❤÷❛ ❝â tr♦♥❣ s→❝❤✳ ❱➻ ✈➟②✱ s➩ t❤ü❝ sü ❝â ➼❝❤ ♥➳✉ ♥❣÷í✐ ✤å❝ ữủ
t

ố s ữủ t r ợ ♠ư❝ ✤➼❝❤ ❝â t❤➸ ❞ị♥❣ ❧➔♠ ❣✐→♦ tr➻♥❤
✤↕✐ sè ❝❤♦ ❝❤♦ ❝→❝ ❧ỵ♣ ❝❛♦ ❤å❝ ❤♦➦❝ ❞ị♥❣ ❧➔♠ s→❝❤ t❤❛♠ ỳ
s ồ t ỵ t❤✉②➳t ✈➔ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤✳ ❚✉② ♥❤✐➯♥✱
✈➻ ❝→❝ ❦❤→✐ ♥✐➺♠ ✤➲✉ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tø ✤➛✉✱ ♥➯♥ ♥â ❝ơ♥❣ ❝â t❤➸ ❜ê ➼❝❤
❝❤♦ t➜t ❝↔ ♥❤ú♥❣ ❛✐ ♠✉è♥ ❤å❝ t❤➯♠ ✈➲ ✤↕✐ sè✳
❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❣✐ó♣ ❝❤♦ ✤å❝ ❣✐↔ ♥❤➟♥ ✤÷đ❝ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➲ ✤↕✐ sè
✤↕✐ ❝÷ì♥❣ ❜➡♥❣ ♠ët ♥❣æ♥ ♥❣ú ❤✐➺♥ ✤↕✐ tr♦♥❣ ♠ët ❝✉è♥ s→❝❤ ♥❤ä ❧➔ ♠ët ✈✐➺❝
❧➔♠ ❦❤â tr→♥❤ ❦❤ä✐ ❝â ♥❤✐➲✉ t❤✐➳✉ sât✳ ❱➻ ✈➟②✱ t→❝ ❣✐↔ ♠♦♥❣ ♠✉è♥ ♥❤➟♥
✤÷đ❝ ♥❤ú♥❣ ♥❤➟♥ ①➨t✱ ❣â♣ ỵ ừ ỗ ồ ỳ t❤✐➳✉
sât ❝õ❛ ❝✉è♥ s→❝❤ ♥➔②✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ P●❙✳ ❚❙❑❍✳ ▲➯ ❚✉➜♥ ❍♦❛ ✤➣ ✤å❝ ❦ÿ
t♦➔♥ ❜ë t õ õ ỵ qỵ ✤➸ ❝✉è♥ s→❝❤ ✤÷đ❝ tèt
❤ì♥✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ●❙✳ ❱❙✳ ◆❣✉②➵♥ ❱➠♥ ✣↕♦ ✤➣ q✉❛♥ t➙♠
✤➳♥ ❜ë s→❝❤ ❝❛♦ ❤å❝ ❝õ❛ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ❝→♠ ì♥ ❍ë✐ ỗ ồ ỹ
t ồ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✤➣ ❣✐ó♣ ✤ï ✤➸ ❝✉è♥ s→❝❤
✤÷đ❝ ①✉➜t ❜↔♥✳

❍➔ ◆ë✐✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✵✷


●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✻

▲í✐ tü❛ ❝❤♦ t→✐ ❜↔♥ ❧➛♥ t❤ù ♥❤➜t
❚♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❝õ❛ ❝✉è♥ s→❝❤ ✤÷đ❝ t→✐ ❜↔♥ ❧➛♥ ♥➔② ❤♦➔♥ t♦➔♥ trị♥❣ ✈ỵ✐
❧➛♥ ①✉➜t ❜↔♥ ✤➛✉ t✐➯♥✳ ❚→❝ ❣✐↔ ❝❤➾ ❝❤➾♥❤ sû❛ ♠ët sè ❧é✐ s♦↕♥ t❤↔♦ ✈➔ ✐♥ ➜♥✳
▼ët ✈➔✐ ❞✐➵♥ ✤↕t ✈➲ t♦→♥ ữủ t ờ ỹ õ ỵ ừ ỗ ♥❣❤✐➺♣
♥❤➡♠ ❣✐ó♣ ♥❣÷í✐ ✤å❝ ❞➵ ❤✐➸✉ ❤ì♥✳ ◗✉❛ ✤➙② t→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ P●❙✳ ❚❙✳

▲➯ ❚❤❛♥❤ ◆❤➔♥ ✈➲ ♥❤ú♥❣ õ ỵ sỷ ờ tr ①✐♥ ❝↔♠
ì♥ P●❙✳ ❚❙❑❍✳ P❤↕♠ ❍✉② ✣✐➸♥ ✤➣ ❣✐ó♣ ✤ï ✈➲ ❝→❝ t❤õ tư❝ ❤➔♥❤ ❝❤➼♥❤ ✤➸
❝✉è♥ s→❝❤ ✤÷đ❝ t→✐ ❜↔♥ ❧➛♥ ♥➔②✳

❍➔ ◆ë✐✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✵✻
❚→❝ ❣✐↔


ữỡ

ỡ ữủ ỵ tt t ủ
r ữỡ ✤➛✉ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ sì ❧÷đ❝ ✈➲ t➟♣
❤đ♣✱ →♥❤ ①↕ ✈➔q✉❛♥ ❤➺✱ ♥❤➡♠ ♠ư❝ ✤➼❝❤ tố t ỵ tt
ỳ ữủ ũ tr s✉èt ❜➔✐ ❣✐↔♥❣ ♥➔②✳ P❤➛♥ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣ ❜➔♥ ✈➲ ❝→❝
❞↕♥❣ t÷ì♥❣ ✤÷ì♥❣ ❦❤→❝ ♥❤❛✉ ❝õ❛ t✐➯♥ ✤➲ ❝❤å♥✳ ❱➻ ❝❤÷❛ t➻♠ t❤➜② t➔✐ ❧✐➺✉
❜➡♥❣ t✐➳♥❣ ✈✐➺t ♥➔♦ ❝â ❝❤ù♥❣ ♠✐♥❤ ✤➛② ✤õ ❝❤♦ ❝→❝ t÷ì♥❣ ✤÷ì♥❣ ♥➔②✱ ♥➯♥
❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ♠ët ❝❤ù♥❣ ♠✐♥❤ ✤➸ ❜↕♥ ✤å❝ t❤❛♠ ❦❤↔♦ t❤➯♠✳

§

✶✳ ❚➟♣ ❤đ♣ ✈➔ ❝→❝ ♣❤➨♣ t♦→♥ tr➯♥ t➟♣ ❤đ♣

✶✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❚➟♣ ❤đ♣ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ t♦→♥ ❤å❝✱ ♥❤÷♥❣
❧↕✐ ❧➔ ♠ët ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛✳ ▼ët ❝→❝❤ trü❝ q✉❛♥✱ t❛ ❝â t❤➸
❤✐➸✉ ♠ët t➟♣ ❤đ♣ ♥❤÷ ❧➔ sü tư t➟♣ ♥❤ú♥❣ ✈➟t✱ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ ❤❛② ♥❤ú♥❣
❦❤→✐ ♥✐➺♠ t♦→♥ ❤å❝ ✳✳✳ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♠ët ❤❛② ♥❤✐➲✉ t➼♥❤ ❝❤➜t ❝❤✉♥❣✳
❚❛ t❤÷í♥❣ sû ❞ö♥❣ ❝→❝ ❝❤ú ❝→✐ ▲❛ t✐♥❤ A, B, C, ..., X, Y, Z ❤♦➦❝ ❝❤ú
❝→✐ ❍② ▲↕♣ ❝ê ♥❤÷ Γ, Ω, Λ, ... ✤➸ ❝❤➾ ♠ët t➟♣ ❤ñ♣✳
❈→❝ ✈➟t ❝õ❛ ♠ët t➟♣ ❤ñ♣ X ❣å✐ ❧➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣ ✤â✳ ▼ët
♣❤➛♥ tû x ❝õ❛ t➟♣ ❤ñ♣ X ữủ ỵ x X.

tt ❝→❝ ♣❤➛♥ tû ❝õ❛ ♠ët t➟♣ ❤ñ♣ X ✤➲✉ ❧➔ ♣❤➛♥ tû ❝õ❛ ♠ët t➟♣
❤ñ♣ Y t❤➻ t❛ ♥â✐ t➟♣ ❤ñ♣ X ❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣ Y ỵ
X Y Y X. ❚r÷í♥❣ ❤đ♣ X ⊆ Y ✈➔ Y ⊆ X t❤➻ t❛ ♥â✐ r➡♥❣ t➟♣
❤ñ♣ X ❜➡♥❣ t➟♣ ❤ñ♣ Y ỵ X = Y. X Y ✈➔ X = Y t❤➻ X
✤÷đ❝ ❣å✐ ❧➔ t➟♣ ủ tỹ sỹ ừ Y ỵ X ⊂ Y.
✼


●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✽

❳→❝ ✤à♥❤ ♠ët t➟♣ ❤ñ♣ ❧➔ ①→❝ ✤à♥❤ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♥â✳ ❈â ♥❤✐➲✉
❝→❝❤ ✤➸ ①→❝ ✤à♥❤ ♠ët t➟♣ ❤đ♣✳ ✣ì♥ ❣✐↔♥ ♥❤➜t ❧➔ ❧✐➺t ❦➯ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû
t➟♣ ❤ñ♣ ✤â ✈➔ ✤➸ tr♦♥❣ ❤❛✐ ❞➜✉ ♠â❝ {...}✳ ❈→❝❤ t❤æ♥❣ ❞ư♥❣ t❤ù ❤❛✐ ❧➔ ♠ỉ
t↔ ♠ët t➟♣ ❤đ♣ q✉❛ ❝→❝ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣
✤â✳ ❈❤➥♥❣ ❤↕♥ t❛ ✈✐➳t X = {x | P (x)} õ r X t ủ ỗ tt
❝→❝ ♣❤➛♥ tû x t❤♦↔ ♠➣♥ ♠➺♥❤ ✤➲ P (x).
❚➟♣ ❤đ♣ ❦❤ỉ♥❣ ❝❤ù❛ ♠ët ♣❤➛♥ tû ♥➔♦ ✤÷đ❝ ❣å✐ ❧➔ t ủ rộ ỵ
.

