Quan hệ hai ngôi và một số bài toán liên quan

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
---------------------------

BÙI THỊ THU THỦY

QUAN HỆ HAI NGƠI
VÀ MỘT SỐ BÀI TỐN LIÊN QUAN

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
---------------------------

BÙI THỊ THU THỦY

QUAN HỆ HAI NGƠI
VÀ MỘT SỐ BÀI TỐN LIÊN QUAN
Chun ngành: Phương pháp Toán sơ cấp
Mã số: 8 46 01 13

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
TS. Trần Nguyên An

THÁI NGUYÊN - 2019

▼ö❝ ❧ö❝
▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✶✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✷✳ ✣↕✐ sè tê ❤ñ♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

❈❤÷ì♥❣ ✷✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✈➔ ♠ët sè ❜➔✐ t♦→♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶✳ ✣➳♠ ♠ët sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳ ⑩♥❤ ①↕ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✸✳ P❤➙♥ ❤♦↕❝❤✱ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✹✳ ✣➳♠ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✈➔ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ❜➢❝ ❝➛✉ ✳ ✳ ✳ ✳ ✳ ✳

✶✺
✷✸
✷✼
✸✷

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✐


▼ð ✤➛✉
❈❤♦ A✱ B ❧➔ ❝→❝ t➟♣ ❤ñ♣✳ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tø t➟♣ A ✤➳♥ t➟♣ B
❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ t➟♣ t➼❝❤ ✣➲ ❝→❝ A × B ✳ ✣➦❝ ❜✐➺t✱ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐
tø A ✤➳♥ A ✤÷đ❝ ❣å✐ ❧➔ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ tr➯♥ A✳ ◆➳✉ R ❧➔ ♠ët q✉❛♥
❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣ A ✈➔ (a, b) ∈ R t❤➻ t❛ ❦➼ ❤✐➺✉ aRb ✭✤å❝ ❧➔ a ❝â q✉❛♥

❤➺ R ✈ỵ✐ b✮✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤→❝ ♥❤❛✉ ừ
t ồ số ố ồ ồ ỵ tt ỗ t ồ t
ởt trữớ ủ ❜✐➺t ❝õ❛ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳ ❈→❝ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤✐➸♥
❤➻♥❤ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ♣❤ê t❤ỉ♥❣ ❧➔ ✧q✉❛♥ ❤➺ t q ỗ
ữ q ợ ỡ q ❤➺ s♦♥❣ s♦♥❣✧✱ ❤➔♠ sè✱ ✳✳✳✳ ❚❛ t❤÷í♥❣ q✉❛♥
t➙♠ ✤➳♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ①↕ ✭r❡❢❧❡①✐✈❡✮✱ ✤è✐ ①ù♥❣
✭s②♠♠❡tr✐❝✮✱ ❜➢❝ ❝➛✉ ✭tr❛♥s✐t✐✈❡✮✱ ❜➜t ✤è✐ ①ù♥❣ ✭❛s②♠♠❡tr✐❝✮✱ ♣❤↔♥ ✤è✐ ①ù♥❣
✭❛♥t✐s②♠♠❡tr✐❝✮✱ ❜➜t ♣❤↔♥ ①↕ ✭✐rr❡❢❧❡①✐✈❡✮✳ ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔
t➻♠ ❤✐➸✉ ♠ët sè ❜➔✐ t♦→♥ tê ❤đ♣ ✈➲ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐✳ ❚➔✐ ❧✐➺✉ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥
✈➠♥ ❧➔ ❣✐↔✐ ♠ët sè ❜➔✐ t➟♣ tr♦♥❣ ❬✼❪✱ ❬✷❪ ✈➔ ❜➔✐ ❜→♦ ❬✻❪✳
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥
t❤ù❝ ❝❤✉➞♥ ❜à ỵ tt q ổ q tữỡ ✤÷ì♥❣✱ q✉❛♥ ❤➺
t❤ù tü✱ →♥❤ ①↕ ✈➔ ♠ð ✤➛✉ ✈➲ ỵ tt tờ ủ tự
ữỡ ữ ố ợ t tự ừ ữỡ tự ợ
õ ự ử tr♦♥❣ ❣✐↔✐ t♦→♥ ♣❤ê t❤ỉ♥❣✳ ❈❤÷ì♥❣ ♥➔② ❝❤õ ②➳✉ t❤❛♠
❦❤↔♦ t❤❡♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✷❪✳
❈❤÷ì♥❣ ✷ t❤❡♦ t➔✐ ❧✐➺✉ ❬✻✱ ✼❪ ❧➔ ❝❤÷ì♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔②
✈➲ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳ ❇➢t ✤➛✉ ❧➔ ❜➔✐ t♦→♥ ✤➳♠
♠ët sè q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t✳ ❈ơ♥❣ ❝➛♥ ♣❤↔✐ ♥â✐ t❤➯♠ r➡♥❣ q✉❛♥ ❤➺ ❤❛✐
♥❣æ✐ ①✉➜t ♣❤→t tø ♥❤ú♥❣ ✈➜♥ ✤➲ tr♦♥❣ t sỡ ữ ỵ tt t
ỵ tt ỗ ❞÷✧ ♥❤÷♥❣ ✈➻ ❦❤✉ỉ♥ ❦❤ê ❝õ❛ ❧✉➟♥ ✈➠♥ t→❝ ❣✐↔ ❝❤➾ ❦❤❛✐ t❤→❝
♠ët sè ❜➔✐ t♦→♥ sì ❝➜♣ ❧✐➯♥ q✉❛♥ t tờ ủ ởt ữ ỵ ừ



✈➠♥ ❧➔ t→❝ ❣✐↔ ❝è ❣➢♥❣ t➻♠ ❤✐➸✉ ♥❤✐➲✉ ❝→❝❤ ❣✐↔✐✱ ❝→❝❤ t✐➳♣ ❝➟♥ ❦❤→❝ ♥❤❛✉ ❝õ❛
♠ët ❜➔✐ t♦→♥✱ ♠ët ✈➜♥ ✤➲✳ ✣➳♠ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❧➔ →♥❤ ①↕ ✈➔ ❝→❝ tr÷í♥❣
❤đ♣ ✤➦❝ ❜✐➺t ✭✤ì♥ →♥❤✱ s♦♥❣ →♥❤✱ t♦➔♥ →♥❤✮ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♠ư❝ t❤ù
❤❛✐ ❝õ❛ ❝❤÷ì♥❣✳ ❱✐➺❝ ự số t ủ ỵ t t ❤✐➸✉ sè ❙t✐r❧✐♥❣
❧♦↕✐ ❤❛✐ ✈➔ ❜➔✐ t♦→♥ ✤➳♠ sè ♣❤➙♥ ❤♦↕❝❤ ♠ët t➟♣ ❤đ♣✳ ❱➜♥ ✤➲ ♥➔② ✤÷đ❝ tr➻♥❤

❜➔② tr♦♥❣ ♠ư❝ t❤ù ❜❛ ❝õ❛ ❝❤÷ì♥❣✳ ▼ư❝ ❝✉è✐ ❝õ❛ ❝❤÷ì♥❣ t➻♠ ❤✐➸✉ sè q✉❛♥ ❤➺
t÷ì♥❣ ✤÷ì♥❣✱ sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ ✭❧✐➯♥ ❤➺ ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü✮ t❤❡♦ ❜➔✐ ❜→♦
❬✻❪✳ ú ỵ r số q tữỡ ữỡ tr t n ♣❤➛♥ tû ❝❤➼♥❤ ❧➔ sè ♣❤➙♥
❤♦↕❝❤✱ sè ❇❡❧❧ t❤ù n✳
❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✱ tỉ✐ ♥❤➟♥ ✤÷đ❝ sü ữợ ú ù
t t ừ r ❆♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐
◆❣✉②➯♥✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛②✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ qỵ t ổ ợ
ồ õ ❤å❝ ❚♦→♥ ❦❤â❛ ✶✶❇ ✭✷✵✶✼✲✷✵✶✾✮ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝
✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➣ tr✉②➲♥ t❤ö ✤➳♥ ❝❤♦ tỉ✐ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤
♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳
▲í✐ ❝✉è✐ ❝ò♥❣✱ t→❝ ❣✐↔ ♠✉è♥ ❞➔♥❤ ✤➸ tr✐ ➙♥ ❜è ♠➭ ✈➔ ❣✐❛ ✤➻♥❤ ✈➻ ✤➣
❝❤✐❛ s➫ ♥❤ú♥❣ ❦❤â ❦❤➠♥ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳
❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✽ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾
❚→❝ ❣✐↔

❇ò✐ ❚❤à ❚❤✉ ❚❤õ②

✷


❈❤÷ì♥❣ ✶

❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à
✶✳✶✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tø t➟♣ A ✤➳♥ t➟♣ B ❧➔ ♠ët t➟♣

❝♦♥ ❝õ❛ t➟♣ t➼❝❤ ✣➲ ❝→❝ A × B ✳ ✣➦❝ ❜✐➺t✱ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ tø A ✤➳♥ A
✤÷đ❝ ❣å✐ ❧➔ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ A✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ♠ët q✉❛♥ ❤➺ ❤❛✐
♥❣æ✐ tr➯♥ ♠ët t➟♣ A ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ t➟♣ A2✳

❚❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❝→❝ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ❜➡♥❣ ❝→❝ ❝❤ú ❝→✐ R ✭❤❛②
S, T, U, V, . . .✮✳ ◆➳✉ R ❧➔ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣ A ✈➔ (a, b) ∈ R
t❤➻ t❛ ❦➼ ❤✐➺✉ aRb ✭✤å❝ ❧➔ a ❝â q✉❛♥ ❤➺ R ✈ỵ✐ b✱ ❤♦➦❝ ♥â✐ t➢t ❧➔ a R b✮✳ ❑❤✐
(a, b) ∈
/ R t❤➻ t❛ ✈✐➳t aRb ✭✤å❝ ❧➔ a ❦❤æ♥❣ õ q R ợ b tữớ
q t ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ❝õ❛ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ●✐↔ sû R ⊆ A × A ❧➔ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐✳ ◗✉❛♥ ❤➺ ❤❛✐
♥❣ỉ✐ R ✤÷đ❝ ❣å✐ ❧➔
✭✐✮ P❤↔♥ ①↕ ✭r❡❢❧❡①✐✈❡✮ ♥➳✉ ∀a ∈ A, ((a, a) ∈ R)❀
✭✐✐✮ ✣è✐ ①ù♥❣ ✭s②♠♠❡tr✐❝✮ ♥➳✉ ∀a, b ∈ A, ((a, b) ∈ R t❤➻ (b, a) ∈ R)❀
✭✐✐✐✮ ❇➢❝ ❝➛✉ ✭tr❛♥s✐t✐✈❡✮ ♥➳✉ ∀a, b, c ∈ A, ((a, b) ∈ R ∧ (b, c) ∈ R t❤➻
(a, c) ∈ R)❀
✭✐✈✮ ❇➜t ✤è✐ ①ù♥❣ ✭❛s②♠♠❡tr✐❝✮ ♥➳✉ ∀a, b ∈ A, ((a, b) ∈ R t❤➻ (b, a) ∈/ R)❀
✭✈✮ P❤↔♥ ✤è✐ ①ù♥❣ ✭❛♥t✐s②♠♠❡tr✐❝✮ ♥➳✉ ∀a, b ∈ A, [((a, b) ∈ R∧(b, a) ∈ R)
t❤➻ a = b]❀
✭✈✐✮ ❇➜t ♣❤↔♥ ①↕ ✭✐rr❡❢❧❡①✐✈❡✮ ♥➳✉ ∀a ∈ A, ((a, a) ∈/ R)✳
❱➼ ❞ö ✶✳✶✳✸✳ ❈❤♦ A = {1, 2, 3}✳ ❳➨t ❝→❝ q✉❛♥ ❤➺
R1 = {(1, 1), (2, 2), (3, 3)}✱
✸