t tr➯♥ t➟♣ ❤ñ♣✳
✶✮ ❍ñ♣✳ ❍ñ♣ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ X Y, ỵ X Y, t ủ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐

X ∪ Y = {x | x ∈ X ❤♦➦❝ x ∈ Y }.
✷✮ ●✐❛♦✳ ●✐❛♦ ❝õ❛ ❤❛✐ t ủ X Y, ỵ X Y, ❧➔ t➟♣ ❤đ♣ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐
X ∩ Y = {x | x ∈ X ✈➔ x ∈ Y }.
✸✮ ❚➼❝❤ ❉❡s❝❛rt❡s✳ ❚➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ X ✈➔ Y, ỵ X ìY,

t ủ ữủ

X × Y = {z = (x, y) | x ∈ X, y ∈ Y }.
✹✮ ❍✐➺✉✳ ❍✐➺✉ ❝õ❛ ❤❛✐ t➟♣ ủ X Y, ỵ X \ Y, t➟♣ ❤đ♣ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐
X \ Y = {x | x X x
/ Y }.

ú ỵ ❈→❝ ♣❤➨♣ t♦→♥ ❤ñ♣✱ ❣✐❛♦✱ t➼❝❤ ❉❡s❝❛rt❡s ❤♦➔♥ t♦➔♥ ❝â t
rở ởt ồ tũ ỵ t ủ {(Xi ) | i ∈ I}, ð ✤➙② I ❧➔ ♠ët t➟♣ ❝❤➾
sè ♥➔♦ ✤â✳ ❑❤✐ ✤â t❛ ①→❝ ✤à♥❤✿

Xi = {x | ∃i ∈ I, x ∈ Xi }.
i∈I

Xi = {x | x ∈ Xi , ∀i ∈ I}.
i∈I

Xi = {z = (xi )i∈I | xi ∈ Xi , ∀i ∈ I}.
i∈I


ữỡ ỡ ữủ ỵ tt t ủ



t t t X n ỵ t srts ừ n
t ủ X.

Đ

ũ ợ t➟♣ ❤ñ♣✱ →♥❤ ①↕ t❤✉ë❝ ✈➔♦ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠
❝ì ❜↔♥ ♥❤➜t ❝õ❛ t♦→♥ ❤å❝✳

✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳
✭✐✮ ▼ët →♥❤ ①↕ f : X −→ Y tø t➟♣ ❤ñ♣ X ✤➳♥ t➟♣ ❤đ♣ Y ❧➔ ♠ët ♣❤➨♣
t÷ì♥❣ ù♥❣ ♠é✐ ♠ët ♣❤➛♥ tû x ∈ X ❞✉② ♥❤➜t ♠ët ♣❤➛♥ tû f (x) ∈ Y.
❚➟♣ ❤đ♣ X ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ỗ ừ f t ủ Y ồ ❧➔
t➟♣ ✤➼❝❤ ❝õ❛ →♥❤ ①↕ f.
✭✐✐✮ f : X −→ Y ữủ ồ ỡ ợ tỷ x, x X tũ
ỵ f (x) = f (x ), s✉② r❛ x = x . ✣➦❝ ❜✐➺t ❦❤✐ X ⊆ Y t❤➻ ✤ì♥ →♥❤
f : X −→ Y ①→❝ ✤à♥❤ ❜ð✐ f (x) = x, ∀x X ữủ ồ ú
tỹ ỵ ❤✐➺✉ ❧➔ X → Y.
✭✐✐✐✮ f : X −→ Y ữủ ồ t ợ ộ tỷ tũ ỵ y Y
ổ tỗ t t t ởt ♣❤➛♥ tû x ∈ X s❛♦ ❝❤♦ f (x) = y.
✭✐✈✮ f : X −→ Y ✤÷đ❝ ❣å✐ ❧➔ s♦♥❣ →♥❤ ♥➳✉ f ✈ø❛ ❧➔ ✤ì♥ →♥❤ ✈ø❛ ❧➔ t♦➔♥
→♥❤✳
✭✈✮ ❈❤♦ f : X −→ Y ✈➔ g : Y −→ Z ❧➔ ♥❤ú♥❣ →♥❤ ①↕✱ t❛ ❣å✐ →♥❤ ①↕
h : X −→ Z ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ h(x) = g(f (x)), ∀x ∈ X ❧➔ →♥❤ ①↕ ❤ñ♣
t❤➔♥❤ ❝õ❛ f g ỵ h = g f.

ú ỵ
f : X −→ Y ✈➔ A ❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ X. ❚❛ ❣å✐ t➟♣
❤ñ♣ f (A) ⊆ Y ①→❝ ✤à♥❤ ❜ð✐ f (A) = {f (x) ∈ Y | x ∈ A} ❧➔ ↔♥❤ ❝õ❛ A
q✉❛ →♥❤ ①↕ f. ❱➟② →♥❤ ①↕ f ❧➔ t♦➔♥ →♥❤ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (X) = Y.
❜✮ ❈❤♦ B ❧➔ ởt t tũ ỵ ừ Y, t ồ t ❤đ♣ f −1 (B) ⊆ X, ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐ f −1 (B) = {x ∈ X | f (x) ∈ B}, ❧➔ ♥❣❤à❝❤ ↔♥❤ ❝õ❛ B q✉❛

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✶✵

→♥❤ ①↕ f. ❇➙② ❣✐í✱ ❣✐↔ sû f ❧➔ ♠ët s♦♥❣ →♥❤✳ ❑❤✐ ✤â✱ ✈➻ f (X) = Y, t❛
❧✉ỉ♥ ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ởt f 1 ữ s ợ y Y tũ
ỵ tỗ t x X s f (x) = y, t❛ ①→❝ ✤à♥❤ f −1 (y) = x. ❉ü❛ ✈➔♦
t➼♥❤ ✤ì♥ →♥❤ ❝õ❛ f t❛ ❞➵ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ f −1 ✤÷đ❝ ①→❝ ✤à♥❤
♥❤÷ tr➯♥ ❧➔ ♠ët →♥❤ ①↕✱ ❣å✐ ❧➔ →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ f. ❑❤✐ ✤â t❛ t❤➜②
♥❣❛② r➡♥❣ f −1 ◦ f = 1X ✈➔ f ◦ f −1 = 1Y , ð 1X ữủ ỵ
ỗ t tr t➟♣ ❤ñ♣ X, tù❝ 1X (x) = x, ∀x ∈ X.

✷✳✸✳ ❇ê ✤➲✳ ❈❤♦ f : X −→ Y ✈➔ g : X −→ Z ❧➔ ❤❛✐ →♥❤ ①↕ ❣✐ú❛ ❝→❝ t➟♣
❤ñ♣✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❧➔ tữỡ ữỡ
ỗ t ởt h : Y −→ Z s❛♦ ❝❤♦ g = h ◦ f.
✭✐✐✮ ❱ỵ✐ tỷ x1 , x2 X tũ ỵ ♥➳✉ f (x1 ) = f (x2 ) s✉② r❛ g(x1 ) =
g(x2 ).

❈❤ù♥❣ ♠✐♥❤✳ (i) =⇒ (ii)✿ ●✐↔ sû f (x1 ) = f (x2 ). ❚ø g = h ◦ f t❛ s✉② r❛
g(x1 ) = h ◦ f (x1 ) = h(f (x1 )) = h(f (x2 )) = h ◦ f (x2 ) = g(x2 ).
(ii) =⇒ (i)✿ ❳➨t t÷ì♥❣ ù♥❣ h : Y −→ Z ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
✲ ◆➳✉ y ∈ f (X)✱ tự tỗ t x X s f (x) = y, ❦❤✐ ✤â t❛ ✤➦t
h(y) = g(x).
✲ ◆➳✉ y ∈
/ f (X)✱ t❛ ❝❤å♥ ♠ët ♣❤➛♥ tû z ∈ Z ố rỗ t h(y) = z.
s r❛ tø ❣✐↔ t❤✐➳t ❝õ❛ ✭✐✐✮ r➡♥❣ t÷ì♥❣ ù♥❣ tr➯♥ ❧➔ ♠ët →♥❤ ①↕✱
❤ì♥ ♥ú❛ tø ❝→❝❤ ①➙② ❞ü♥❣ t❛ ❝â g = h ◦ f ✳
❇ê ✤➲ ✷✳✸ ❣✐ó♣ t❛ ♥❤➟♥ ✤÷đ❝ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ✤ì♥ ❣✐↔♥ ❦❤✐ ♥➔♦ ♠ët
→♥❤ ①↕ ❧➔ ✤ì♥ →♥❤✱ t♦➔♥ →♥❤ ❤❛② s♦♥❣ →♥❤ ữ s

ỵ f : X Y ❧➔ ♠ët →♥❤ ①↕ ❣✐ú❛ ❤❛✐ t➟♣ ❤ñ♣✳ ❑❤✐ ✤â
❝→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ ✤ó♥❣✿
✭✐✮ f ❧➔ ✤ì♥ →♥❤ tỗ t ởt g : Y −→ X s❛♦ ❝❤♦
g ◦ f = 1X .
f t tỗ t↕✐ ♠ët →♥❤ ①↕ h : Y −→ X s❛♦ ❝❤♦
f ◦ h = 1Y .