R2 = {(1, 1), (1, 2), (1, 3)}✱

R3 = A × A✱

R4 = {(2, 2), (3, 3), (1, 2)}✳ ❚❛
R1 R2 R3 R4
P❤↔♥ ①↕ T F T F
✣è✐ ①ù♥❣ T F T F
❇➢❝ ❝➛✉ T T T T

❝â ✈ỵ✐ ❚ ỵ r ỵ s

ử A = {1, 2, 3, 4}✳ ❳➨t ❝→❝ q✉❛♥ ❤➺✳
R1 = {(1, 1), (2, 2), (3, 3), (2, 1), (4, 3), (3, 2)}✱
R2 = A × A✱

R3 = {(1, 1), (2, 2), (3, 3), (2, 1), (4, 3), (4, 1), (3, 2)}✱

R4 = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (4, 3), (3, 4)}✳

P❳ ✣❳ P P

R1
R2
R4
R5

F
T
T
T

F
T
T
T

F
F

F
F

T
F
T
F

F
F
F
F

F
T
T
T

ú ỵ t t tt P P❤↔♥ ①↕✱ ✣❳ ❂ ✣è✐ ①ù♥❣✱ P✣❳ ❂ P❤↔♥ ✤è✐
①ù♥❣✱ ❇P❳ ❂ ❇➜t ♣❤↔♥ ①↕✱ ❇❈ ❂ ❇➢❝ ❝➛✉✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ✭✐✮ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣ A ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥
❤➺ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ♥â ❝â ❝→❝ t➼♥❤ ❝❤➜t ♣❤↔♥ ①↕✱ ✤è✐ ①ù♥❣ ✈➔ ❜➢❝ ❝➛✉✳ ❚❤❡♦
tr✉②➲♥ t❤è♥❣✱ ❝→❝ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ t❤÷í♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ ❞➜✉ ∼ .
✭✐✐✮ ❈❤♦ ∼ ❧➔ ♠ët q✉❛♥ ❤➺ tữỡ ữỡ tr t A. ợ ộ a A, t ồ
ợ tữỡ ữỡ ừ a ố ợ q t÷ì♥❣ ✤÷ì♥❣ ∼✱ ❦➼ ❤✐➺✉ ❜ð✐ [a]∼ ✭❤❛②
[a]✱ ❤❛② a✱ ❤❛② C(a)✮✱ ✤â ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ A ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
[a] = {b ∈ A | b ∼ a}.

ỗ tớ t ủ tt ợ tữỡ ữỡ ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ A ✤÷đ❝
❣å✐ ❧➔ t➟♣ t❤÷ì♥❣ ❝õ❛ A t❤❡♦ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ∼✱ ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔

A/ ∼ . ◆❤÷ ✈➟②✱ t❛ ❝â ❜✐➸✉ ❞✐➵♥
A/ ∼ = {[a] | a ∈ A}.

❱➼ ❞ö ✶✳✶✳✻✳ ❈❤♦ m ❧➔ ♠ët sè tü ♥❤✐➯♥ ❧ỵ♥ ❤ì♥ ✶✳ ❚r➯♥ t➟♣ Z ❝→❝ sè ♥❣✉②➯♥
t❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ ổ R ữ s ợ ồ a, b Z, t❛ ♥â✐
aRb ⇔ m|(a − b).
✹


ữủ ồ q ỗ ữ t ổ m ỏ ồ q
ỗ ữ m a ỗ ữ b t ổ m t❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❧➔
a ≡ b (mod m).

❚❛ t❤➜② ✤â ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ Z✳ ợ a Z ợ tữỡ
ữỡ ừ a ữủ a ữủ ồ ởt ợ t ữ t ổ
m ợ a tữỡ ừ Z ố ợ q ỗ ữ m
ữủ ❦➼ ❤✐➺✉ ❜ð✐ Zm ✈➔ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝→❝ ợ t ữ t ổ m
t ủ ợ t❤➦♥❣ ❞÷ ♠♦❞✉❧♦ m✮✳ ❈❤♦ a ∈ Z✱ ❦❤✐ ✤â t❛ ❝â
a = {b ∈ Z | b ≡ a (mod m)} = {b ∈ Z | b − a

❝❤✐❛ ❤➳t ❝❤♦ m}.

❱ỵ✐ ♠é✐ a ∈ Z ✤➣ ❝❤♦✱ t❛ ❧✉æ♥ ❝â ❜✐➸✉ ❞✐➵♥ a = mq+r tr♦♥❣ ✤â 0 r m1
t ỵ ợ ữ ❑❤✐ ✤â b − a = b − mq − r✱ ♥➯♥ t❛ ❝â
a = {b ∈ Z | b−mq−r ❝❤✐❛

❤➳t ❝❤♦ m} = {b ∈ Z

| b−r ❝❤✐❛

❤➳t ❝❤♦ m} = r.

❍ì♥ ♥ú❛✱ ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ i, j s❛♦ ❝❤♦ 0 ≤ i < j ≤ m − 1 t❛ ❧✉æ♥ ❝â
0 < j − i < m ♥➯♥ j − i ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ m✳ ❉♦ ✤â i ≡ j (mod m)✱ ♥➯♥
i = j. t t Zm ỗ m tỷ ổ ởt ❦❤→❝ ♥❤❛✉ ♥❤÷ s❛✉✿
Zm = {0, 1, . . . , m 1}.

ú ỵ r a = mq + r t❤➻ a = r. ❱➻ t❤➳ ✈ỵ✐ q1, . . . , qm m số
tý ỵ t❛ ❧✉æ♥ ❝â
Zm = {q1 m, q2 m + 1, . . . , qm m + m − 1}.

❈❤➥♥❣ ❤↕♥ Z3 = {0, 1, 2} = {6, −2, 8}✳
✣à♥❤ ỵ ữợ t ỵ ừ q tữỡ ữỡ rữợ
t ỵ ú t ❝➛♥ ❦❤→✐ ♥✐➺♠ s❛✉✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ❈❤♦ A ❧➔ ♠ët t➟♣ ❤ñ♣✳ ❚❛ ❣å✐ ♠ët ♣❤➙♥ ❤♦↕❝❤ ✭❤❛② ♠ët
sü ❝❤✐❛ ❧ỵ♣✮ tr➯♥ A ❧➔ ♠ët ♣❤➨♣ ♣❤➙♥ ❝❤✐❛ t➟♣ A t❤➔♥❤ ♠ët ❤å ❝→❝ t➟♣ ❝♦♥
❦❤→❝ ré♥❣ {Ai}i∈I t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿
✭✐✮ Ai ∩ Aj = ∅ ✈ỵ✐ ♠å✐ i, j ∈ I, i = j.
✭✐✐✮ A = Ai.
i∈I

✺


ỵ A ởt q tữỡ ✤÷ì♥❣ tr➯♥ t➟♣ A✳ ❑❤✐ ✤â ❝→❝

♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✳
✭✐✮ [a] = ∅ ✈ỵ✐ ♠å✐ a ∈ A.
✭✐✐✮ A = [a].
a∈A

✭✐✐✐✮ [a] = [b] ❤♦➦❝ [a] ∩ [b] = ∅ ✈ỵ✐ ♠å✐ a, b ∈ A.
❱➻ t❤➳ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ∼ ①→❝ ✤à♥❤ ♠ët ♣❤➙♥ ❤♦↕❝❤ tr➯♥ A. ◆❣÷đ❝
❧↕✐✱ ♥➳✉ {Ai}i∈I ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ tr➯♥ A t❤➻ tỗ t t ởt q
tữỡ ữỡ tr A s ộ Ai ởt ợ tữỡ ữỡ
✶✳✶✳✾✳ ✭✐✮ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ ♠ët t➟♣ ❤đ♣ ✤÷đ❝ ❣å✐ ❧➔
q✉❛♥ ❤➺ t❤ù tü ♥➳✉ ♥â ❝â ❝→❝ t➼♥❤ ❝❤➜t ♣❤↔♥ ①↕✱ ♣❤↔♥ ✤è✐ ①ù♥❣✱ ✈➔ ❜➢❝ ❝➛✉✳
◗✉❛♥ ❤➺ t❤ù tü t❤÷í♥❣ ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ ❞➜✉ ✧≤✧ ✭✤å❝ ❧➔ ✧♥❤ä ❤ì♥ ❤♦➦❝
❜➡♥❣✧✮✳ ❑❤✐ a ≤ b t❤➻ t❛ ❝ô♥❣ ✈✐➳t b ≥ a.
✭✐✐✮ ❑❤✐ tr➯♥ ♠ët t➟♣ ❤ñ♣ A ❝â ♠ët q✉❛♥ ❤➺ t❤ù tü ≤ t❤➻ t❛ ♥â✐ A ❧➔ ♠ët
t➟♣ ❤đ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ð✐ ≤✳
❱➼ ❞ư ✶✳ ✭✐✮ ◗✉❛♥ ❤➺ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ ≤ t❤ỉ♥❣ t❤÷í♥❣ ✭t❛ ✤➣ ❜✐➳t ð ♣❤ê
t❤ỉ♥❣✮ ❧➔ q✉❛♥ ❤➺ t❤ù tü tr➯♥ ❝→❝ t➟♣ N✱ Z✱ Q✱ ✈➔ R✳
✭✐✐✮ ◗✉❛♥ ❤➺ ❜❛♦ ❤➔♠ ⊆ tr➯♥ t➟♣ 2A ✭t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ A✮ ❧➔ ♠ët
q✉❛♥ ❤➺ t❤ù tü✳
✭✐✐✐✮ ◗✉❛♥ ❤➺ ❝❤✐❛ ❤➳t ✧⑤✧ tr➯♥ t➟♣ N∗ = N \ {0} ❧➔ ♠ët q✉❛♥ ❤➺ tự tỹ
t A t tũ ỵ ừ t➟♣ N∗✳ ❑❤✐ ✤â q✉❛♥ ❤➺ ❝❤✐❛ ❤➳t ✧⑤✧ tr➯♥
t➟♣ A ❝ơ♥❣ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü tr➯♥ A✳
▼ư❝ ❝✉è✐ ❣✐ỵ✐ t❤✐➺✉ ❧ỵ♣ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t ❧➔ →♥❤ ①↕✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵✳ ❈❤♦ R ❧➔ q✉❛♥ ❤➺ ✷ ♥❣æ✐ tø A ✤➳♥ B ✳ ❑❤✐ ✤â ♠✐➲♥
ừ R R ỵ D(R) ữủ ✤à♥❤ ♥❣❤➽❛ ❧➔ t➟♣
{x|x ∈ A; ∃y ∈ A, (x, y) R}.

ừ R R ỵ im(R) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ t➟♣
{y|y ∈ B, ∃x ∈ A, (x, y) ∈ R}.

✻


❱➼ ❞ö ✶✳✶✳✶✶✳ ❈❤♦ A = {4, 5, 7, 8, 9} ✈➔ B = {16, 18, 20, 22}✱

R = {(4, 16), (4, 20), (5, 20), (8, 16), (9, 18)}✳

❑❤✐ ✤â R ❧➔ q✉❛♥ ❤➺ ✷ ♥❣æ✐ tø
A ✤➳♥ B ✱ D(R) = {4, 5, 8, 9}✱ im(R) = {16, 18, 20}✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✷✳ ✭✐✮ ❈❤♦ A✱ B ❧➔ ❝→❝ t➟♣ ❦❤→❝ ré♥❣✳ ▼ët q✉❛♥ ❤➺ ❤❛✐
♥❣æ✐ f tø A ✤➳♥ B ✤÷đ❝ ❣å✐ ❧➔ ♠ët →♥❤ ①↕ ♥➳✉
✭✶✮ D(f ) = A ✭tù❝ ❧➔ ∀a ∈ A, ∃b ∈ B, (a, b) ∈ f ✮✱
✭✷✮ ❱ỵ✐ ♠å✐ (a, b), (a, b) ∈ f, a = a ❦➨♦ t❤❡♦ b = b.
▼ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣ ♠ët →♥❤ ①↕ f tø t➟♣ A ✤➳♥ t➟♣ B ❧➔ ♠ët q✉②
t➢❝ ❝❤♦ t÷ì♥❣ ù♥❣ ♠é✐ ♣❤➛♥ tû a ∈ A ✈ỵ✐ ♠ët ♣❤➛♥ tû ❞✉② ♥❤➜t b ∈ B. ❑❤✐
✤â t❛ ✈✐➳t f (a) = b✱ t❛ ❣å✐ b ❣å✐ ❧➔ ↔♥❤ ❝õ❛ ♣❤➛♥ tû a ❜ð✐ →♥❤ ①↕ f ❀ ✈➔ t❛ ❣å✐
a ❧➔ ♠ët t↕♦ ↔♥❤ ❝õ❛ ♣❤➛♥ tû b✳ ❚➟♣ A ữủ ồ t ỗ t B ồ
t ❝õ❛ →♥❤ ①↕ f ✳ ✣➸ ❞✐➵♥ t↔ →♥❤ ①↕ f ♥❤÷ tr➯♥ ♥❣÷í✐ t❛ ❦➼ ❤✐➺✉✿
f
A→
− B, a → f (a) = b, ❤♦➦❝
f : A → B, a → f (a) = b, ❤♦➦❝
f :A→B
a −→ f (a) = b.

q ữợ r õ ởt ré♥❣ tø t➟♣ ∅ ✤➳♥ t➟♣ B ❜➜t ❦➻✳
✭✐✐✐✮ ❈❤♦ →♥❤ ①↕ f : A → B, a → f (a)✳ ❚❛ ❣å✐ t➟♣ ❤đ♣ ❝♦♥ G(f ) ❝õ❛
A × B ①→❝ ✤à♥❤ ❜ð✐
G(f ) = {(a, f (a)) | a A}

ỗ t ừ f
❍❛✐ →♥❤ ①↕ ✤÷đ❝ ❣å✐ ❧➔ ❜➡♥❣ ♥❤❛✉ ♥➳✉ ❝❤ó♥❣ õ ỗ
ỗ t õ ❦❤→❝✱ ❝❤♦ f : A → B ✈➔ g : A → B ❧➔ ❤❛✐
→♥❤ ①↕✱ ❦❤✐ ✤â f = g ♥➳✉ A = A , B = B ✈➔ f (a) = g(a) ✈ỵ✐ ♠å✐ a ∈ A.
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✸✳ ❈❤♦ f : A −→ B, a → b = f (a) ❧➔ ♠ët →♥❤ ①↕✳

✭✐✮ f ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ →♥❤ ♥➳✉ f (a) = f (a ) ❦➨♦ t❤❡♦ a = a ✈ỵ✐ ♠å✐
a, a ∈ A.

✭✐✐✮ f ữủ ồ t ợ ồ b B t tỗ t a A
f (a) = b.

✭✐✐✐✮ f ✤÷đ❝ ❣å✐ ❧➔ s♦♥❣ →♥❤ ♥➳✉ ♥â ✈ø❛ ❧➔ ✤ì♥ →♥❤ ✈ø❛ ❧➔ t♦➔♥ →♥❤✳
✼


✶✳✷✳ ✣↕✐ sè tê ❤ñ♣
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ ✭◗✉② t➢❝ ❝ë♥❣✮✳ ●✐↔ sû ❝→❝ ✈✐➺❝ A1, A2, ..., Am ❝â t❤➸ ❧➔♠

t÷ì♥❣ ù♥❣ ❜➡♥❣ n1, n2, ...nm ❝→❝❤ ✈➔ ❣✐↔ sû ổ õ õ t
ỗ tớ ✤â sè ❝→❝❤ ❧➔♠ ♠ët tr♦♥❣ m ✈✐➺❝ ✤â ❧➔ n1 + n2 + ... + nm.
◗✉② t➢❝ ❝ë♥❣ ✤÷đ❝ t ữợ t ủ ữ s
◆➳✉ A1, ..., Am ❧➔ ❝→❝ t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ ✤ỉ✐ ♠ët rí✐ ♥❤❛✉✱
❦❤✐ ✤â✿
|A1 ∪ ... ∪ Am | = |A1 | + ... + |Am−1 | + |Am |.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸ ✭◗✉② t➢❝ ♥❤➙♥✮✳ ●✐↔ sû ♠ët ♥❤✐➺♠ ✈ư ♥➔♦ ✤â ✤÷đ❝ t→❝❤

r❛ ❧➔♠ ❤❛✐ ✈✐➺❝✳ ❱✐➺❝ t❤ù ♥❤➜t ❝â t❤➸ ❧➔♠ ❜➡♥❣ n1 ❝→❝❤✱ ✈✐➺❝ t❤ù ❤❛✐ ❝â t❤➸
❧➔♠ ❜➡♥❣ n2 ❝→❝❤ s❛✉ ❦❤✐ ✈✐➺❝ t❤ù ♥❤➜t ✤➣ ✤÷đ❝ ❧➔♠✱ ❦❤✐ ✤â s➩ ❝â n1n2 ❝→❝❤
t❤ü❝ ❤✐➺♥ ♥❤✐➺♠ ✈ư ♥➔②✳
◗✉② t➢❝ ♥❤➙♥ t❤÷í♥❣ ✤÷đ❝ ♣❤→t ❜✐➸✉ tê♥❣ q✉→t ❜➡♥❣ ♥❣ỉ♥ ♥❣ú t➟♣
❤đ♣ ♥❤÷ s❛✉✿
▼➺♥❤ ✤➲ ✶✳✷✳✹✳ ❈❤♦ n t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ A1, A2, ..., An (n ≥ 2). ❑❤✐ ✤â
|A1 × A2 × ... × An | = |A1 |.|A2 |...|An |.

❙❛✉ ✤➙② t❛ ♥❤➢❝ ❧↕✐ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ tê ❤đ♣ t❤❡♦ ❬✷❪✳ ❈❤♦
A1 , . . . , Am ❧➔ ❝→❝ t t S t ởt sỡ ỗ s ①➳♣ ✭❝â t❤➸ ❧➔ trü❝ q✉❛♥
❤➻♥❤ ❤å❝ ❤♦➦❝ ❝â t❤➸ trứ tữủ ữủ ổ t ữợ q✉② t➢❝ s➢♣
①➳♣✮✱ ❝á♥ R1, . . . , Rn ❧➔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤➦t ❝→❝ r➔♥❣
❜✉ë❝ ❧➯♥ sü s➢♣ ①➳♣ ❝→❝ ♣❤➛♥ tû ừ A1, . . . , Am t sỡ ỗ S ✳ ❑❤✐ ✤â ♠ët
s➢♣ ①➳♣ ❜➜t ❦ý ❝→❝ ♣❤➛♥ tû ❝õ❛ A1, . . . , Am t❤❡♦ sì ỗ S tọ
R1, . . . , Rn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝➜✉ ❤➻♥❤ tê ❤đ♣ tr➯♥ ❝→❝ t➟♣ A1, . . . , Am✳
◆➳✉ ❝→❝ ♣❤➛♥ tû ❝õ❛ A1, . . . , Am ✤➲✉ t❤✉ë❝ t➟♣ A t❤➻ ❝➜✉ ❤➻♥❤ tê ❤ñ♣ tr➯♥
A1 , . . . , Am t❤÷í♥❣ ✤÷đ❝ ❣å✐ ♥❣➢♥ ❣å♥ ❧➔ ❝➜✉ ❤➻♥❤ tê ❤đ♣ tr➯♥ t➟♣ A✳
❱➼ ❞ư ✷✳ sỷ A1 t ỗ ồ s ỳ A2 t ỗ ồ
s ừ ởt ợ
ỡ ỗ s S ồ ♠é✐ ❤➔♥❣ ❝â ✻ ✈à tr➼✧❀
✣✐➲✉ ❦✐➺♥ R1✿ ✸ ✈à tr➼ ✤➛✉ ❝õ❛ ❤➔♥❣ ✶ ♣❤↔✐ ❧➔ ♥ú✱ ✸ ✈à tr➼ s❛✉ ❝õ❛ ❤➔♥❣
✶ ♣❤↔✐ ❧➔ ♥❛♠❀
✽


✣✐➲✉ ❦✐➺♥ R2✿ ð ❤➔♥❣ ✷✱ ♥❛♠ ✈➔ ♥ú❛ ✤÷đ❝ ①➳♣ ✈➔♦ ❝→❝ ✈à tr➼ ①❡♥ ❦➩
♥❤❛✉✱ ♥❤÷♥❣ ✈à tr➼ ✤➛✉ t✐➯♥ ♣❤↔✐ ❧➔ ♥❛♠❀
✣✐➲✉ ❦✐➺♥ R2✿ ✸ ✈à tr➼ ✤➛✉ ❝õ❛ ❤➔♥❣ ✸ ♣❤↔✐ ❧➔ ♥❛♠✱ ✸ ✈à tr➼ s❛✉ ❝õ❛
❤➔♥❣ ✸ ♣❤↔✐ ❧➔ ♥ú❀
❑❤✐ ✤â ♠é✐ ❝→❝❤ s➢♣ ①➳♣ t❤➔♥❤ ❤➔♥❣ ❝õ❛ ❝→❝ ❤å❝ s✐♥❤ tø A1 ✈➔ A2
t sỡ ỗ S tọ R1, R2, R3 ❧➔ ♠ët ❝➜✉ ❤➻♥❤ tê ❤đ♣✳
●❤✐ ❝❤ó ✶✳✷✳✺ ✭❈❤➾♥❤ ❤ñ♣ ❝â ❧➦♣✮✳ ●✐↔ sû A ❧➔ ♠ët ❤ñ♣ ❤ú✉ ❤↕♥ ✈ỵ✐ |A| = n✱
❝á♥ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ ❝ơ♥❣ ❣✐↔ sû
A1 , A2 , . . . , Ak ❧➔ ❝→❝ ♣❤➛♥ tû ❝õ❛ A✱
S sỡ ỗ s ở õ tự tỹ ỗ k t❤➔♥❤ ♣❤➛♥ (x1 , x2 , . . . , xk )✧✱
R ❧➔ ✤✐➲✉ ❦✐➺♥ s➢♣ ①➳♣ ✧x1 ∈ A1 , x2 ∈ A2 , . . . , xk ∈ Ak ✧✳
❑❤✐ ✤â ♠é✐ ❝➜✉ ❤➻♥❤ tê ❤ñ♣ tr➯♥ A1, A2, . . . , Ak t❤❡♦ S t❤ä❛ ♠➣♥ R
✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤➾♥❤ ❤đ♣ ❝â ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A✳

❉➵ t❤➜② r➡♥❣ ♠é✐ ❝❤➾♥❤ ❤ñ♣ ❝â ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❝â t❤➸
❝♦✐ ❧➔ ♠ët ♣❤➛♥ tû ❝õ❛ t➼❝❤ ✣➲ ❝→❝ A1 ×. . .×Ak ✳ ❉♦ ✤â✱ t ỵ số
ủ õ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❜➡♥❣ Akn✱ t❤➻ Akn = |A1 ×A2 ×. . .×Ak |✳
❚❤❡♦ q✉② t➢❝ ♥❤➙♥ t❛ ❝â
|A1 × A2 × . . . × Ak | = |A1 ||A2 | . . . |Ak | = nk .