ữỡ ỡ ữủ ỵ tt t ủ



f s tỗ t →♥❤ ①↕ g : Y −→ X ✈➔
h : Y −→ X s❛♦ ❝❤♦ g ◦ f = 1X ✈➔ f h = 1Y .
ự ừ ỵ ❞➵ ❞➔♥❣ ✤÷đ❝ s✉② r❛ tø ❇ê ✤➲ ✭✷✳✸✮✱ ❝❤ó♥❣ tỉ✐ ①❡♠
♥❤÷ ❧➔ ♠ët ❜➔✐ t➟♣ ✤ì♥ ❣✐↔♥ ❝❤♦ ♥❣÷í✐ ✤å❝✳

§

✸✳ ◗✉❛♥ ❤➺

❚r♦♥❣ t✐➳t ♥➔② t❛ s➩ ①➨t ❤❛✐ ❧♦↕✐ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ q✉❛♥ trå♥❣ ♥❤➜t✱ ✤â ❧➔
q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✈➔ q✉❛♥ ❤➺ t❤ù tü✳

✸✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤ñ♣✳ ❚❛ ♥â✐ r➡♥❣ Ω ❧➔ ♠ët q✉❛♥ ❤➺
n✲♥❣æ✐ tr➯♥ X ♥➳✉ Ω ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ t➼❝❤ ❉❡s❝❛rt❡s X n . ✣➦❝ ❜✐➺t✱ ❦❤✐ Ω
❧➔ ♠ët q✉❛♥ ❤➺ ✷✲♥❣æ✐ tr➯♥ X, t❤❛② ✈➻ ✈✐➳t (a, b) ∈ Ω ♥❣÷í✐ t❛ ✈✐➳t ❧➔ aΩb.

✸✳✷✳ ❱➼ ❞ư✳ ❚r➯♥ t➟♣ ❤đ♣ N t➜t ❝↔ ❝→❝ sè tü ♥❤✐➯♥ t❛ ①→❝ ✤à♥❤ q✉❛♥ ❤➺

✷✲♥❣ỉ✐ Ω ♥❤÷ s❛✉✿
✶✮ Ω = {(n1 , n2 ) ∈ N2 | n1 , n2 ✤➲✉ ❧➔ ♥❤ú♥❣ sè ❝❤➤♥}. ❚❛ ♥❤➟♥ t❤➜②
r➡♥❣✱ tø n1 Ωn2 s✉② r❛ n2 Ωn1 , ♥❤÷♥❣ nΩn ❧➔ ổ ú ợ ồ số
n. r trữớ ủ t❛ ♥â✐ r➡♥❣ Ω ❧➔ ♠ët q✉❛♥ ❤➺ ✷✲♥❣æ✐ ✤è✐
①ù♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ♣❤↔♥ ①↕✳
✷✮ Ω = {(n1 , n2 ) ∈ N2 | n1 ❝❤✐❛ ❤➳t ❝❤♦ n2 }. ❚❛ ❞➵ ♥❤➟♥ t❤➜② r➡♥❣ nΩn
✈ỵ✐ ♠å✐ n ∈ N, ♥❤÷♥❣ tø n1 Ωn2 ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ s✉② r❛ n2 Ωn1 . ❱➟②
tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② q✉❛♥ ❤➺ ✷✲♥❣ỉ✐ Ω ❧➔ ♣❤↔♥ ①↕ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔
✤è✐ ①ù♥❣✳
✸✮ Ω = {(n1 , n2 ) N2 | ữợ số ợ ♥❤➜t (n1 , n2 ) = 1} ∪ {(1, 1)}.
❘ã r➔♥❣ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ♠ỵ✐ ♥➔② ❧➔ ♣❤↔♥ ①↕ ✈➔ ✤è✐ ①ù♥❣✱ ♥❤÷♥❣ tø
n1 Ωn2 ✈➔ n2 Ωn3 ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ s✉② r❛ n1 Ωn3 ✭2Ω6 ✈➔ 6Ω3 ♥❤÷♥❣
t❛ ❦❤æ♥❣ ❝â 2Ω3✮✳ ❚❛ ♥â✐ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr♦♥❣ ✈➼ ❞ư ♥➔② ❧➔ ♣❤↔♥
①↕✱ ✤è✐ ①ù♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ❜➢❝ ❝➛✉✳ ❉➵ t❤➜② r➡♥❣ ❝→❝ q✉❛♥ ❤➺ ❤❛✐
♥❣æ✐ tr♦♥❣ ❝→❝ ✈➼ ❞ö ✭✶✮ ✈➔ ✭✷✮ ✤➲✉ ❧➔ ❜➢❝ ❝➛✉✳
❱➼ ❞ö ✸✳✷ ❝❤♦ t❛ t❤➜② ❝â r➜t ♥❤✐➲✉ q✉❛♥ ❤➺ ổ tú tr ởt t
ủ trữợ ✤➙② ❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ❤❛✐ ❧♦↕✐ q✉❛♥ ❤➺ ✤➦❝ ❜✐➺t q✉❛♥
trå♥❣ tr♦♥❣ ✤↕✐ sè✳


●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✶✷

✸✳✸✳ ✣à♥❤ ♥❣❤➽❛✳ ▼ët q✉❛♥ ❤➺ ✷✲♥❣ỉ✐ Ω tr➯♥ t➟♣ ❤đ♣ X ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥
❤➺ t÷ì♥❣ ✤÷ì♥❣✱ ♥➳✉ ♥â t❤♦↔ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✳
✭✐✮ P❤↔♥ ①↕✿ xΩx, ∀x ∈ X.
✭✐✐✮ ✣è✐ ①ù♥❣✿ xΩy =⇒ yΩx, ∀x, y ∈ X.
✭✐✐✐✮ ❇➢❝ ❝➛✉✿ xΩy, yΩz =⇒ xΩz, ∀x, y, z ∈ X.
❑❤✐ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ Ω ✤➣ ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ X ✱ t❤❛② ✈➻ ✈✐➳t xΩy

♥❣÷í✐ t❛ t❤÷í♥❣ ✈✐➳t x ∼ y.

✸✳✹✳ ❈❤ó þ✳ ❈❤♦ Ω ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ ❤ñ♣ X ✈➔ x ∈ X.
❚❛ ❣å✐ t➟♣ ❤ñ♣

Ω(x) = {y X | y x}
ợ tữỡ ✤÷ì♥❣ ❝õ❛ x t❤❡♦ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ Ω. ❉➵ t❤➜② r➡♥❣✿
✲ Ω(x) = ∅✱ ✈➻ x ∈ Ω(x).
✲

x∈X

Ω(x) = X.

✲ ∀x, y ∈ X, ❤♦➦❝ Ω(x) = Ω(y) ❤♦➦❝ Ω(x) ∩ Ω(y) = ∅. ❚❤➟t ✈➟②✱ ♥➳✉
z ∈ Ω(x) ∩ Ω(y), t❛ s✉② r❛ z ∼ x ✈➔ z ∼ y. ❉♦ q✉❛♥ ❤➺ Ω ❝â t➼♥❤ ✤è✐
①ù♥❣ ✈➔ ❜➢❝ ❝➛✉✱ ♥➯♥ x ∼ y ✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä ❤♦➦❝ Ω(x) = Ω(y) ❤♦➦❝
Ω(x) ∩ Ω(y) = ∅, ∀x, y ∈ X.
❱➟② t❛ ♥❤➟♥ ✤÷đ❝ ♠ët ♣❤➙♥ ❤♦↕❝❤ ừ X q ợ tữỡ ữỡ
(x). ủ tt ợ tữỡ ữỡ ữủ ỵ X/Ω ✈➔ ❣å✐
❧➔ t➟♣ ❤đ♣ t❤÷ì♥❣ ❝õ❛ X q✉❛ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ Ω. ❍ì♥ ♥ú❛ t❛ ❝â t❤➸
①→❝ ✤à♥❤ ♠ët →♥❤ ①↕ π : X −→ X/Ω, π(x) = Ω(x), ∀x ∈ X ✈➔ ❣å✐ ♥â ❧➔
→♥❤ ①↕ ❝❤➼♥❤ t➢❝ s✐♥❤ ❜ð✐ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ Ω.

✸✳✺✳ ✣à♥❤ ♥❣❤➽❛✳ ▼ët q✉❛♥ ❤➺ ✷✲♥❣ỉ✐ Ω tr➯♥ ♠ët t➟♣ ❤đ♣ X ✤÷đ❝ ❣å✐ ❧➔
q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ ♥➳✉ q✉❛♥ ❤➺ ✤â ❧➔ ♣❤↔♥ ①↕✱ ❜➢❝ ❝➛✉ ✈➔ ♣❤↔♥ ✤è✐
①ù♥❣ ✭♥❣❤➽❛ ❧➔✱ tø xΩy, yΩx =⇒ x = y, ∀x, y ∈ X).
❑❤✐ tr➯♥ t➟♣ ❤ñ♣ X ❝â ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ Ω t❤➻ t❛ ♥â✐ X ❧➔
♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ð✐ Ω. ❚❤ỉ♥❣ tữớ ữớ t ũ ỵ
ởt q ❤➺ t❤ù tü ❜ë ♣❤➟♥✳ ❍❛✐ ♣❤➛♥ tû x, y X ữủ ồ s

s ữủ ố ợ q ❤➺ t❤ù tü ❜ë ♣❤➟♥ ≤ ♥➳✉ ❤♦➦❝ x ≤ y ❤♦➦❝ y ≤ x.