❱➻ ✈➟②

k

An = nk .

●❤✐ ❝❤ó ✶✳✷✳✻ ✭❈❤➾♥❤ ❤đ♣ ❦❤ỉ♥❣ ❧➦♣✮✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❤ú✉ ợ
|A| = n

ỏ k ởt số ữỡ sỷ
A1 = A
S sỡ ỗ s ở õ tự tỹ ỗ k t (x1 , x2 , . . . , xk )✧✱
R ❧➔ ✤✐➲✉ ❦✐➺♥ s➢♣ ①➳♣ ✧x1 ∈ A1 , x2 ∈ A1 , . . . , xk ∈ A1 ✧✳
❑❤✐ ✤â ♠é✐ ❝➜✉ ❤➻♥❤ tê ❤ñ♣ tr➯♥ A1 t❤❡♦ S t❤ä❛ ♠➣♥ tr➯♥ R ✤÷đ❝ ❣å✐
❧➔ ♠ët ❝❤➾♥❤ ❤đ♣ ❦❤ỉ♥❣ ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A✳ ❈❤➾♥❤ ❤ñ♣ ổ
tữớ ỡ ữủ ồ ủ ỵ ❤✐➺♥ sè ❝→❝ ❝❤➾♥❤ ❤ñ♣ ❝❤➟♣ k
❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❜➡♥❣ Akn✳
Akn =

n!
, ♥➳✉ k
(n − k)!
✾

n.


●❤✐ ❝❤ó ✶✳✷✳✼ ✭❚ê ❤đ♣ ❦❤ỉ♥❣ ❧➦♣✮✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ ✈ỵ✐ |A| = n✱

❝á♥ k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ ❣✐↔ sû
A1 = A✱
S sỡ ỗ s t õ k tỷ {x1 , x2 , . . . , xk }✧✱
R ❧➔ ✤✐➲✉ ❦✐➺♥ s➢♣ ①➳♣ ”x1 ∈ A1 , x2 ∈ A1 , . . . , xk ∈ A1 ”✳
❑❤✐ ✤â ♠é✐ ❝➜✉ ❤➻♥❤ tê ❤ñ♣ tr➯♥ A1 t❤❡♦ S t❤ä❛ ♠➣♥ R ✤÷đ❝ ❣å✐ ❧➔
♠ët tê ❤đ♣ ❦❤ỉ♥❣ ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A✳ ❚ê ❤đ♣ ❦❤ỉ♥❣ ❧➦♣ t❤÷í♥❣
✤÷đ❝ ❣å✐ ✤ì♥ ❣✐↔♥ ❧➔ tê ❤đ♣✳
◆❤÷ ✈➟②✱ ♠ët tê ❤ñ♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❝â t❤➸ ✤÷đ❝ ①❡♠ ♥❤÷
❧➔ ♠ët t➟♣ ❝♦♥ ❧ü❝ ❧÷đ♥❣ k ❝õ❛ A✳ ❑➼ ❤✐➺✉ sè ❝→❝ tê ❤ñ♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥
tû ❝õ❛ A ❜➡♥❣ nk ✱
♥➳✉ 0 k n,
0,
♥➳✉ k > n.
●❤✐ ❝❤ó ✶✳✷✳✽ ✭✣❛ t➟♣✮✳ ▼ët sü tö t➟♣ ❝→❝ ✈➟t ❝â ❜↔♥ ❝❤➜t tũ ỵ tr
õ õ t õ ỳ t ổ t ữủ ợ õ t ữ
sỹ ❧➦♣ ❧↕✐ ❝õ❛ ❝ị♥❣ ♠ët ✈➟t✮✱ ✤÷đ❝ ❣å✐ ❧➔ ✤❛ t➟♣ ❤ñ♣ ❤❛② ♥❣➢♥ ❣å♥ ❧➔ ✤❛
t➟♣✳ ❈→❝ ✈➟t tr♦♥❣ ✤❛ t➟♣ ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ♣❤➛♥ tû✳ ❚❛ ❝ơ♥❣ ❞ị♥❣ ❝→❝
♣❤÷ì♥❣ ♣❤→♣ ①→❝ ✤à♥❤ t➟♣ ❤đ♣ ✤➸ ①→❝ t ữ ố ợ t
t số tỷ ổ t ữủ ợ ♥❤❛✉✱ sè ❧÷đ♥❣ ❝→❝
♣❤➛♥ tû ❝õ❛ ♠ët ✤❛ t➟♣ A ❝ơ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❧ü❝ ❧÷đ♥❣ ❝õ❛ A ✈➔ ✤÷đ❝ ỵ
|A|
ử A = {a, a, a, b, c, c} ❧➔ ♠ët ✤❛ t➟♣ ✈ỵ✐ |A| = 6✳
❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ❤✐➸♥ ♥❤✐➯♥ ♠é✐ t➟♣ ❝ô♥❣ ❧➔ ✤❛ t➟♣✱ ♥❤÷♥❣ ♥❣÷đ❝ ❧↕✐✱
♠ët ✤❛ t➟♣ ❝â t❤➸ ❦❤ỉ♥❣ ❧➔ t➟♣ ❤đ♣✳ ❈❤➥♥❣ ❤↕♥✱ ✤❛ t➟♣ A ð tr➯♥ ❦❤ỉ♥❣ ❧➔
t➟♣ ❤ñ♣✳

◆➳✉ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♠ët ✤❛ t➟♣ A ✤➲✉ ❧➔ ♣❤➛♥ tû ❝õ❛ ♠ët t➟♣ B ✱ t❤➻
t❛ s➩ ♥â✐ r➡♥❣ A ❧➔ ✤❛ t➟♣ tr➯♥ B ✳ ❈❤➥♥❣ ❤↕♥✱ ✤❛ t➟♣ A ð tr➯♥ ❧➔ ♠ët ✤❛ t➟♣
tr➯♥ t➟♣ B = {a, b, c}✳
●❤✐ ❝❤ó ✶✳✷✳✾ ✭❚ê ❤đ♣ ❧➦♣✮✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ ✈ỵ✐ |A| = n✱ ❝á♥
k ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ ❣✐↔ sû A1 , A2 , . . . , Ak tỷ ừ A
S sỡ ỗ s➢♣ ①➳♣ ✧✤❛ t➟♣ ❝â k ♣❤➛♥ tû {x1 , x2 , . . . , xk }✧✱
n
=
k

=

n!
k!(n−k)! ,

✶✵


❧➔ ✤✐➲✉ ❦✐➺♥ s➢♣ ①➳♣ ”x1 ∈ A1, x2 ∈ A2, . . . , xk ∈ Ak ”✳
❑❤✐ ✤â✱ ♠é✐ ❝➜✉ ❤➻♥❤ tê ❤ñ♣ tr➯♥ A1, A2, . . . , Ak t❤❡♦ S t❤ä❛ ♠➣♥ R
✤÷đ❝ ❣å✐ ❧➔ ♠ët tê ❤ñ♣ ❝â ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A✳
◆❤÷ ✈➟②✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ♠ët tê ❤đ♣ ❝â ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛
A ❝â t ởt t ỹ ữủ k ợ ❝→❝ ♣❤➛♥ tû ✤➲✉ t❤✉ë❝ A✳ ❑➼ ❤✐➺✉
sè ❝→❝ tê ❤ñ♣ ❝â ❧➦♣ ❝❤➟♣ k ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❜➡♥❣ CRnk ✳ ❚❛ ❝ô♥❣ ♥❤➟♥
①➨t r➡♥❣ ♥➳✉ A = {a1, a2, . . . , an}✱ t❤➻ ♠ët t B ỹ ữủ k ợ
tỷ t❤✉ë❝ A ❤♦➔♥ t♦➔♥ ✤÷đ❝ ①→❝ ✤à♥❤ ♥➳✉ sè ❧➛♥ ①✉➜t ❤✐➺♥ tr♦♥❣ B ❝õ❛
♠é✐ ai, i = 1, 2, . . . , n✱ ✤÷đ❝ ①→❝ ✤à♥❤✳ ❚❛ ❝â
R

k

CRnk = Cn+k−1
.

●❤✐ ❝❤ó ✶✳✷✳✶✵ ✭❍♦→♥ ✈à ❦❤ỉ♥❣ ❧➦♣✮✳ ●✐↔ sû A ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥ ✈ỵ✐
|A| = n✳

❑❤✐ ✤â ♠ët ❝❤➾♥❤ ❤ñ♣ ❝❤➟♣ n ❝õ❛ n ♣❤➛♥ tû A ✤÷đ❝ ❣å✐ ❧➔ ♠ët
❤♦→♥ ✈à ❦❤ỉ♥❣ ❧➦♣ ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A✳ ❍♦→♥ ✈à ❦❤ỉ♥❣ ❧➦♣ t❤÷í♥❣ ✤÷đ❝ ❣å✐
✤ì♥
ỵ số ❝õ❛ n ♣❤➛♥ tû ❝õ❛ A ❧➔ Pn t❤➻ t❤❡♦ ✤à♥❤
♥❣❤➽❛ t❛ ❝â
Pn = Ann =

n!
n!
=
= n!.
(n − n)!
0!

●❤✐ ❝❤ó ✶✳✷✳✶✶ ✭❍♦→♥ ✈à ❝â ❧➦♣✮✳ ●✐↔ sû A = {a1, a2, . . . , an} ❧➔ ♠ët t➟♣

❤ú✉ ❤↕♥ ❧ü❝ ❧÷đ♥❣ n✱ ❝á♥ m1, m2, . . . , mn ❧➔ n sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ✤➣ ❝❤♦
s❛♦ ❝❤♦ ➼t ♥❤➜t ❝â ♠ët mi = 0✳ ❚❛ ❝ô♥❣ ❣✐↔ sû m = m1 + m2 + . . . + mn✱
A1 , A2 , . . . , Am tỷ ừ A
S sỡ ỗ s ở õ tự tỹ ỗ m t (x1 , x2 , . . . , xm )✱✧
R1 ❧➔ ✤✐➲✉ ❦✐➺♥ ✧x1 ∈ A1 , x2 ∈ A2 , . . . , xm ∈ Am ✧✱
R2 ❧➔ ✤✐➲✉ ❦✐➺♥✧a1 ①✉➜t ❤✐➺♥ ð ✤ó♥❣ m1 t❤➔♥❤ ♣❤➛♥✱ a2 ①✉➜t ❤✐➺♥ ð
✤ó♥❣ m2 t❤➔♥❤ ♣❤➛♥✱ ✳ ✳ ✳ ✱ an ①✉➜t ❤✐➺♥ ð ✤ó♥❣ mn t❤➔♥❤ ♣❤➛♥✧✳
❑❤✐ ✤â ❝➜✉ ❤➻♥❤ tê ❤ñ♣ A1, . . . , Am t❤❡♦ S t❤ä❛ ♠➣♥ R1, R2 ✤÷đ❝ ❣å✐

❧➔ ♠ët ❤♦→♥ ✈à ❝â ❧➦♣ ❝õ❛ ❝→❝ ♣❤➛♥ tû a1, . . . , an ❝õ❛ t➟♣ A ✈ỵ✐ t❤❛♠ sè ❧➦♣
❧➔ m1, . . . , mn✳
❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❝❤➾♥❤ ❤ñ♣ ❝â ❧➦♣ ✈➔ ❤♦→♥ ✈à ❝â ❧➦♣ t❛ t❤➜② ♥❣❛②
r➡♥❣ ♠ët ❤♦→♥ ✈à ❝â ❧➦♣ ❝õ❛ ❝→❝ ♣❤➛♥ tû a1, . . . , an ❝õ❛ t➟♣ A ✈ỵ✐ t❤❛♠ sè
❧➦♣ ❧➔ m1, m2, . . . , mn ❝❤➼♥❤ ❧➔ ♠ët ❝❤➾♥❤ ❤ñ♣ ❧➦♣ ❝❤➟♣ m ❝õ❛ n ♣❤➛♥ tû ❝õ❛
✶✶


t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ R2✳ ❈ô♥❣ t❤➜② ♥❣❛② r➡♥❣ ♠ët ❤♦→♥ ✈à ❝â ❧➦♣ ❝õ❛ ❝→❝
♣❤➛♥ tû a1, a2, . . . , an ❝õ❛ t➟♣ A ✈ỵ✐ t❤❛♠ sè ❧➦♣ ❧➔ m1 = m2 = . . . = mn = 1
❝❤➼♥❤ ❧➔ ♠ët ❤♦→♥ ✈à ❦❤æ♥❣ ❧➦♣ ❝❤➟♣ n ừ tỷ A
ỵ số ❝â ❧➦♣ ❝õ❛ ❝→❝ ♣❤➛♥ tû a1, a2, . . . , an ✈ỵ✐ t❤❛♠
sè ❧➦♣ m1, m2, . . . , mn ❧➔ m ,mm,...,m ✳ ✣➸ t➼♥❤ sè ♥➔② t❛ ❝♦✐ ♠ët ❤♦→♥ ✈à ❝â
❧➦♣ tr➯♥ ❧➔ ♠ët ❝→❝❤ t❤ü❝ ❤✐➺♥ ❤➔♥❤ ✤ë♥❣ H ✧t↕♦ r❛ ❤♦→♥ ✈à õ ỗ
n t H1 , H2 , . . . , Hn s❛✉ ✤➙②✿
●✐❛✐ ✤♦↕♥ H1✿ t↕♦ r❛ m1 t❤➔♥❤ ♣❤➛♥ a1 ❝❤♦ ❤♦→♥ ✈à ❝â ❧➦♣✳ ❘ã r➔♥❣ ❧➔
♠é✐ t➟♣ ❝♦♥ B1 = {i ∈ {1, . . . , m} |xi = a} ❝õ❛ t➟♣ {1, 2, . . . , m} tữỡ ự
ợ ú ởt tỹ ❱➻ ✈➟② ❝â mm ❝→❝❤ t❤ü❝ ❤✐➺♥
H1 ✳
●✐❛✐ ✤♦↕♥ H2✿ t↕♦ r❛ m2 t❤➔♥❤ ♣❤➛♥ a2 ❝❤♦ ❤♦→♥ ✈à ❝â ❧➦♣✳ ❱➻ ✤➣ ❝❤å♥
r❛ m1 t❤➔♥❤ ♣❤➛♥ ✤➸ ❧➔♠ t❤➔♥❤ ♣❤➛♥ a1 ♥➯♥ ❝❤➾ ❝á♥ ❧↕✐ m − m1 t❤➔♥❤ ♣❤➛♥
❝â t❤➸ ❞ò♥❣ ✤➸ ❝❤å♥ r❛ ❧➔♠ t❤➔♥❤ ♣❤➛♥ a2✳ ▲➟♣ ❧✉➟♥ ♥❤÷ ð ❣✐❛✐ ✤♦↕♥ H1 t❛
❝â m−1
❝→❝❤ t❤ü❝ ❤✐➺♥ ❣✐❛✐ ✤♦↕♥ H2✳
m
A

1

2

n

1

2

....................................

●✐❛✐ ✤♦↕♥ Hn✿ ❚↕♦ r❛ mn t❤➔♥❤ ♣❤➛♥ ❧➔ an ❝❤♦ ❤♦→♥ ✈à ❝â ❧➦♣✳ ❱➻ ð
(n 1) trữợ t ồ r (m1 + . . . + mn−1 ) t❤➔♥❤ ♣❤➛♥ ♥➯♥ ❝❤➾
❝á♥ ❧↕✐ m − (m1 + . . . + mn−1) t❤➔♥❤ ♣❤➛♥ ❝â t❤➸ ❞ò♥❣ ✤➸ ❧➔♠ t❤➔♥❤ ♣❤➛♥
)
an ✳ ❉♦ ✤â m−(m +...+m
❝→❝❤ t❤ü❝ ❤✐➺♥ ❣✐❛✐ ✤♦↕♥ Hn✳
m
❚❤❡♦ ỵ t õ
1

n1

n

m
m1 , m2 , . . . , mn
=

=

m

m1

m − m1
m2

m − (m1 + . . . + mn−1 )
mn

(m − m1 )!
(m − m1 − . . . − mn−1 )!
m!
.
···
m1 !(m − m1 )! m2 !(m − m1 − m2 )!
m!(m − m1 − . . . − mn−1 )!

❙✉② r❛

m
m1 , m2 , . . . , mn

=

m!
.
m1 !m2 ! . . . mn !

t q tr õ t t ữợ ữ s
ừ ởt t ủ
ỵ ❈❤♦ ❝→❝ sè tü ♥❤✐➯♥ m1, m2, . . . , mn s❛♦ ❝❤♦ m1 + m2 +

. . . + mn = m. ❑❤✐ ✤â sè ♣❤➙♥ ❤♦↕❝❤ ♠ët t ủ A ỗ m tỷ



♥❤❛✉ t❤➔♥❤ ❤đ♣ rí✐ r↕❝ ❝õ❛ n t➟♣ ❝♦♥ B1, B2, . . . , Bn✱ ✈ỵ✐ sè ♣❤➛♥ tû t tự
tỹ m1, m2, . . . , mn, ỵ ❤✐➺✉ m ,mm,...,m ✈➔ ❜➡♥❣
1

2

n

m!
.
m1 !m2 ! . . . mn !

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ t❤ü❝ ❤✐➺♥ ❝→❝ ♣❤➙♥ ❤♦↕❝❤ ✤➣ ♠æ t↔ tr➯♥ ✤➙② ❝õ❛ t➟♣
A t❤➔♥❤ n t➟♣ ❝♦♥ B1 , B2 , . . . , Bn ♥❤÷ s❛✉✿ ❚❛ ❧➜② ♠ët t➟♣ ❝♦♥ B1 ❜➜t ❦ý
❝❤ù❛ m1 ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣ ❆ ✭✤✐➲✉ ♥➔② ❝â t❤➸ t❤ü❝ ❤✐➺♥ t❤❡♦ mm ❝→❝❤✮✱
1
tr♦♥❣ m − m1 ♣❤➛♥ tû ❝á♥ ❧↕✐✱ t❛ ❧➜② ♠ët t➟♣ ❝♦♥ B2 ❝❤ù❛ m2 ♣❤➛♥ tû ✭✤✐➲✉
− m1
♥➔② ❝â t❤➸ t❤ü❝ ❤✐➺♥ t❤❡♦ m m
❝→❝❤✮✱ ✳ ✳ ✳ ✳ ❑❤✐ ✤â✱ t❤❡♦ q✉② t➢❝ ♥❤➙♥✱
2
sè t➜t ❝↔ ❝→❝ ❝→❝❤ ❝❤å♥ ❝→❝ t➟♣ ❝♦♥ B1, B2, . . . , Bn ❧➔
m − m1
m − m1 − m2 − . . . − mn−1
...
m2

mn
m!
(m − m1 )!
(m − m1 − m2 )!
=
×
×
m1 !(m − m1 )! m2 !(m − m1 − m2 )! m3 !(m − m1 − m2 − m3 )!
(m − m1 − m2 − . . . − mn )!
× ... ×
mn !(m − m1 − m2 − . . . − mn )!
m!
=
.
m1 !m2 ! . . . mn !
m
m1

❱➼ ❞ö ✶✳✷✳✶✸✳ ●✐↔ sû ❝â m ❝❤ú ❝→✐✱ tr♦♥❣ ✤â ❝â m1 ❝❤ú a1✱ m2 ❝❤ú a2 ✳ ✳ ✳ ✱

❝❤ú an ✭tr♦♥❣ ✤â m1 + m2 + . . . + mn = m✮✳ ❚ø ❝→❝ ❝❤ú ❝→✐ ✤â ❝â t❤➸
❧➟♣ ✤÷đ❝ ❜❛♦ ♥❤✐➯✉ ✧tø✧ ❦❤→❝ ♥❤❛✉ ✭tø õ tứ ổ õ
ộ tứ ỗ m ❝❤ú ❝→✐ ✤➣ ❝❤♦❄
❚❛ ✤→♥❤ sè ✈à tr➼ ❝→❝ ❝❤ú ❝→✐ tr♦♥❣ ♠é✐ tø ❜ð✐ ❝→❝ sè 1, 2, . . . , m✳ ▼é✐ ✧tø✧
✤÷đ❝ ①→❝ ✤à♥❤ ❤♦➔♥ t♦➔♥ ❜ð✐ t➟♣ ❤ñ♣ Bi ❝→❝ sè ❤✐➺✉ ❝õ❛ ❝→❝ ✈à tr➼ t↕✐ ✤â ❝â
❝❤ú ai. ❉♦ ✤â sè ✧tø✧ ❦❤→❝ ♥❤❛✉ ❧➟♣ ♥➯♥ tø ❝→❝ ❝❤ú ❝→✐ ✤➣ ❝❤♦ ❜➡♥❣ sè ❝→❝❤
♠➔ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ t➟♣ ❤ñ♣ A = {1, 2, . . . , m} ữợ ❞↕♥❣ ❤đ♣ rí✐ r↕❝ ❝õ❛
❝→❝ t➟♣ ❝♦♥ B1, B2, . . . , Bn. ỵ t số ✧tø✧ ❝➛♥ t➻♠ ❜➡♥❣

mn

m
m1 , m2 , . . . , mn

=

m!
.
m1 !m2 ! . . . mn !