ữỡ ỡ ữủ ỵ tt t ủ



A ❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣ X ✈➔ x X. õ r x
ởt ữợ ✭❝➟♥ tr➯♥✮ ❝õ❛ t➟♣ A tr♦♥❣ t➟♣ X ♥➳✉ x ≤ a (a ≤ x), ∀a ∈ A.
✣➦❝ ❜✐➺t✱ ♠ët ♣❤➛♥ tû x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ❝ü❝ ✤↕✐ ✭❝ü❝ t✐➸✉✮ ❝õ❛
t➟♣ ❤ñ♣ X, ♥➳✉ x ❧➔ tr ữợ t ừ t {x} tr X.
◗✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ ≤ tr➯♥ t➟♣ ❤ñ♣ X ✤÷đ❝ ❣å✐ ❧➔ t✉②➳♥ t➼♥❤ ♥➳✉
❤❛✐ ♣❤➛♥ tû tị② ỵ ừ X s s ữủ ợ ởt q✉❛♥ ❤➺ t❤ù tü
t✉②➳♥ t➼♥❤ tr➯♥ X ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ ❤➺ t❤ù tü tèt ♥➳✉ ♠å✐ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝
ré♥❣ ❝õ❛ X ✤➲✉ ❝❤ù❛ ♠ët ♣❤➛♥ tû ❝ü❝ t✐➸✉✳

✸✳✻✳ ❱➼ ❞ư✳
✶✮ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤đ♣✱ t➟♣ ❤đ♣ 2X = {A | A ⊆ X} ✤÷đ❝ ❣å✐ ❧➔ t➟♣
❤đ♣ ❝→❝ ❜ë ♣❤➟♥ ❝õ❛ X ✭❞➵ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣✱ ♥➳✉ X ❝â n ♣❤➛♥
tû t❤➻ 2X ❝â 2n ♣❤➛♥ tû✱ ✤✐➲✉ ♥➔② ❣✐↔✐ t❤➼❝❤ t↕✐ s❛♦ t❛ ❧↕✐ ũ ỵ
ữ tr ởt q ≤ tr➯♥ 2X , ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥
❤➺ ❜❛♦ ❤➔♠ ♥❤÷ s❛✉✿ A ≤ B ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A ⊆ B. ❉➵ ❞➔♥❣ ❝❤ù♥❣
♠✐♥❤ ✤÷đ❝ r➡♥❣ q✉❛♥ ❤➺ ♥➔② ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ 2X .
❍ì♥ ♥ú❛✱ ♥➳✉ X ❝❤ù❛ ➼t ♥❤➜t ✷ ♣❤➛♥ tû x = y t❤➻ q✉❛♥ ❤➺ ✤â ❦❤ỉ♥❣
❜❛♦ ❣✐í ❧➔ ♠ët q✉❛♥ ❤➺ t✉②➳♥ t➼♥❤✱ ✈➻ {x} ❦❤æ♥❣ s♦ s→♥❤ ữủ ợ
{y}.
tự tỹ tổ tữớ tr t ❤ñ♣ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ ❩ ❧➔
♠ët q✉❛♥ ❤➺ t❤ù tü t✉②➳♥ t➼♥❤✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü
tèt ✭❝❤➥♥❣ ❤↕♥✱ t➟♣ ❤ñ♣ {... − 2, −1, 0} ❦❤æ♥❣ ❝â ♣❤➛♥ tû ❝ü❝ t✐➸✉✮✳
✸✮ ◗✉❛♥ ❤➺ t❤ù tü t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ sè tü ♥❤✐➯♥ N

❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü t✉②➳♥ t➼♥❤✱ ❤ì♥ ♥ú❛ ♥â ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü
tèt✳

§

✹✳ ❚➟♣ ❤đ♣ t÷ì♥❣ ✤÷ì♥❣

✹✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❍❛✐ t➟♣ ❤đ♣ X ✈➔ Y ữủ ồ tữỡ ữỡ ỵ
X Y tỗ t ởt s f : X −→ Y. ❑❤✐ ✤â t❛ ❝ô♥❣ ♥â✐ r➡♥❣
X ✈➔ Y õ ũ ỹ ữủ

ú ỵ ó r q t ủ X tữỡ ữỡ ợ t ủ Y ✧
t❤♦↔ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿


●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✶✹

✲ P❤↔♥ ①↕ ✿ X ∼ X ✱ ✈➻ 1X : X −→ X ❧➔ ♠ët s♦♥❣ →♥❤✳
✲ ✣è✐ ①ù♥❣✿ X ∼ Y =⇒ Y ∼ X ✱ ✈➻ f −1 : Y −→ X ❝ô♥❣ ❧➔ ♠ët s♦♥❣
→♥❤✳
✲ ❇➢❝ ❝➛✉✿ X ∼ Y, Y ∼ Z =⇒ X ∼ Z ✱ ✈➻✱ tø f : X −→ Y ✈➔
g : Y −→ Z ❧➔ s♦♥❣ →♥❤✱ s✉② r❛ h = g ◦ f : X −→ Z ❝ô♥❣ ❧➔ ♠ët s♦♥❣
→♥❤✳
❱➟②✱ ♥➳✉ ❝❤♦ ♠ët ❤å ❝→❝ t➟♣ ❤ñ♣ Σ ♥➔♦ ✤â t❤➻ q✉❛♥ ❤➺ ∼ ①→❝ ✤à♥❤ tr➯♥ Σ
❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ t❤❡♦ ♥❣❤➽❛ ❝õ❛ ✭✸✳✸✮✳

✹✳✸✳ ❇ê ✤➲✳ P❤➨♣ ❧➜② t➼❝❤ ❉❡s❝❛rt❡s ✈➔ ❤đ♣ ❧➔ ❜↔♦ t♦➔♥ t➼♥❤ t÷ì♥❣ ✤÷ì♥❣✳
◆❣❤➽❛ ❧➔✱ ♥➳✉ X ∼ X1 ✈➔ Y ∼ Y1 t❤➻ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣✳

✭✐✮ X × Y ∼ X1 × Y1 .
✭✐✐✮ ●✐↔ t❤✐➳t t❤➯♠ r➡♥❣ X ∩ Y = X1 ∩ Y1 = ∅ t❤➻ X ∪ Y ∼ X1 Y1 .

ự tt tỗ t ❝→❝ s♦♥❣ →♥❤ f : X −→ X1 ✈➔ g :
Y −→ Y1 . ❚❛ ①➙② ❞ü♥❣ ♥❤ú♥❣ →♥❤ ①↕ ợ ữ s
: X ì Y X1 ì Y1 , φ(x, y) = (f (x), g(y)), ∀x ∈ X, ∀y ∈ Y.
f (z), ♥➳✉ z ∈ X,
g(z), ♥➳✉ z ∈ Y.

ϕ : X ∪ Y −→ X1 ∪ Y1 , ϕ(z) =

❉➵ ❦✐➸♠ tr❛ t❤➜② r➡♥❣ φ, ϕ ❧➔ ♥❤ú♥❣ s♦♥❣ →♥❤✱ ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳

✹✳✹✳ ❈❤ó þ✳ ▼ët ❝→❝❤ t÷ì♥❣ tü t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ tê♥❣ q✉→t
❝õ❛ ❇ê ✤➲ ✹✳✸ ❝❤♦ ♥❤✐➲✉ t➟♣ ❤đ♣ ♥❤÷ s❛✉✿ ❈❤♦ (Xi )i∈I ✈➔ (Yi )i∈I ❧➔ ❤❛✐ ❤å
❝→❝ t➟♣ ❤đ♣ ✈ỵ✐ I ❧➔ ♠ët t➟♣ ❝❤➾ sè ♥➔♦ ✤â ✭❝â t❤➸ ❝â ✈æ ❤↕♥ ♣❤➛♥ tû✮✳ ●✐↔
sû r➡♥❣ Xi ∼ Yi , ∀i ∈ I. ❑❤✐ ✤â

Xi ∼
i∈I

Yi .
i∈I

❍ì♥ ♥ú❛✱ ♥➳✉ Xi ∩ Xj = Yi ∩ Yj = ∅, ∀i, j ∈ I, i = j, t❤➻

X∼
i∈I

Y.

i∈I


ữỡ ỡ ữủ ỵ tt t ủ



ỵ trrst X, Y t ủ X tữỡ
ữỡ ợ ởt t ủ ừ Y Y tữỡ ữỡ ợ ởt t ủ
ừ X t X tữỡ ữỡ ợ Y.
ự tt tỗ t↕✐ X1 ⊆ X ✈➔ Y1 ⊆ Y s❛♦ ❝❤♦ X1 ∼
Y, Y1 ∼ X. ●✐↔ sû f : Y −→ X1 ❧➔ ♠ët s♦♥❣ →♥❤✳ ✣➦t X2 = f (Y1 ). ❱➻
X ∼ Y1 , Y1 ∼ X2 , s r X X2 . tỗ t ởt s♦♥❣ →♥❤ g : X −→ X2 .
✣➦t X0 = X ✈➔ q✉❛ ❝æ♥❣ t❤ù❝ tr✉② ❝❤ù♥❣ Xn+1 = g(Xn−1 ) t❛ ♥❤➟♥ ✤÷đ❝
♠ët ❞➣② ✈ỉ ❤↕♥ ❝→❝ t➟♣ ❤đ♣ ỗ
X = X0 X1 X2 X3 ... .
❉ü❛ ✈➔♦ ❝→❝❤ ①➙② ❞ü♥❣ ❝õ❛ Xn ✈➔ t➼♥❤ s♦♥❣ →♥❤ ❝õ❛ g ❞➵ ❞➔♥❣ s✉② r❛ r➡♥❣

(∗)

Xn−1 \ Xn ∼ Xn+1 \ Xn+2 , n = 1, 2, ... .

✤➸ ❣å♥ ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ t❛ ✤➦t t✐➳♣

D = X1 ∩ X2 ∩ X3 ∩ ... ;
D1 = (X \ X1 ) ∪ (X2 \ X3 ) ∪ (X4 \ X5 ) ∪ ... ;
D2 = (X2 \ X3 ) ∪ (X4 \ X5 ) ∪ (X6 \ X7 ) ∪ ... .
⑩♣ ❞ư♥❣ ❇ê ✤➲ ✭✹✳✸✮✱ ❈❤ó þ ✹✳✹ ✈➔ ✭✯✮ t❛ ✤÷đ❝

(∗∗)

D1 ∼ D2 .

❇➙② ❣✐í t❛ ❞➵ t❤➜② ❝→❝ t➟♣ ❤ñ♣ X, X1 ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝ ♥❤÷ s❛✉

X = D ∪ D1 ∪ [(X1 \ X2 ) ∪ (X3 \ X4 ) ∪ (X5 \ X6 ) ∪ ...],
X1 = D ∪ D2 ∪ [(X1 \ X2 ) ∪ (X3 \ X4 ) ∪ (X5 \ X6 ) ∪ ...].
▲↕✐ ✤➦t

D3 = (X1 \ X2 ) ∪ (X3 \ X4 ) ∪ (X5 \ X6 ) ∪ ... ,
t❛ ♥❤➟♥ ✤÷đ❝

X = D1 ∪ (D ∪ D3 ),
X1 = D2 ∪ (D ∪ D3 ).
ởt ỳ ử ú ỵ ✭✯✯✮ t❛ s✉② r❛ X ∼ X1 . ❚❤❡♦ ❣✐↔ t❤✐➳t
❜❛♥ ✤➛✉ t❤➻ X1 ∼ Y. ❱➟② X ∼ Y ỵ ữủ ự t


✶✻

§

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✺✳ ❚✐➯♥ ✤➲ ❝❤å♥ ✈➔ ❝→❝ ♠➺♥❤ ✤➲ t÷ì♥❣ ✤÷ì♥❣

❚✐➯♥ ✤➲ s❛✉ ✤➙② ❣✐ú ♠ët trỏ rt q trồ tr ỵ tt t ủ
❜✐➺t ❧➔ ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➟♣ ❤đ♣ ✈ỉ ❤↕♥✳

✺✳✶✳ ❚✐➯♥ ✤➲ ❝❤å♥✳ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤ñ♣ tũ ỵ õ ổ tỗ t ởt

: 2X −→ X s❛♦ ❝❤♦ ϕ(A) ∈ A, ∀A ⊆ X, A = ∅. ⑩♥❤ ①↕ ϕ ✤÷đ❝
❣å✐ ❧➔ →♥❤ ①↕ ❝❤å♥ tr➯♥ t➟♣ ❤đ♣ X.
❚r♦♥❣ ♠ët t❤í✐ ❣✐❛♥ trữợ rt t ồ ố
ồ ữ ởt ỵ ố ❝❤ù♥❣ ♠✐♥❤ ♥â✳ ❱✐➺❝ ♥➔② ✤➣
✤÷❛ ✤➳♥ ♥❤ú♥❣ tr❛♥❤ ❝➣✐ ❧➙✉ ❞➔✐✱ ✤➦❝ ❜✐➺t ✤➣ ✤➦t r❛ ❝❤♦ ❧♦❣✐❝ t♦→♥ ỵ
tt t ủ ỳ rt õ ✈➔ q✉❛♥ trå♥❣✳ ▼➣✐ ✤➳♥ ❦❤✐
♥❣÷í✐ t❛ ♥❤➟♥ r❛ r➡♥❣✱ õ ỵ ỡ ừ t ồ ❝â t❤➸
❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝❤➦t ❝❤➩ ♥➳✉ ♥❣÷í✐ t❛ ❝ỉ♥❣ ♥❤➟♥ ❚✐➯♥ ✤➲ ❝❤å♥ ♥❤÷ ❧➔
♠ët t✐➯♥ ✤➲✳ ❱➔ ❤ì♥ ỳ ú ỏ tữỡ ữỡ ợ ồ Pt
s❛✉ ❝ò♥❣ ♥➔② ✤➣ ❝❤➜♠ ❞ùt ♠å✐ tr❛♥❤ ❝➣✐ ①✉♥❣ q✉❛♥❤ ✈✐➺❝ ❝æ♥❣ ♥❤➟♥
❚✐➯♥ ✤➲ ❝❤å♥ ❤❛② ❦❤æ♥❣✳ ❙❛✉ ✤➙② t s ữ r ởt số ỵ q trồ
tr ỵ tt t ủ tữỡ ữỡ ợ ồ
X ❧➔ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü✳ ▼ët t➟♣ ❤đ♣ ❝♦♥ A ❝õ❛ X ✤÷đ❝
❣å✐ ❧➔ ♠ët ①➼❝❤ ❝õ❛ X, ♥➳✉ A ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ ❝õ❛ t➟♣ ❤đ♣ X
❧➟♣ t❤➔♥❤ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t✉②➳♥ t➼♥❤✳
▼ët ①➼❝❤ A ❝õ❛ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü X ✤÷đ❝ ❣å✐ ❧➔ ①➼❝❤ ❝ü❝
✤↕✐ ♥➳✉ ♥â ❦❤ỉ♥❣ ❧➔ t➟♣ ❤đ♣ ❝♦♥ ❝õ❛ ❜➜t ❦ý ♠ët ừ X.

ỵ s tữỡ ữỡ ợ ồ
ỵ r ồ t ủ õ t ữủ s tự tỹ tốt
ỵ sr ộ ừ ởt t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü
❧✉ỉ♥ ♥➡♠ tr♦♥❣ ♠ët ①➼❝❤ ❝ü❝ ✤↕✐✳
✭✐✐✐✮ ✭❇ê ✤➲ ❑✉r❛t♦✇s❦✐✲❩♦r♥✮ ◆➳✉ ♠é✐ ①➼❝❤ ❝õ❛ ♠ët t➟♣ ❤đ♣ ❦❤ỉ♥❣
ré♥❣ ✤÷đ❝ s➢♣ t❤ù tü X ✤➲✉ ❝â ❝➟♥ tr➯♥✱ t❤➻ X ❝❤ù❛ ➼t ♥❤➜t ♠ët ♣❤➛♥
tû ❝ü❝ ✤↕✐✳
✭✐✈✮ ✭❇ê ✤➲ ❚❡✐❝❤♠☎
✉❧❧❡r✲❚✉❦❡②✮ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤ñ♣ ✈➔ X ❧➔ ♠ët


ữỡ ỡ ữủ ỵ tt t ủ

ồ ổ ré♥❣ ♥❤ú♥❣ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ X ❝â t➼♥❤ ❝❤➜t✿ ♠ët t➟♣ ❤ñ♣
❝♦♥ A ❝õ❛ X t❤✉ë❝ ✈➔♦ ❤å X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠å✐ t➟♣ ❤ñ♣ ❝♦♥✱ ❤ú✉ ❤↕♥
♣❤➛♥ tû ❝õ❛ A t❤✉ë❝ X. ❑❤✐ ✤â X ❝❤ù❛ ➼t ♥❤➜t ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐
t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ tr t ủ
ự s ự ỵ t ữủ ỗ s ồ
= = = == ồ
ồ = (i) rữợ ❤➳t t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët ✈➔✐ t❤✉➟t ♥❣ú ♠ỵ✐✱ ❝➛♥
t❤✐➳t ❝❤♦ ❝❤ù♥❣ ♠✐♥❤✳ ▼ët t➟♣ ❤ñ♣ ❝♦♥ B ❝õ❛ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ tèt A
✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤♦↕♥ ừ A, ợ b B tũ ỵ t {x ∈ A | x ≤ b} ⊆ B.
❇➙② ❣✐í✱ ❣✐↔ sû B ❧➔ ♠ët ✤♦↕♥ ❝õ❛ A ✈➔ B = A. A \ B = , tỗ t
tû ❝ü❝ t✐➸✉ ❜ tr♦♥❣ t➟♣ ❤ñ♣ ♥➔②✳ ❚❛ ❞➵ ❞➔♥❣ s✉② r❛ B = {x ∈ A | x ≤
b ✈➔ x = b}. ❑❤✐ ✤â t❛ ♥â ✤♦↕♥ B ✤÷đ❝ s✐♥❤ ❜ð✐ b tr♦♥❣ t➟♣ ❤đ♣ A ✈➔ ỵ
B = [A, b].
r ự ỵ X ởt t ủ tũ ỵ
❝❤å♥ t❛ ❝â ♠ët →♥❤ ①↕ ϕ ①→❝ ✤à♥❤ tr➯♥ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❜ë ♣❤➟♥ ❝õ❛
X, s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ t➟♣ ❤đ♣ ❝♦♥ ❦❤→❝ ré♥❣ Y ❝õ❛ X ①→❝ ✤à♥❤ ♠ët ♣❤➛♥ tû
ϕ(Y ) ∈ Y. ❚❛ ❣å✐ ♠ët t➟♣ ❤ñ♣ ❝♦♥ A ❝õ❛ X ❧➔ tèt✱ ♥➳✉ ♥â ❧➔ ♠ët t➟♣ ❤đ♣
✤÷đ❝ s➢♣ t❤ù tü tèt ✈➔ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû a ∈ A ❧✉ỉ♥ ❝â