❈❤➥♥❣ ❤↕♥✱ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ ♠✐♥❤ ❤å❛ ❝ư t❤➸✿
✶✸


✐✮ ❙è ❝→❝ ✧tø✧ t❤✉ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ❝→❝ ❝❤ú ❝→✐ ❝õ❛ tø ✬♥❤❛♥❤✬ ❜➡♥❣
5!
= 30.
2!2!1!

✐✐✮ ❙è ❝→❝ tứ õ t ữủ tứ ỳ ỗ ✹ ❝❤ú a✱ ✹ ❝❤ú b✱ ✷ ❝❤ú c✱
12!
✷ ❝❤ú d✮ ❜➡♥❣ 4!4!2!2!
✳
●❤✐ ❝❤ó ✶✳✷✳✶✹ ✭❈ỉ♥❣ t❤ù❝ ✤❛ t❤ù❝✮✳ ✧❈ỉ♥❣ t❤ù❝ ♥❤à t❤ù❝ ◆❡✇t♦♥✧ ❧➔
sü ❦❤❛✐ tr✐➸♥ ❝õ❛ ❜✐➸✉ t❤ù❝ (a + b)n tr♦♥❣ ✤â a, b ∈ R ✈➔ n ∈ N∗✳ ✧❈æ♥❣
t❤ù❝ ✤❛ t❤ù❝✧ ❧➔ sü ❦❤❛✐ tr✐➸♥ ❝õ❛ ❜✐➸✉ t❤ù❝ (a1 + a2 + . . . + am)n tr♦♥❣ ✤â
a1 , a2 , . . . , am ∈ R ✈➔ n ∈ N∗ ✳
❙ü ❦❤❛✐ tr✐➸♥ ❝õ❛ (a1 + a2 + . . . + am)n ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ s❛✉ ✤➙②✱
❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ✤❛ t❤ù❝
(a1 + . . . + am )n =

n1 ,...,nm ∈N,

m
i=1

ni =n

=
n1 ,...,nm ∈N,

m
i=1

ni =n

✶✹

m
m1 , m2 , . . . , mn

an1 1 . . . anmm

n!
an1 1 . . . anmm .
n1 ! . . . nm !


❈❤÷ì♥❣ ✷

◗✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✈➔ ♠ët sè ❜➔✐ t♦→♥

✷✳✶✳ ✣➳♠ ♠ët sè q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t
▼ư❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ ♠ö❝ ♥➔② ❧➔ ✤➳♠ ♠ët sè q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t
♥❤÷ sè q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ♣❤↔♥ số q ổ ố ự rữợ
t t❛ ✤÷❛ r❛ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚❤❡♦ ỵ tt tờ ủ số t
n!
k tỷ ừ t➟♣ n ❧➔ nk =
. ❚❛ ✤÷❛ r❛ ♠ët ❝❤ù♥❣ ♠✐♥❤ sì ❝➜♣
k!(n − k)!
❦❤↔♥❣ ✤à♥❤ ♥➔② ❞ü❛ ✈➔♦ ❝→❝ q✉② t➢❝ ✤➳♠ ❝ì ❜↔♥✳
❇ê ✤➲ ✷✳✶✳✶✳ ❙è t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ k ♣❤➛♥ tû ❝õ❛ ♠ët t➟♣ ❤ñ♣ n tỷ
k!(nn! k)! = nk .
ự ỵ ❤✐➺✉ sè t➟♣ ❝♦♥ k ♣❤➛♥ tû ❝õ❛ ♠ët t➟♣ n ♣❤➛♥ tû ❧➔ Unk .
▼✉è♥ ❞ü♥❣ ♠ët t➟♣ ❝♦♥ k ♣❤➛♥ tû ❝õ❛ t➟♣ ❤ñ♣ A t❛ ❝â t❤➸ ❣❤➨♣ t❤➯♠ ✈➔♦
♠ët t➟♣ ❝♦♥ k − 1 ♣❤➛♥ tû ❝õ❛ A ♠ët tr♦♥❣ n − (k − 1) = n − k + 1 ♣❤➛♥
tû ❝á♥ ❧↕✐✳
❱➻ ❝â Unk−1 t➟♣ ❝♦♥ k − 1 ♣❤➛♥ tû ✈➔ t❛ ❝â t❤➸ ❜ê s✉♥❣ t➟♣ ❝♦♥ ➜②
t❤➔♥❤ ♠ët t➟♣ ❝♦♥ k ♣❤➛♥ tû t❤❡♦ n − k + 1 ❝→❝❤✱ ♥➯♥ ❧➔♠ ♥❤÷ ✈➟② t❛ t❤✉
✤÷đ❝ (n − k + 1)Unk−1 t➟♣ ❝♦♥ k ♣❤➛♥ tû ❝õ❛ A✳ ◆❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ t➜t ❝↔
❝→❝ t➟♣ ❝♦♥ ✤➲✉ ❦❤→❝ ♥❤❛✉✱ ✈➻ t❛ ❝â t❤➸ ♥❤➟♥ ♠ët t➟♣ ❝♦♥ k ♣❤➛♥ tû t❤❡♦ k
❝→❝❤✱ ❝ö t❤➸ ❧➔ ❧➜② ♠é✐ ♠ët tr♦♥❣ k ♣❤➛♥ tû ❝õ❛ ♥â ❣❤➨♣ t❤➯♠ k − 1 ♣❤➛♥
tû ❝á♥ ❧↕✐✳ ❱➻ ✈➟② sè (n − k + 1)Unk−1 ✈ø❛ t➻♠ ✤÷đ❝ ð tr➯♥ ❣➜♣ k ❧➛♥ sè Unk
❝→❝ t➟♣ ❝♦♥ k ♣❤➛♥ tû ❝õ❛ A✳ ❉♦ ✤â t❛ ❝â ✤➥♥❣ t❤ù❝
(n − k + 1)Unk−1 = kUnk .

❚ø ✤â t❛ s✉② r❛
Unk =

n − k + 1 k−1 n − k + 1 n − k + 2 k−2
Un =
Un = . . .

k
k
k−1
✶✺


(n − k + 1) . . . (n − 1) n
U1 .
k(k − 1) . . . 2

❧➔ sè t➟♣ ❝♦♥ ♠ët ♣❤➛♥ tû ❝õ❛ A✳ ◆â ❜➡♥❣ sè ♣❤➛♥ tû ❝õ❛ A✱ tù❝ ❧➔ n✳
❱➟②
U1n

n(n − 1) . . . (n − k + 1)
1.2 . . . k
n!
n(n − 1) . . . (n − k + 1)(n − k) . . . 3.2.1
=
.
=
1.2 . . . k.(n − k) . . . 3.2.1
k!(n − k)!
Ukn =

❇ê ✤➲ ✷✳✶✳✷ ✭❙è t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ❤ú✉ ❤↕♥✮✳ ❙è t➟♣ ❝♦♥ ❦❤→❝ ♥❤❛✉ ❝õ❛

♠ët t➟♣ A ❤ú✉ ❤↕♥ ♣❤➛♥ tû ❧➔ 2|A|✳
❈❤ù♥❣ ♠✐♥❤✳ ❈→❝❤ ✶✿ ❈❤♦ A ❧➔ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ❚❛ ❧✐➺t ❦➯ ❝→❝ ♣❤➛♥ tû ❝õ❛
A t❤❡♦ ♠ët t❤ù tü ♥➔♦ ✤â✳ ●✐ú❛ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ A ✈➔ ❝→❝ ❞➣② ♥❤à ♣❤➙♥ ❝â

✤ë ❞➔✐ |A| ❝â sü t÷ì♥❣ ù♥❣ ✶✲✶✳ ❈ư t❤➸ ❧➔✱ ♠ët t ừ A ữủ ợ
õ sè ✶ ð ✈à tr➼ t❤ù i ♥➳✉ ♣❤➛♥ tû t❤ù i tr♦♥❣ ❞❛♥❤ s→❝❤ t❤✉ë❝
t➟♣ ❝♦♥ ♥➔②✱ ✈➔ ❧➔ sè ✵ tr♦♥❣ ♥❤ú♥❣ tr÷í♥❣ ❤đ♣ ♥❣÷đ❝ ❧↕✐✳ ❚❤❡♦ q✉② t➢❝ ♥❤➙♥
❝â 2|A| ❞➣② ♥❤à ♣❤➙♥ ✤ë ❞➔✐ |A|. ❱➻ ✈➟② |P (A)| = 2|A|.
❈→❝❤ ✷✿ ●å✐ Ta ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣ A ❝❤ù❛ ♣❤➛♥
tû a ∈ A✳ ❍✐➸♥ ♥❤✐➯♥ ♠é✐ t➟♣ ❝♦♥ ♥❤÷ t❤➳ ✤÷đ❝ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤✱ ♥➳✉ t❛
❜✐➳t t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❝á♥ ❧↕✐ ❝õ❛ ♥â ✭trø a✮✳ ❱➻ ✈➟② ❝â ❜❛♦ ♥❤✐➯✉ t➟♣ ❝♦♥
♥❤÷ t❤➳ t❤➻ ❝â ❜➜② ♥❤✐➯✉ t➟♣ ❝♦♥ tr♦♥❣ t➟♣ A = A \ a✳ ❚➟♣ ❤ñ♣ A ♥➔② ❝â
m − 1 ♣❤➛♥ tû✳ ❱➻ ✈➟②✱ ♥➳✉ t❛ ❣å✐ sm ❧➔ sè t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ❝â m
♣❤➛♥ tû✱ t❤➻ |Ta| = sm−1.
●å✐ T a ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❤ì♣ A ❦❤ỉ♥❣ ❝❤ù❛ a✱
t❤➻ |T a| ❝ô♥❣ ❜➡♥❣ sm−1✱ ✈➻ ❝→❝ t➟♣ ❝♦♥ ✤â ❝ô♥❣ ❧➔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣
A = A \ a✳
❱➻ 2A = Ta ∪ T a ✈➔ Ta ∩ T a = ∅✱ ♥➯♥ t❤❡♦ q✉② t➢❝ ❝ë♥❣✱ t❛ ❝â
|2A | = |T (a)| + |T a | = 2sm−1 .

❚ø ✤â s✉② r❛ sm = 2sm−1✳ ⑩♣ ❞ö♥❣ ❧✐➯♥ t✐➳♣ ✤➥♥❣ t❤ù❝ ♥➔②✱ t❛ ✤÷đ❝
sn = 2sn−1 = 22 sn−2 = . . . = 2n−1 s1 .
s1

❧➔ sè t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ❤ñ♣ ❝â ♠ët ♣❤➛♥ tû✳ ◆❤÷♥❣ ♠ët t➟♣ ❤đ♣ ❝â
✶✻


♠ët ♣❤➛♥ tû ❝❤➾ ❝â ❤❛✐ t➟♣ ❝♦♥ ❧➔ t➟♣ ré♥❣ ✈➔ ❝❤➼♥❤ ♥â✳ ❱➟② s1 = 2✳ ❉♦ ✤â
sn = 2n .