a = (X \ [A, a]).
ó r ổ tỗ t t ủ ❝♦♥ tèt tr♦♥❣ X ✱ ✈➻ {ϕ(X)} ❧➔ ♠ët t➟♣ ❤đ♣
❝♦♥ tèt ❝õ❛ X. ❍ì♥ ♥ú❛✱ t❛ ♥❤➟♥ t❤➜② r➡♥❣ ♠å✐ t➟♣ ❤ñ♣ ❝♦♥ tèt ✤➲✉ ❝â
♣❤➛♥ tû ❝ü❝ t✐➸✉ ❧➔ ϕ(X). ❱➟②✱ ♥➳✉ A, B ❧➔ ❤❛✐ t➟♣ ❤ñ♣ tèt ❝õ❛ X t❤➻ ❝❤ó♥❣
❝â ➼t ♥❤➜t ♠ët ✤♦↕♥ ❝❤✉♥❣ ❧➔ {ϕ(X)}. ✣➦t C ❧➔ ❤ñ♣ ❝õ❛ t➜t ❝↔ ❝→❝ ✤♦↕♥
❝❤✉♥❣ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ ♥➔②✳ ❉➵ t❤➜② r➡♥❣ C ❧➔ ♠ët ✤♦↕♥ ❝❤✉♥❣ ❝õ❛ ❝↔ ❤❛✐
t➟♣ ❤ñ♣ A ✈➔ B. ●✐↔ sû r➡♥❣ C ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ ❝↔ A ✈➔ B. ❱➻ C ❧➔ t➟♣
❤ñ♣ tèt ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B, ♥➯♥ t❤❡♦ ♥❤➟♥ ①➨t ð ♣❤➛♥ ✤➛✉ ❝❤ù♥❣ ♠✐♥❤

C = [A, ϕ(X \ C)] = [B, ϕ(X \ C)]. ❱➟② C = C ∪ {ϕ(X \ C)} ❧➔ ♠ët ✤♦↕♥
❝❤✉♥❣ ❝õ❛ A ✈➔ B ❝❤ù❛ t❤ü❝ sü ✤♦↕♥ C. ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❝ü❝
✤↕✐ ❝õ❛ C. ❱➟②✱ tr♦♥❣ ❤❛✐ t➟♣ ❤ñ♣ tèt A ✈➔ B ♣❤↔✐ ❝â ♠ët t➟♣ ❤ñ♣ ❧➔ ✤♦↕♥
❝õ❛ t➟♣ ❤ñ♣ ớ ợ ỵ D ủ ừ t➜t ❝↔ ❝→❝ t➟♣ ❤ñ♣ ❝♦♥
tèt ❝õ❛ X, t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ D ❝ơ♥❣ ❧➔ ♠ët t➟♣ ❤đ♣ ❝♦♥ tèt ❝õ❛ X. ❚❤➟t
✈➟②✱ ♥➳✉ a, b ❧➔ ❤❛✐ ♣❤➛♥ tû tũ ỵ ừ D, t a, b tr ❤❛✐ t➟♣


✶✽

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

❤ñ♣ ❝♦♥ tèt A, B ❝õ❛ X, s✉② r❛ ❝❤ó♥❣ t❤✉ë❝ ✈➔♦ t➟♣ ❤đ♣ ❧ỵ♥ ❤ì♥✱ ❝❤➥♥❣
❤↕♥ ❧➔ A. ❑❤✐ ✤â t❛ ①→❝ ✤à♥❤ a ≤ b tr➯♥ D ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a ≤ b t❤❡♦ q✉❛♥
❤➺ t❤ù tü t♦➔♥ ♣❤➛♥ tr➯♥ A. ❘ã r➔♥❣ ❝→❝❤ ①→❝ ✤à♥❤ ♥➔② ❧➔♠ D trð t❤➔♥❤
♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t✉②➳♥ t➼♥❤✳ ●✐↔ sû D ❦❤ỉ♥❣ ♣❤↔✐ ữủ s tốt tự
tỗ t ởt t ủ ổ ré♥❣ E ⊆ D s❛♦ ❝❤♦ tr♦♥❣ E ❦❤æ♥❣ ❝â ♣❤➛♥
tû ❝ü❝ t✐➸✉✳ ❚ø ✤➙② s✉② r❛ [E, x] ❝ô♥❣ ❧➔ ♠ët t➟♣ ❤đ♣ ❦❤ỉ♥❣ ❝â ♣❤➛♥ tû
❝ü❝ t✐➸✉ ✈ỵ✐ ộ x E trữợ t ✈➻ [E, x] ❧✉ỉ♥ ♥➡♠
tr♦♥❣ ♠ët t➟♣ ❤đ♣ ❝♦♥ tèt ❝õ❛ X. ❱➟② D ❧➔ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü
tèt✳ ❍ì♥ ♥ú❛✱ ♥➳✉ a ∈ D t❤➻ a ♣❤↔✐ ♥➡♠ tr♦♥❣ ♠ët t➟♣ ❤ñ♣ ❝♦♥ tèt A ♥➔♦
✤â✳ ❉♦ ✤â t❛ ♥❤➟♥ ✤÷đ❝

a = ϕ(X \ [A, a]) = ϕ(X \ [D, a]).
✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ D ❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥ tèt ❝õ❛ X. ●✐↔ sû D = X.
❑❤✐ ✤â t➟♣ ❤ñ♣ D = D ∪ {ϕ(X \ D)} s➩ ❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥ tèt ❝❤ù❛ t❤ü❝
sü D. ❑➳t ❧✉➟♥ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❝→❝❤ ①➙② ❞ü♥❣ ❝õ❛ D. ❱➟② D = X, tù❝
t➟♣ ❤ñ♣ X ✤➣ ✤÷đ❝ s➢♣ t❤ù tü tèt✳

(i) =⇒ (ii)✿ ❈❤♦ A ❧➔ ♠ët ①➼❝❤ ❝õ❛ t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥ X. ◆➳✉

A = X t❛ ❦❤æ♥❣ ❝á♥ ❣➻ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ♥ú❛✳ ❚r→✐ ❧↕✐✱ ♥➳✉ B = X \ A = ∅,
❞ü❛ ✈➔♦ ✭✐✮ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t tr➯♥ B ❝â ♠ët t❤ù tü ✤÷đ❝ s➢♣ tốt ú ỵ
r tự tỹ t ở ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥ ❝õ❛ X ❤↕♥ ❝❤➳ tr➯♥
B. ❚❛ s➩ ♣❤➙♥ ❤♦↕❝❤ B t❤➔♥❤ ❤❛✐ t➟♣ ❤ñ♣ ❝♦♥ C, D ♥❤÷ s❛✉✿ P❤➛♥ tû ❝ü❝
t✐➸✉ b ∈ B s tở C b s s ữủ ợ ồ ♣❤➛♥ tû ❝õ❛ A, ❝á♥ ♥❣÷đ❝
❧↕✐ t❛ ❝❤♦ b ∈ D. x ởt tỷ tũ ỵ ừ B ✈➔ ❣✐↔ sû r➡♥❣ ♠å✐
♣❤➛♥ tû ❝õ❛ [B, x] t tở C D rỗ õ t x C
x s s ữủ ợ ồ ♣❤➛♥ tû ❝õ❛ A ✈➔ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû ❝õ❛ [B, x] ✭t❤❡♦
q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ ❝õ❛ X ✮✱ ♥❣÷đ❝ ❧↕✐ t❤➻ t❛ ❝❤♦ x ∈ D. ❳➨t t➟♣ ❤ñ♣
A = A ∪ C. ❘ã r➔♥❣ A ❧➔ ♠ët ①➼❝❤ ❝õ❛ X ✈➔ ❧➔ ♠ët ①➼❝❤ ❝ü❝ ✤↕✐ ❝❤ù❛ A,
✈➻ ♠é✐ ♣❤➛♥ tû ❝õ❛ D ❦❤æ♥❣ s♦ s→♥❤ ữủ t t ợ ởt tỷ ừ C.
(ii) = (iii) x ởt tỷ tũ ỵ ừ t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ ❜ë ♣❤➟♥
X. ◆➳✉ x ❧➔ ❝ü❝ ✤↕✐ t❤➻ ♠➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ●✐↔ sû x ❦❤æ♥❣
♣❤↔✐ ❧➔ ♣❤➛♥ tû ❝ü❝ ✤↕✐✳ ❑❤✐ ✤â✱ t❤❡♦ ỗ ởt tỷ {x}
tr ởt ①➼❝❤ ❝ü❝ ✤↕✐ A ♥➔♦ ✤â ❝õ❛ X. ❚❤❡♦ ❣✐↔ tt tỗ t ởt
tr b ừ A, tự a ≤ b, ∀a ∈ A. ◆➳✉ b ❦❤æ♥❣ ♣❤↔✐ ❧➔ tỷ ỹ
tỗ t ởt tỷ c = b s❛♦ ❝❤♦ b ≤ c. ❚ø ✤➙② s✉② r❛ a ≤ c, ∀a ∈ A. ❱➟②


ữỡ ỡ ữủ ỵ tt t ủ



A {c} ❧➔ ♠ët ①➼❝❤ ♠ỵ✐ t❤ü sü ❝❤ù❛ A. ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❝ü❝
✤↕✐ ❝õ❛ A, ❞♦ ✤â b ❧➔ ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐ ❝õ❛ X.
(iii) =⇒ (iv)✿ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈➔ X ♠ët
t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ X t❤♦↔ ♠➣♥ ❣✐↔ t❤✐➳t ❝õ❛ ✭✐✈✮✳ ❈❤ó þ r➡♥❣✱ X
✈ỵ✐ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ ⊆ trð t❤➔♥❤ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳
✣➸ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ X ❝â ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣
♠å✐ ①➼❝❤ tr♦♥❣ X ✤➲✉ ❝â ❝➟♥ tr➯♥ tr♦♥❣ X. ❇➙② ❣✐í ❣å✐ V ❧➔ ❤đ♣ ❝õ❛ t➜t ❝↔