❈→❝❤ ✸✿ ❈❤♦ t➟♣ A ❝â n ♣❤➛♥ tû✳ ❳➨t t➟♣ ❤đ♣ Y = {0, 1}✳ ❱ỵ✐ ♠é✐ t➟♣
❝♦♥ B ❝õ❛ A✱ t❛ ①→❝ ✤à♥❤ ♠ët →♥❤ ①↕ f : A → Y ♥❤÷ s❛✉✿ ❈❤♦ x ∈ A✱ ♥➳✉
x ∈ B t❤➻ t❛ ✤➦t f (x) = 1✱ ❝á♥ ♥➳✉ x ∈

/ B t❤➻ t❛ ✤➦t f (x) = 0 ữ ự
ợ ộ t B ❝õ❛ A✱ ❝â ♠ët →♥❤ ①↕ f tø A tỵ✐ Y ✳
✣↔♦ ❧↕✐✱ ♥➳✉ f ❧➔ ♠ët →♥❤ ①↕ tø A tỵ✐ Y t❤➻ ù♥❣ ✈ỵ✐ ♥â ❝â ♠ët t➟♣
B ừ A ỗ tt tỷ x ∈ A s❛♦ ❝❤♦ f (x) = 1✳
❙ü t÷ì♥❣ ù♥❣ ➜② ❣✐ú❛ t➟♣ ❤ñ♣ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣ A ✈➔ t➟♣ ❤đ♣
❝→❝ →♥❤ ①↕ tø A tỵ✐ X rã r➔♥❣ ❧➔ 1 − 1✳ ❉♦ ✤â sè t➟♣ ❝♦♥ ❝õ❛ A✱ tù❝ ❧➔ |2A|✱
❜➡♥❣ sè →♥❤ ①↕ tø t➟♣ ❤đ♣ A ❝â n ♣❤➛♥ tû tỵ✐ t➟♣ ❤đ♣ Y ❝â ❤❛✐ ♣❤➛♥ tû✳
❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷ sè ♥➔② ❧➔ 2m✳ ❱➟② |2A| = 2m.
❈→❝❤ ✹✿ ❈→❝ t➟♣ ❝♦♥ ừ A ỗ t tỷ ♣❤➛♥ tû✱ ✳✳✳✱
n ♣❤➛♥ tû✳ ▼é✐ k ∈ {0, 1, ..., n}✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✶ ❝â nk t➟♣ ❝♦♥ k ♣❤➛♥ tû
❝õ❛ A✳ ❱➟② sè t➟♣ ❝♦♥ ❝õ❛ A ❧➔
n
n
n
+
+ ··· +
0
1
n

= (1 + 1)n = 2n

t❤❡♦ ❝æ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ✤❛ t❤ù❝✳
❚❛ ✤÷❛ r❛ ♠ët ù♥❣ ❞ư♥❣ ❝õ❛ ❦➳t q✉↔ tr➯♥ tr♦♥❣ ❣✐↔✐ t♦→♥ sì ❝➜♣✳
❇➔✐ t♦→♥ ✷✳✶✳✸ trữợ số ữỡ
m

T =
k=0

n

k
k
Cn+k
Cm+k
+
.
2m+k k=0 2n+k

ữợ ự tờ ❝➛♥ t➼♥❤ ❜➡♥❣ ✶✱ tù❝ ❧➔✿
m

k=0

n

k
k
Cn+k
Cm+k
+
= 1.
2n+k+1 k=0 2m+k+1

❈→❝ ❧ô② t❤ø❛ ❝õ❛ ✷ ❝❤♦ t❛ ❧✐➯♥ t÷ð♥❣ ✤➳♥ sè t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ❤ñ♣
k
❚r♦♥❣ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ t➟♣ S = {1, 2, . . . , m + n + 1} ❞➵ t❤➜② Cn+k
2m−k
t➟♣ ❞↕♥❣ {a1, a2, . . . , an+i} , (1 < i m + 1) tr♦♥❣ ✤â (a1 < a2 < . . . < an+i)

n
✈➔ an+1 = n + k + 1 ✈ỵ✐ 0 k m ✭❉♦ ❝â Cn+k
❝→❝❤ ❝❤å♥ n ♣❤➛♥ tû
(a1 , a2 , . . . , an )✮ tø t➟♣ {1, 2, . . . , n + k}✱ ✶ ❝→❝❤ ❝❤å♥ an+1 = n + k + 1
✶✼


✈➔m
k=0

❝→❝❤ ❝❤å♥ t➟♣ ❝♦♥ ❝õ❛ t➟♣{n + k + 1, . . . , n + m + 1}✳ ◆❤÷ ✈➟②
k
Cn+k
2m−k ❧➔ sè t➟♣ ❝♦♥ ❝õ❛ S ❝â ♥❤✐➲✉ ❤ì♥ n ♣❤➛♥ tû✳

2m−k

n

k
❚÷ì♥❣ tü✱ Cm+k
2n−k ❧➔ sè t➟♣ ❝♦♥ ❝õ❛ S ❝â ♥❤✐➲✉ ❤ì♥ m ♣❤➛♥ tû✱
k=0
❝ơ♥❣ tù❝ ❧➔ msè t➟♣ ❝♦♥ ❝õ❛ Sn ❦❤æ♥❣ ❝â q✉→ n ♣❤➛♥ tû✳
k
k
❱➟② Cn+k
2m−k +
Cm+k
2n−k ❧➔ sè t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ S ✱ tù❝

k=0
k=0
❧➔ 2m+n+1✳ ✣â ❧➔ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳
❇➔✐ t♦→♥ ✷✳✶✳✹ ✭❆P▼❖ ✶✾✾✽✮✳ ●✐↔ sû Fk ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❜ë (A1, A2, . . . , Ak )
tr♦♥❣ ✤â Ai (i = 1, 2, . . . , k) ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛ {1, 2, . . . , n} ✭❝→❝ t➟♣
A1 , A2 , . . . , Ak ❝â t❤➸ trò♥❣ ♥❤❛✉✮✳ ❍➣② t➼♥❤

|A1 ∪ A2 ∪ . . . ∪ Ak |.

Sn =
(A1 ,A2 ,...,Ak )Fk

ữợ õ 2n t ❝♦♥ ❝õ❛ {1, 2, . . . , n} ♥➯♥ ❝â 2nk ❜ë (A1, A2, . . . , Ak )✳
❱ỵ✐ ♠é✐ k✲❜ë (A1, A2, . . . , Ak ) ❝õ❛ t➟♣ {1, 2, . . . , n − 1} t❛ ❝â t❤➸ t❤➯♠ ❤♦➦❝
❦❤æ♥❣ t❤➯♠ n ✈➔♦ t➟♣ Ai ✤➸ ✤÷đ❝ k✲❜ë (A1, A2, . . . , Ak ) ❝õ❛ {1, 2, . . . , n}
ợ ú ỵ r số kở (A1, A2, . . . , Ak ) ❝õ❛ t➟♣ {1, 2, . . . , n − 1} ❧➔ 2(n−1)k
✈➔ ❝â 2k − 1 ❝→❝❤ t❤➯♠ n ✈➔♦ k✲❜ë (A1, A2, . . . , Ak ) ❝õ❛ t➟♣ {1, 2, . . . , n − 1}
t❤➻ t❛ ❝â✿ Sn = 2k Sn−1 + (2k − 1).2k(n−1)✳
❉➵ t❤➜②✿ S1 = 2k − 1✳ ❚ø ✤➜② ❜➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝✿
Sn = n.2k(n−1) (2k − 1)✳
▼➺♥❤ ✤➲ ✷✳✶✳✺✳ ●✐↔ sû A ởt t ủ ỗ n tỷ sè ❝→❝
❝➦♣ (U, V )✱ ð ✤â U, V ⊆ A✱ t❤ä❛ ♠➣♥ U ❦❤æ♥❣ ❧➔ t➟♣ ❝♦♥ t❤ü❝ sü ❝õ❛ V ✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✷ sè t➟♣ ❝♦♥ ❝õ❛ t➟♣ A ❧➔ 2n✳ ❱➟② sè ❝→❝ ❝➦♣
(U, U ) 2n ì 2n ợ ộ số tỹ k t V ỗ k tû✱ sè
❝→❝ t➟♣ ❝♦♥ U ❝õ❛ V ✤ó♥❣ ❜➡♥❣ 2k ✳ ❚ø ✤➙② s✉② r❛ sè ❝→❝ ❝➦♣ (U, V )✱ ð ✤â
U, V ⊆ A✱ t❤ä❛ ♠➣♥ U ❧➔ t➟♣ ❝♦♥ t❤ü❝ sü ❝õ❛ V ✤ó♥❣ ❜➡♥❣
n

k=0

n
k

2k − 2n = 3n − 2n .

❉♦ ✈➟②✱ sè ❝→❝ ❝➦♣ (U, V )✱ ð ✤â U, V
sü ❝õ❛ V ✤ó♥❣ ❜➡♥❣

⊆ A✱ t❤ä❛ ♠➣♥ U

❦❤æ♥❣ ❧➔ t➟♣ ❝♦♥ t❤ü❝

2n .2n − [3n − 2n ] = 4n − 3n − 2n .
✶✽


◆❤➟♥ ①➨t ✷✳✶✳✻✳ ❈❤♦ A = {a1, a2, ..., an}✳ õ t õ t t
AìA

ữợ tr➟♥ ♠➔ ♣❤➛♥ tû ❞á♥❣ ith ❝ët t❤ù j th ❧➔ (ai, aj )✱ 1 ≤

i, j ≤ n.

▼➺♥❤ ✤➲ ✷✳✶✳✼✳ ❈❤♦ A = {a1, a2, ..., an} ✈➔ B = {b1, b2, ..., bm}✳ ❙è q✉❛♥

❤➺ ❤❛✐ ♥❣æ✐ tø A ✤➳♥ B ❧➔ 2mn.
❈❤ù♥❣ ♠✐♥❤✳ ❙è ♣❤➛♥ tû A × B ❧➔ mn✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tø A ✤➳♥ B ❧➔
sè t➟♣ ❝♦♥ ❝õ❛ t➟♣ A × B ✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷ t❛ ❝â sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tø
A ✤➳♥ B ❧➔ 2mn .
❍➺ q✉↔ ✷✳✶✳✽✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣ A ❧➔

2

2n .

❈❤ù♥❣ ♠✐♥❤✳ ❙è ♣❤➛♥ tû A × A ❧➔ n2✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ A ❧➔ sè t➟♣
❝♦♥ ❝õ❛ t➟♣ A × A✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷ t❛ ❝â sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣
A ❧➔ 2n .
▼➺♥❤ ✤➲ ✷✳✶✳✾✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ①↕ tr➯♥
t➟♣ A ❧➔ 2n −n.
❈❤ù♥❣ ♠✐♥❤✳ tỷ A ì A ữợ tr➟♥✳ ▼é✐ q✉❛♥ ❤➺ ❤❛✐
♥❣æ✐ ♣❤↔♥ ①↕ tr➯♥ t➟♣ A ♣❤↔✐ ❝❤ù❛ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❝õ❛
♠❛ tr➟♥✳ ◆❣♦➔✐ r❛ ♥â ❝â t❤➸ ❝❤ù❛ t❤➯♠ ♠ët t➟♣ ❝♦♥ ❝→❝ ♣❤➛♥ tû ♥➡♠ ♥❣♦➔✐
✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤✳ ❙è ♣❤➛♥ tû ♥➡♠ ♥❣♦➔✐ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❧➔ n2 − n✳ ❉♦ ✤â
t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✳✷✱ sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ①↕ tr➯♥ t➟♣ A ❧➔ 2n −n.
●✐↔ sû A = {a1, ..., an}✳ ❇✐➸✉ ❞✐➵♥ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ R ❜ð✐ ♠❛ tr➟♥ (aij ) s❛♦
❝❤♦
1 ♥➳✉ (ai , aj ) ∈ R
aij =
0 ♥➳✉ (ai , aj ) ∈
/R
❚❛ ❝â R ❝â t➼♥❤ ❝❤➜t ♣❤↔♥ ①↕ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ aij = 1 ✈ỵ✐ ♠å✐ 2n = 1, ..., n
❝á♥ n2 − n ♣❤➛♥ tû ❦❤→❝ ❝â t❤➸ ❧➔ ✵ ❤♦➦❝ ✶
2

2

2

✶✾




∗ ∗ ... ∗
1 ∗ . . . ∗

∗ 1 . . . ∗

✳✳ ✳✳ ✳ ✳ ✳ ✳✳ 

∗ ∗ ... 1

1
∗

∗

✳
✳
∗

❈â n2 − n ♣❤➛♥ tû ❝❤ù❛ ❞➜✉ ộ tr tữỡ ự ợ t ừ t➟♣
n2 − n ♣❤➛♥ tû✳ ❉♦ ✤â ❝â 2n −n ♠❛ tr➟♥ ❤❛② 2n −n q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❝â t➼♥❤
❝❤➜t ♣❤↔♥ ①↕✳
▼➺♥❤ ✤➲ ✷✳✶✳✶✵✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✤è✐ ①ù♥❣
tr➯♥ t➟♣ A ❧➔ 2(n(n+1))/2✳
❈❤ù♥❣ ♠✐♥❤✳ ❇✐➸✉ ❞✐➵♥ ♣❤➛♥ tû A × A ữợ tr t A ì A
t ❜❛ t➟♣✿ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤✱ ❝→❝ tỷ tr t
ữợ (i > j) ❝→❝ ♣❤➛♥ tû ð ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥ (i < j)✳ ❚r♦♥❣ ♠é✐

q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❝â t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣✱ ♥➳✉ ♣❤➛♥ tû (ai, aj ) tø ♠❛ tr t
ữợ tở q ổ t tø (aj , ai) tø ♠❛ tr➟♥ t❛♠ ❣✐→❝
tr➯♥ ❝ô♥❣ t❤✉ë❝ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳ ❈â (n2 − n)/2 ♣❤➛♥ tỷ tr t
ữợ tữỡ ự ợ (n2 − n)/2 ♣❤➛♥ tû ð ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥✳ ộ q
ổ ố ự ỗ ởt t ❝♦♥ ❝õ❛ t➟♣ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣
❝❤➨♦ ❝❤➼♥❤ ✈➔ t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ ♣❤➛♥ tû ð ♠❛ tr➟♥ t ữợ ũ
ợ tỷ tr tữỡ ù♥❣ ð ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥✮✳ ❚❤❡♦ ▼➺♥❤
✤➲ ✷✳✶✳✷ sè t➟♣ ❝♦♥ ❝õ❛ t➟♣ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ❧➔ 2n✱ sè t➟♣
❝♦♥ ❝õ❛ t➟♣ ❝→❝ ♣❤➛♥ tỷ tr t ữợ 2(n n)/2 ❚❤❡♦ q✉② t➢❝
♥❤➙♥ t❛ ❝â sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✤è✐ ①ù♥❣ tr➯♥ ❧➔ ❧➔
2

2

2

2n .2(n

2

−n)/2

= 2(n

2

+n)/2

.

❇ê ✤➲ ✷✳✶✳✶✶✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ✤è✐ ①ù♥❣

tr➯♥ t➟♣ A ❧➔ 2n.3(n −n)/2✳
❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ R ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ t➟♣ A✳ ◆➳✉
♣❤➛♥ tû (ai, aj ) ð ♠❛ tr t ữợ (i = j) tở q t❤➻ ♣❤➛♥ tû
2

✷✵


tø ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥ ❦❤æ♥❣ t❤✉ë❝ q✉❛♥ ❤➺ ữủ õ
õ ợ ộ ❝➦♣ i = j
✶✳ (ai, aj ) ∈ R ✈➔ (aj , ai) ∈/ R,
✷✳ (ai, aj ) ∈/ R ✈➔ (aj , ai) ∈ R,
✸✳ (ai, aj ) ∈/ R ✈➔ (aj , ai) ∈/ R.
❙è ❝➦♣ (i, j), (i = j) ❧➔ n 2−n ♥➯♥ sè ❝→❝❤ ❝❤å♥ ❝➦♣ (i, j) t❤ä❛ ♠➣♥ ✶✮✱ ✷✮✱ ✸✮
❧➔ 3 ✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ R ❝â t❤➸ ❝â ❤♦➦❝ ❦❤æ♥❣ ❝â ❝→❝ ♣❤➛♥ tû t↕✐ ✤÷í♥❣
❝❤➨♦ ♥➯♥ sè ❝→❝❤ ❝❤å♥ ♣❤➛♥ tû t↕✐ ✤÷í♥❣ ❝❤➨♦ ❧➔ 2n✳ ❱➟② t❤❡♦ q✉② t➢❝ ♥❤➙♥
sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ t➟♣ A ❧➔ 2n.3(n −n)/2✳
❚❛ tr➻♥❤ ❜➔② ♠ët ❧➟♣ ❧✉➟♥ ❦❤→❝✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ tr➯♥ A ❝❤♦ t÷ì♥❣
ù♥❣ 1 1 ợ tr (aij )nìn
(aj , ai )

2

n2 −n
2

2

♥➳✉ (ai, aj ) ∈ R
0 ♥➳✉ ((ai , aj ) ∈
/R
❱➻ ✈➟② t❛ ✤➳♠ sè ♠❛ tr➟♥ ❇♦♦❧❡ ù♥❣ ✈ỵ✐ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ♣❤↔♥ ✤è✐ ①ù♥❣✳ ●✐↔
sû R ❧➔ q✉❛♥ ❤➺ ✷ ♥❣æ✐ ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ A ❝â ♠❛ tr➟♥ ❇♦♦❦❧❡ (aij )m×n✳
❈→❝ ♣❤➛♥ tû tr ữớ õ t ồ tũ ỵ tr {0, 1} ♥➯♥ ❝â
2n ❝→❝❤ ❝❤å♥✳ ❳➨t ❝→❝ ♣❤➛♥ tû ữớ ợ tỷ aij
t tr➯♥ (i < j) ♥➳✉ aij = 1 t❤➻ aji = 0✱ ♥➳✉ aij = 0 t❤➻ aji ❝â t❤➸ ❧➔
✵ ❤♦➦❝ ✶✳ ❳➨t ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥✱ sè ♣❤➛♥ tû ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥ ❧➔
n
n
n −n
=
✳
●✐↔
sû
0
≤
k
≤
✳ ❈❤å♥ k sè ✶ ð ♠❛ tr➟♥ t❛♠ ❣✐→❝ tr➯♥✱ ❝â
2
2
2
aij =

1

2

( n2 )
k

❝→❝❤ ❝❤å♥ ✈à tr➼ sè ✶✱ ❝á♥ ❧↕✐

n
− k ✈à tr➼ sè ✵✳ P❤➛♥ tû ✤è✐ ①ù♥❣ ✈ỵ✐
2
 
n
 −k
✤è✐ ①ù♥❣ ✈ỵ✐ sè ✵ ❧➔ ✵ ❤♦➦❝ ✶✳ õ 2 2

số tr t ữợ ❧➔ ✵✱
❝→❝❤ ❝❤å♥ ♠❛ tr➟♥ t❛♠ ❣✐→❝ ♥❤÷ ✈➟②✳ ❱➟② ❝â
 

n
 
2  n  n−k
 2 2 k
k=0
k

 

✷✶


❈→❝❤ s➢♣ ①➳♣ ♣❤➛♥ tû tr➯♥ ❝→❝ ♠❛ tr➟♥ t❛♠ ❣✐→❝✳ ❱➟② ❝â

 

n
 


n
2
 −k
n
 2 2 2
2n (
k=0
k
 

♠❛ tr➟♥ ❤❛② q✉❛♥ ❤➺ ♣❤↔♥ ✤è✐ ①ù♥❣✳ ❚❛ ❝â
 

n
 


n
n
n
2
 
 
 −k

n
 2 2 2
3 2 = (1 + 2) 2 =
k=0
k
 

 

 

 

n
n2 −n
2n .3 2 = 2n 3 2 .

♥➯♥ sè q✉❛♥ ❤➺ ♣❤↔♥ ✤è✐ ①ù♥❣ ❧➔
❇ê ✤➲ ✷✳✶✳✶✷✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✈ø❛ ✤è✐ ①ù♥❣
✈ø❛ ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ t➟♣ A ❧➔ 2n✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû R ❧➔ q✉❛♥ ❤➺ ✷ ♥❣æ✐ ✈ø❛ ✤è✐ ①ù♥❣ ✈ø❛ ♣❤↔♥ ✤è✐ ①ù♥❣✳
◆➳✉ (ai, aj ) ∈ R✱ (i = j) t❤➻ (aj , ai) ∈ R ✈➻ R ✤è✐ ①ù♥❣✳ ◆❤÷♥❣ ✈➻ R ♣❤↔♥
✤è✐ ①ù♥❣ ♥➯♥ ai = aj , i = j ✳ ✣✐➲✉ ♥➔② ❧➔ ổ ỵ õ R ổ ự
tỷ (ai, aj ), i = j ✳ ❱➟②
 

R ⊆ {(a, a)|a ∈ A}.

❉➵ t❤➜② ❦❤✐ ✤â R ✈ø❛ ✤è✐ ①ù♥❣ ✈ø❛ ♣❤↔♥ ✤è✐ ①ù♥❣✳ ▼➔ |{(a, a)|a ∈ A}| = n
♥➯♥ sè q✉❛♥ ❤➺ ✷ ♥❣æ✐ ✈ø❛ ✤è✐ ①ù♥❣ ✈ø❛ ♣❤↔♥ ✤è✐ ①ù♥❣ ❧➔ 2n✳

❇ê ✤➲ ✷✳✶✳✶✸✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✈ø❛ ✤è✐ ①ù♥❣
✈ø❛ ❜➜t ✤è✐ ①ù♥❣ tr➯♥ t➟♣ A ❧➔ ✶✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû R ❧➔ q✉❛♥ ❤➺ ✤è✐ ①ù♥❣ ✈➔ ❜➜t ✤è✐ ①ù♥❣ tr➯♥ A✳ ❱ỵ✐ ♠å✐
a ∈ A✱ t❛ ❝â (a, a) ∈
/ R ✈➻ R ❜➜t ✤è✐ ①ù♥❣✳ ữỡ tỹ tỷ (a, b) ợ
a = b ❝ơ♥❣ ❦❤ỉ♥❣ t❤✉ë❝ R✳ ❱➟② R = ∅ ❧➔ q✉❛♥ ❤➺ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ ✈ø❛
✤è✐ ①ù♥❣ ✈ø❛ ❜➜t ✤è✐ ①ù♥❣✳
❇ê ✤➲ ✷✳✶✳✶✹✳ ❈❤♦ A = {a1, a2, ..., an}✳ ❙è q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✈ø❛ ♣❤↔♥ ①↕
✈ø❛ ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ t➟♣ A ❧➔ 3(n −n)/2✳
2

✷✷