❝→❝ t➟♣ ❤ñ♣ tr♦♥❣ ♠ët ①➼❝❤ ❝õ❛ X. ❘ã r➔♥❣ V ❧➔ ♠ët ❝➟♥ tr➯♥ ❝õ❛ ①➼❝❤ ♥➔②
tr♦♥❣ t➟♣ ❤ñ♣ 2X . ❱✐➺❝ ❝á♥ ❧↕✐ ❝õ❛ t❛ ❧➔ ❝❤➾ r❛ V ∈ X. ●✐↔ sû {v1 , ..., vn }
❧➔ ♠ët t➟♣ ❤ñ♣ ❝♦♥✱ ❤ú✉ ❤↕♥ ♥➔♦ ✤â ❝õ❛ V. ❉♦ ♠é✐ vi t❤✉ë❝ ✈➔♦ ♠ët t➟♣
❤ñ♣ Ai õ tr t t tỗ t ♠ët t➟♣ ❤ñ♣✱ ❝❤➥♥❣ ❤↕♥
A1 , ❝❤ù❛ t➜t ❝↔ ♥❤ú♥❣ t➟♣ ❝á♥ ❧↕✐✳ ❙✉② r❛ {v1 , ..., vn } ∈ X. ❚❤❡♦ ❣✐↔ t❤✐➳t
❝õ❛ ✭✐✈✮ t❛ ✤✐ ✤➳♥ V ∈ X.
(iv) =⇒ ❚✐➯♥ ✤➲ ❝❤å♥✿ ❈❤♦ X ❧➔ ♠ët t ủ tũ ỵ t t ủ X tt
❝→❝ ♣❤➛♥ tû ❝õ❛ ♥â ❧➔ ♥❤ú♥❣ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ X t❤♦↔ ♠➣♥ ❚✐➯♥ ✤➲ ❝❤å♥✳
❘ã r➔♥❣ t➟♣ ❤ñ♣ ♥➔② ❧➔ ❦❤ỉ♥❣ ré♥❣✱ ✈➻ ♠å✐ t➟♣ ❤đ♣ ❝♦♥✱ ❤ú✉ ❤↕♥ ♣❤➛♥ tû
❝õ❛ X ✤➲✉ t❤✉ë❝ X, ❤ì♥ ♥ú❛ X ❧➔ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥
t❤❡♦ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠✳ ❱➜♥ ✤➲ ❝á♥ ❧↕✐ ❧➔ ❝❤ù♥❣ ♠✐♥❤ X ∈ X. ❚❤➟t ✈➟②✱ ❝❤♦
Γ = (As ) ởt tũ ỵ ừ X. t A = s As . ❱➻ As t❤♦↔ ♠➣♥ ❚✐➯♥ ✤➲
❝❤å♥ ♥➯♥ tr➯♥ õ tỗ t ồ s . õ t❛ ①→❝ ✤à♥❤ tr➯♥ A ♠ët
→♥❤ ①↕ ϕ s❛♦ ❝❤♦ tr➯♥ ♠é✐ As ♥â trị♥❣ ✈ỵ✐ ϕs . ❘ã r➔♥❣ ϕ ❧➔ ♠ët →♥❤ ①↕
❝❤å♥ ❝õ❛ A✳ ❱➟② A ∈ X ❧➔ ♠ët ❝➟♥ tr➯♥ ❝õ❛ ①➼❝❤ Γ tr♦♥❣ X. ỷ ử t
t ợ ự t tữỡ tü ♥❤÷ tr♦♥❣ ✭✐✐✐✮ =⇒ ✭✐✈✮ t❛ s✉②
r❛ tr♦♥❣ X ❝â ➼t ♥❤➜t ♠ët ♣❤➛♥ tû ❝ü❝ ✤↕✐ V. ●✐↔ sỷ V = X, tự tỗ t
ởt tỷ x ∈ X \ V. ❚ø ✤➙② s✉② r❛ ♥❣❛② r➡♥❣ V ∪ {x} ∈ X. ✣✐➲✉ ♥➔②
♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ ❝ü❝ ✤↕✐ ❝õ❛ V. ❱➟② V = X ✈➔ ✤à♥❤ ỵ ữủ ự
ừ

t
X {Ai }i∈I ❧➔ ♥❤ú♥❣ t➟♣ ❤đ♣✳ ❈❤ù♥❣ ♠✐♥❤ ❝→❝ ❝ỉ♥❣ t❤ù❝ s❛✉ ✤➙②✿
✭✐✮ X \ (∩i∈I Ai ) = ∪i∈I (X \ Ai ).


✷✵

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✭✐✐✮ X \ (∪i∈I Ai ) = ∩i∈I (X \ Ai ).
✷✮ ❈❤♦ f : X −→ Y ❧➔ ♠ët →♥❤ ①↕ ✈➔ C, D ❧➔ ❤❛✐ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ Y.
❈❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❧➔ ✤ó♥❣✳
✭✐✮ f −1 (C ∪ D) = f −1 (C) ∪ f −1 (D).
✭✐✐✮ f −1 (C ∩ D) = f −1 (C) ∩ f −1 (D).
✭✐✐✐✮ f −1 (Y \ C) = X \ f −1 (C).
✸✮ ❈❤♦ f : X −→ Y ✈➔ g : Y −→ Z ❧➔ ♥❤ú♥❣ s♦♥❣ →♥❤✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
g ◦ f ❧↕✐ ❧➔ s♦♥❣ →♥❤ ✈➔ (g ◦ f )−1 = f −1 ◦ g −1 .
✹✮ ❈❤♦ f : X −→ X ❧➔ ♠ët →♥❤ ①↕ ①→❝ ✤à♥❤ tr➯♥ t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ X.
❈❤ù♥❣ ♠✐♥❤ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ f ❧➔ ✤ì♥ →♥❤✳
✭✐✐✮ f ❧➔ t♦➔♥ →♥❤✳
✭✐✐✐✮ f ❧➔ s♦♥❣ →♥❤✳
✺✮ ❍➣② ①→❝ ✤à♥❤ sè ❝→❝ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ ♠ët t➟♣ ❝â n ♣❤➛♥ tû✳
✻✮ ❚➼♥❤ sè ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣ 2X ❜✐➳t r➡♥❣ X ❝â n ♣❤➛♥ tû✳
✼✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ t➟♣ ❤ñ♣ sè tü ♥❤✐➯♥ ◆ ✈➔ t➟♣ ❤ñ♣ sè ỳ t õ ũ
ỹ ữủ
ỵ X Y ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ →♥❤ ①↕ tø t➟♣ ❤đ♣ Y ✈➔♦ t➟♣ ❤đ♣
X. ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ✹ t ủ tũ ỵ A, B, C, D t t➼♥❤ ❝❤➜t
A ∼ B ✈➔ C ∼ D t❛ ❧✉æ♥ õ AC B D .
ỵ 2X t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ t➟♣ ❤ñ♣ ❝♦♥ ❝õ❛ t➟♣ ❤đ♣ X. ❈❤ù♥❣
♠✐♥❤ r➡♥❣ ❧ü❝ ❧÷đ♥❣ ❝õ❛ ❤❛✐ t➟♣ ❤đ♣ X ✈➔ 2X ❧➔ ❦❤→❝ ♥❤❛✉✳
✶✵✮ ❈❤♦ ≥ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ t➟♣ ❤ñ♣ X. ❈❤ù♥❣ ♠✐♥❤
❝→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✐✮ ▼é✐ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ Y ❝õ❛ X ❝❤ù❛ ➼t ♥❤➜t ♠ët ♣❤➛♥ tû ❝ü❝
t✐➸✉✳


ữỡ ỡ ữủ ỵ tt t ủ
ộ ①➼❝❤ ❣✐↔♠ ❝→❝ ♣❤➛♥ tû ❝õ❛ X

x1 ≥ x2 ≥ ... xn ...
ứ tự tỗ t ởt sè tü ♥❤✐➯♥ k s❛♦ ❝❤♦ xk = xk+1 = ....

✷✶


✷✷

●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐


ữỡ

õ
ỵ tt õ tở ởt tr ỵ tt ữủ t tr sợ
t rt ú ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ♥❤➜t tr♦♥❣ ✤↕✐ sè✳ ◆❣♦➔✐
❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ♥❤â♠ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣
♥➔②✱ t❛ s➩ ✤÷❛ t❤➯♠ ❦❤→✐ ♥✐➺♠ ♣❤↕♠ trị✱ ❦❤ỉ♥❣ ♥❤ú♥❣ ♥❤➡♠ ❧➔♠ ❣å♥ ❤ì♥
❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➲ ♥❤â♠ tü ❞♦✱ t➼❝❤ ✈➔ ✤è✐ t➼❝❤ tr♦♥❣ ♥❤â♠✱ ♠➔ ♥â ❝á♥ r➜t
❤ú✉ ➼❝❤ ❝❤♦ t➜t ❝↔ ❝→❝ ❝❤÷ì♥❣ ✈➲ s❛✉ tr♦♥❣ ❜➔✐ ❣✐↔♥❣ ♥➔②✳

§

✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ư ✈➲ ♥❤â♠

✶✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ▼ët t➟♣ ❤đ♣ G ✤÷đ❝ ồ ởt õ tỗ t ởt
tứ t➼❝❤ ❉❡s❝❛rt❡s G × G ✈➔♦ G ✭↔♥❤ ❝õ❛ ♣❤➛♥ tỷ (a, b) G ì G, ợ
a, b ỳ tỷ tũ ỵ ừ G, q t ỵ ab
õ G s ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❤â♠ ♥❤➙♥✮ t❤♦↔ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙②✳

✭G1 ✮ ❑➳t ❤ñ♣✿ a(bc) = (ab)c, ∀a, b, c G.
G2 õ ỡ ỗ t↕✐ ♠ët ♣❤➛♥ tû e ∈ G s❛♦ ❝❤♦ ae = ea = a, ∀a ∈ G.
✭G3 ✮ ❈â ♥❣❤à❝❤ ợ ộ tỷ a G ổ tỗ t↕✐ ♠ët ♣❤➛♥ tû
b ∈ G s❛♦ ❝❤♦ ab = ba = e. P❤➛♥ tû ab ✤÷đ❝ ❣å✐ ❧➔ t➼❝❤ ❝õ❛ a ✈➔ b ✈➔
→♥❤ ①↕ ①→❝ ✤à♥❤ t➼❝❤ ð tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ t♦→♥ tr➯♥ ♥❤â♠ ♥❤➙♥ G.
P❤➛♥ tû e tr♦♥❣ (G2 ) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ G. P❤➛♥ tû b
tr♦♥❣ (G3 ) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ a tr♦♥❣ G ỵ
a1 . õ ởt õ G trữợ t ỵ tỷ ỡ



●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐

✷✹

❧➔ e ♥❤÷ ð tr➯♥✳ ◆➳✉ ❝â ♥❤✐➲✉ ♥❤â♠ G, H, ... t❛ s➩ ❞ò♥❣ ỵ
eG , eH , ... ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ ❝→❝ ♥❤â♠ G, H, ... ◆➳✉ ♣❤➨♣
t♦→♥ tr➯♥ G t❤♦↔ ♠➣♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥
✭G4 ✮ ●✐❛♦ ❤♦→♥✿ ab = ba, ∀a, b ∈ G, t❤➻ ♥❤â♠ G ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ❆❜❡❧✳
◆❤â♠ ❆❜❡❧ ♥❤✐➲✉ ❦❤✐ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ❣✐❛♦ ❤♦→♥✳
❚❤ỉ♥❣ t❤÷í♥❣ ♥❣÷í✐ t❛ q✉❡♥ ✈✐➳t ♣❤➨♣ t♦→♥ tr➯♥ ♠ët ♥❤â♠ ❆❜❡❧ t❤❡♦
❧è✐ ❝ë♥❣✿ a + b ✈➔ ❣å✐ ❧➔ tê♥❣ ❝õ❛ a ✈➔ b tr G. õ tữỡ ự
ợ tỷ ỡ e tr õ tỷ ổ ỵ ❤✐➺✉ ✵✱
✈➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ a−1 s➩ ❧➔ ♣❤➛♥ tỷ ố ỵ a tr ởt
õ ở
ởt õ G ✤÷đ❝ ❣å✐ ❧➔ ❤ú✉ ❤↕♥ ❤❛② ✈ỉ ❤↕♥ ♥➳✉ t➟♣ ❤đ♣ G ❧➔ ❤ú✉ ❤↕♥
❤❛② ✈ỉ ❤↕♥ ♣❤➛♥ tû✳ ❚r÷í♥❣ ❤ñ♣ ♥❤â♠ G ❧➔ ❤ú✉ ❤↕♥ t❤➻ sè ♣❤➛♥ tû ừ G
ữủ ồ ừ õ õ ỵ ❤✐➺✉ ❧➔ | G |✳

✶✳✷✳ ❚➼♥❤ ❝❤➜t✳ ❚❛ s➩ ✤÷❛ r❛ ð ✤➙② ♥❤ú♥❣ t➼♥❤ ❝❤➜t ✤ì♥ ❣✐↔♥ ♥❤➜t ❝õ❛

♠ët ♥❤â♠ G✿
✶✮ P❤➛♥ tû ✤ì♥ ✈à e ❝õ❛ G ✤÷đ❝ ①→❝ ✤à♥❤ ❞✉② ♥❤➜t✳
❚❤➟t ✈➟②✱ ♥➳✉ e ❝ô♥❣ ❧➔ ♠ët ♣❤➛♥ tû ✤ì♥ ✈à✱ s✉② r❛ e = ee = e .
✷✮ ▼é✐ ♣❤➛♥ tû a ❝õ❛ G ❝❤➾ ❝â ❞✉② ♥❤➜t ♠ët ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ a−1 ✱
❤ì♥ ♥ú❛ e−1 = e, (a−1 )−1 = a ✈➔ (ab)−1 = b−1 a−1 .
❚❤➟t ✈➟②✱ ♥➳✉ b, c ❧➔ ❤❛✐ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ a✱ s✉② r❛

b = be = b(ac) = (ba)c = ec = c.
P❤➛♥ ❝á♥ ❧↕✐ ✤÷đ❝ s r ởt tữỡ tỹ
t ữợ a, b, x ỳ tỷ tũ ỵ ừ G. ❚ø ❝→❝
✤➥♥❣ t❤ù❝ xa = xb ❤♦➦❝ ax = bx ✤➲✉ s✉② r❛ a = b.
❚❤➟t ✈➟②✱ ♥❤➙♥ ✈➔♦ ❜➯♥ tr→✐ ❤❛✐ ✈➳ ❝õ❛ ✤➥♥❣ t❤ù❝ xa = xb ✈ỵ✐ x−1 ,
s✉② r❛

a = ea = (x−1 x)a = x−1 (xa) = x−1 xb = (x−1 x)b = eb = b.


❈❤÷ì♥❣ ✷✳ ◆❤â♠

✷✺

P❤➛♥ ❝á♥ ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✳
✹✮ ❚r♦♥❣ G ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ xa = b ✈➔ ax = b ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳
❚❤➟t ✈➟②✱ x = ba−1 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ ✈➔ ❧➔ ❞✉② ♥❤➜t
❞♦ t➼♥❤ ❝❤➜t ✸✳
✺✮ ❈❤♦ a ∈ G, t❛ ①→❝ ✤à♥❤ a0 = e✱ an = a...a ✭♥✲♣❤➛♥ tû a✮ ✈➔ a−n =
(a−1 )n . ❑❤✐ ✤â t❛ ✤÷đ❝ an am = an+m , (an )m = anm , ❤ì♥ ♥ú❛✱ ♥➳✉ G
❧➔ ❆❜❡❧ t❤➻ (ab)n = an bn , ∀a, b ∈ G.
❈→❝ ❝ỉ♥❣ t❤ù❝ tr➯♥ ✤÷đ❝ s✉② r❛ ❞➵ ❞➔♥❣ tø ✤à♥❤ ♥❣❤➽❛✳

✶✳✸✳ ❱➼ ❞ư✳
✶✮ ❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ Z ✈ỵ✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ❤❛✐ sè ♥❣✉②➯♥
t❤ỉ♥❣ t❤÷í♥❣ ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠ ❆❜❡❧✳ ❈ơ♥❣ ♥❤÷ ✈➟②✱ t➟♣ ❤đ♣ t➜t
❝↔ ❝→❝ sè ❤ú✉ t✛ ❦❤→❝ ❦❤ỉ♥❣ ợ tổ tữớ
t ởt õ
❈❤♦ G ❧➔ ♠ët ♥❤â♠ ✈➔ X ❧➔ ♠ët t➟♣ ❦❤ỉ♥❣ ré♥❣✳ ❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝
→♥❤ ①↕ tø X G ỵ M (X, G), ởt õ ợ t
ữủ ữ s ố ợ tũ ỵ f, g M (X, G), →♥❤
①↕ t➼❝❤ f g ✤÷đ❝ ①→❝ ✤à♥❤ q✉❛ ❝ỉ♥❣ t❤ù❝ (f g)(x) = f (x)g(x), ∀x ∈ X.
❑❤✐ ✤â✱ ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ M (X, G) ❧➔ →♥❤ ①↕ ❝❤♦ ù♥❣ ♠å✐ ♣❤➛♥ tû
❝õ❛ X ❧➯♥ ♣❤➛♥ tû ✤ì♥ ✈à ❝õ❛ G ✈➔ →♥❤ ①↕ ♥❣❤à❝❤ ✤↔♦ f −1 ✤÷đ❝ ①→❝
✤à♥❤ ❜ð✐ (f −1 )(x) = (f (x))−1 , ∀x ∈ X. ❚❛ ❝ơ♥❣ ❞➵ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝
r➡♥❣ M (X, G) ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ G ❧➔ ♠ët ♥❤â♠ ❆❜❡❧✳
✸✮ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ✈✉æ♥❣ ❝➜♣ n ❝â ✤à♥❤ t❤ù❝ ❦❤→❝ ❦❤ỉ♥❣
GL(n,❘✮ tr➯♥ t➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ ❘ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ♠❛ tr➟♥ t❤ỉ♥❣
t❤÷í♥❣ ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠✱ ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ t✉②➳♥ t➼♥❤ ✤➛② ✤õ ❝➜♣
n ✈ỵ✐ ❤➺ sè tr♦♥❣ ❘✳ ❘ã r➔♥❣ ♥❤â♠ ♥➔② ❧➔ ❆❜❡❧ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ n = 1.
✹✮ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ s♦♥❣ →♥❤ tứ X
õ ỵ S(X), ởt õ ♥❤➙♥ ✈ỵ✐ ♣❤➨♣ t♦→♥ ♥❤➙♥
❝❤➼♥❤ ❧➔ ♣❤➨♣ ❧➜② ❤đ♣ t❤➔♥❤ ❤❛✐ →♥❤ ①↕ ✤➣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣
❈❤÷ì♥❣ ✶✱ ✭✭✷✳✶✮✱ ✭✈✮✮✳ P❤➛♥ tû ✤ì♥ ✈à ❝õ❛ S(X) ❧➔ →♥❤ ①↕ ỗ t
1X tỷ ừ ởt s →♥❤ f ∈ S(X) ❝❤➼♥❤ ❧➔ →♥❤