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 NGÔI VÀ MỘT SỐ BÀI TOÁN LIÊN QUAN Chuyên ngành: Phương pháp Toán sơ cấp Mã số: 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♥❣ ✷✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✈➔ ♠ët sè ❜➔✐ t♦→♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳ ✣➳♠ ♠ët sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ⑩♥❤ ①↕ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳ P❤➙♥ ❤♦↕❝❤✱ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳ ✣➳♠ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✈➔ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❜➢❝ ❝➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✸ ✷✼ ✸✷ ❑➳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✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ A✳ ◆➳✉ R ❧➔ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ t➟♣ A ✈➔ (a, b) ∈ R t❤➻ t❛ ❦➼ ❤✐➺✉ aRb ✭✤å❝ ❧➔ a ❝â q✉❛♥ ❤➺ R ✈ỵ✐ b✮✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤→❝ ♥❤❛✉ ❝õ❛ t♦→♥ ❤å❝✿ ✣↕✐ sè✱ ❙è ỵ tt ỗ t t➼♥❤✱ ✳✳✳✳ ▼ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳ ❈→❝ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✤✐➸♥ ❤➻♥❤ 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 số tự ỵ tt q ❤❛✐ ♥❣ỉ✐✱ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✱ q✉❛♥ ❤➺ t❤ù tü✱ ỵ tt tờ ủ ❚✉② ❧➔ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❝❤♦ ❈❤÷ì♥❣ ✷ ♥❤÷♥❣ ố ợ t tự ữỡ ❦✐➳♥ t❤ù❝ ♠ỵ✐ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ 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 ữ ữ ổ ❧✉➟♥ ✈➠♥ t→❝ ❣✐↔ ❝❤➾ ❦❤❛✐ t❤→❝ ♠ët sè ❜➔✐ t♦→♥ ❝➜♣ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜➔✐ t♦→♥ tê ❤đ♣✳ ởt ữ ỵ t ❝è ❣➢♥❣ t➻♠ ❤✐➸✉ ♥❤✐➲✉ ❝→❝❤ ❣✐↔✐✱ ❝→❝❤ t✐➳♣ ❝➟♥ ❦❤→❝ ♥❤❛✉ ❝õ❛ ♠ët ❜➔✐ t♦→♥✱ ♠ët ✈➜♥ ✤➲✳ ✣➳♠ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ❧➔ →♥❤ 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 ✤÷ì♥❣ tr➯♥ t➟♣ n ♣❤➛♥ tû ❝❤➼♥❤ ❧➔ sè ♣❤➙♥ ❤♦↕❝❤✱ sè ❇❡❧❧ t❤ù n✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ tổ ữủ sỹ ữợ ú ù t➟♥ t➻♥❤ ❝õ❛ ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛②✳ ổ ỷ ỡ t qỵ t❤➛② ❝ỉ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ❦❤â❛ ❈❛♦ ❤å❝ ❚♦→♥ ❦❤â❛ ✶✶❇ ✭✷✵✶✼✲✷✵✶✾✮ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ✤➣ tr✉②➲♥ t❤ö ✤➳♥ ❝❤♦ tæ✐ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ▲í✐ ❝✉è✐ ❝ị♥❣✱ t→❝ ❣✐↔ ♠✉è♥ ❞➔♥❤ ✤➸ tr✐ ➙♥ ❜è ♠➭ ✈➔ ❣✐❛ ✤➻♥❤ ✈➻ ✤➣ ❝❤✐❛ s➫ ♥❤ú♥❣ ❦❤â ❦❤➠♥ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✽ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❇ị✐ ❚❤à ❚❤✉ ❚❤õ② ✷ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✣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✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❜➡♥❣ ❝→❝ ❝❤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✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✳ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ 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 õ ợ ỵ r ỵ s 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 t➢t✿ P❳ ❂ P❤↔♥ ①↕✱ ✣❳ ❂ ✣è✐ ①ù♥❣✱ P✣❳ ❂ P❤↔♥ ✤è✐ ①ù♥❣✱ ❇P❳ ❂ ❇➜t ♣❤↔♥ ①↕✱ ❇❈ ❂ ❇➢❝ ❝➛✉✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ✭✐✮ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ 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✉❛♥ ❤➺ t÷ì♥❣ ữỡ tr t A ợ ộ a A, t ợ tữỡ ữỡ 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 tỷ tr A ữủ ❣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✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ R ữ s ợ a, b Z, t ♥â✐ aRb ⇔ m|(a − b) ✹ ◗✉❛♥ ❤➺ ♥➔② ữủ q ỗ ữ t ổ m ỏ q ỗ ữ m a ỗ ữ b t ổ m t tữớ ❦➼ ❤✐➺✉ ❧➔ a ≡ b (mod m) ❚❛ t❤➜② ✤â ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ Z✳ ợ a Z ợ tữỡ ữỡ a ữủ a ữủ ởt ợ t ữ t ổ m ợ a tữỡ Z ố ợ q ỗ ữ ♠♦❞✉❧♦ m ✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐ Zm ✈➔ ✤÷đ❝ ❣å✐ t ợ t ữ t ổ m t ủ ợ t ữ 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 ≤ r ≤ m1 t ỵ ợ ữ õ 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❛♦ ❝❤♦ ≤ i < j ≤ m − t❛ ❧✉æ♥ ❝â < j − i < m ♥➯♥ j − i ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ m✳ ❉♦ ✤â i ≡ j (mod m) i = j t t Zm ỗ m ♣❤➛♥ tû ✤ỉ✐ ♠ët ❦❤→❝ ♥❤❛✉ ♥❤÷ s❛✉✿ Zm = {0, 1, , m − 1} ú ỵ r a = mq + r t a = r ❱➻ t❤➳ ✈ỵ✐ q1, , qm m số tý ỵ t ổ ❝â Zm = {q1 m, q2 m + 1, , qm m + m − 1} ❈❤➥♥❣ ❤↕♥ Z3 = {0, 1, 2} = {6, −2, 8}✳ ỵ ữợ t ỵ q tữỡ ữỡ rữợ t ỵ ❝❤ó♥❣ 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 ✺ ✣à♥❤ ỵ A ởt q tữỡ ữỡ 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 ❧➔ ởt tr A t tỗ t t ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ A s❛♦ ❝❤♦ ♠é✐ Ai ởt ợ tữỡ ữỡ ▼ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ 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➟♣ ❤ñ♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ð✐ ≤✳ ❱➼ ❞ư ✶✳ ✭✐✮ ◗✉❛♥ ❤➺ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ ≤ t❤ỉ♥❣ t❤÷í♥❣ ✭t❛ ✤➣ ❜✐➳t ð ♣❤ê t❤æ♥❣✮ ❧➔ 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✉❛♥ ❤➺ t❤ù tü✳ t A t tũ ỵ t N∗✳ ❑❤✐ ✤â q✉❛♥ ❤➺ ❝❤✐❛ ❤➳t ✧⑤✧ tr➯♥ t➟♣ A ❝ô♥❣ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü tr➯♥ A✳ ▼ư❝ ❝✉è✐ ❣✐ỵ✐ t❤✐➺✉ ❧ỵ♣ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t ❧➔ →♥❤ ①↕✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵✳ ❈❤♦ R ❧➔ q✉❛♥ ❤➺ ✷ ♥❣æ✐ tø A ✤➳♥ B ✳ ❑❤✐ õ R R ỵ ❤✐➺✉ D(R) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ t➟♣ {x|x ∈ A; ∃y ∈ A, (x, y) ∈ R} ❷♥❤ ❝õ❛ R R ỵ im(R) ữủ 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✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ 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 t ỗ t B ❣å✐ ❧➔ 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 ①↕ ré♥❣ tø t➟♣ ∅ ✤➳♥ t➟♣ B ❜➜t ❦➻✳ ✭✐✐✐✮ ❈❤♦ →♥❤ ①↕ f : A → B, a → f (a)✳ ❚❛ ❣å✐ t➟♣ ❤ñ♣ ❝♦♥ G(f ) ❝õ❛ A × B ①→❝ ✤à♥❤ ❜ð✐ G(f ) = {(a, f (a)) | a A} ỗ t ❝õ❛ →♥❤ ①↕ f ✳ ✭✐✈✮ ❍❛✐ →♥❤ ①↕ ✤÷đ❝ ú 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 t ợ b B t tỗ t a A ✤➸ f (a) = b ✭✐✐✐✮ f ✤÷đ❝ ❣å✐ ❧➔ s♦♥❣ →♥❤ ♥➳✉ ♥â ✈ø❛ ❧➔ ✤ì♥ →♥❤ ✈ø❛ ❧➔ t♦➔♥ →♥❤✳ ✼ ❜➜t ❦ý✳ ❑❤✐ ✤â✿ |A1 ∪ A2 ∪ ∪ An | = |Ai1 | − 1≤i1 ≤n |Ai1 ∩ Ai2 | + 1≤i1 ≤i2 ≤n + (−1)k+1 |Ai1 ∩ Ai2 ∩ ∩ Aik | + 1≤i1 ≤i2 ≤ ik ≤n n+1 + (−1) |Ai1 ∩ Ai2 ∩ ∩ Ain | ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ tr➯♥ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ n✳ ❱ỵ✐ n = 1✱ ❝ỉ♥❣ t❤ù❝ tr➯♥ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥â ❝ơ♥❣ ✤ó♥❣ ❝❤♦ n = 2✳ ❚❛ ❝â✿ A1 = (A1 ∩ A2 ) ∪ (A1 \ (A1 ∩ A2 )), A2 = (A1 ∩ A2 ) ∪ (A2 \ (A1 ∩ A2 )), A1 ∪ A2 = (A1 ∩ A2 ) ∪ (A1 \ (A1 ∩ A2 )) ∪ (A1 ∩ A2 ) ∪ (A2 \ (A1 ∩ A2 )) ❍ñ♣ ❝õ❛ ❝→❝ ✈➳ ♣❤↔✐ ❧➔ ❤đ♣ ❝õ❛ ❝→❝ t➟♣ ✤ỉ✐ ♠ët rí✐ ♥❤❛✉✳ ❱➻ ✈➟② t❤❡♦ q✉② t➢❝ ❝ë♥❣✱ |A1 | = |A1 ∩ A2 | + |A1 \ (A1 ∩ A2 )|, |A2 | = |A1 ∩ A2 | + |A2 \ (A1 ∩ A2 )|, |A1 ∪ A2 | = |A1 ∩ A2 | + |A1 \ (A1 ∩ A2 )| + |A2 \ (A1 ∩ A2 )| ❚ø ❜❛ ✤➥♥❣ t❤ù❝ ♥➔② t❛ ♥❤➟♥ ✤÷đ❝ |A1 ∪ A2| = |A1| + |A2| − |A1 ∩ A2| ❱➟② ❝æ♥❣ t❤ù❝ tr➯♥ ✤ó♥❣ ❝❤♦ tr÷í♥❣ ❤đ♣ n = ●✐↔ sû ❝ỉ♥❣ t❤ù❝ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ n = m ✈➔ A1, A2, , Am, Am+1 ❧➔ m + t➟♣ ❤ú✉ ❤↕♥ ❜➜t ❦ý ✤➣ ❝❤♦✳ ❱➻ ❝æ♥❣ t❤ù❝ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ n = ♥➯♥ |A1 ∪ A2 ∪ ∪ Am ∪ Am+1 | = |(A1 ∪ A2 ∪ ∪ Am ) ∪ Am+1 | = |A1 ∪ A2 ∪ ∪ Am | + |Am+1 | − |(A1 ∪ A2 ∪ ∪ Am ) ∩ Am+1 | |Ai1 | − = 1≤i1 ≤m + (−1)k+1 + |Ai1 ∩ Ai2 | + 1≤i1 ≤i2 ≤m |Ai1 ∩ Ai2 ∩ ∩ Aik | + 1≤i1 ≤i2 ≤ ik ≤m m+1 (−1) |Ai1 ∩ Ai2 ∩ ∩ Aim | + |Am+1 | − |(A1 ∪ A2 ∪ ∪ Am ) ∩ Am+1 | ❚ø ❤❛✐ ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r ổ tự tr ỵ ụ ú n = m + ỵ ố t♦➔♥ →♥❤✮✳ ●✐↔ sû A, B ❧➔ ❝→❝ t➟♣ ❤ú✉ ❤↕♥ ✈ỵ✐ |A| = n ✈➔ |B| = k✳ ❑❤✐ ✤â✱ sè t➜t ❝↔ ❝→❝ t♦➔♥ →♥❤ f : A −→ B ❜➡♥❣ k!S(n, k)✱ tr♦♥❣ ✤â k S(n, k) = (−1)k−j k! j=0 k j j n ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ X ❧➔ t➟♣ t➜t ❝↔ ❝→❝ →♥❤ ①↕ f : A −→ B = {m1, , mk }, Xi = {f ∈ X | mi ∈ / f (A)}, i = 1, , k ❚❤❡♦ ỵ ũ trứ số t f : A −→ B ❜➡♥❣✿ |X \ (X1 ∪ ∪ Xk )| = |X| − |X1 ∪ ∪ Xk | = |X| − ( |Xi1 | − 1≤i1 ≤n (−1)k+1 |X1 ∩ + = kn − k + (−1)k−1 1≤i1 ≤i2 ≤n X2 ∩ ∩ Xk |) (k − 1)n + k k−1 k k i (−1)i = i=0 k k (k − 2)n − k k 1n + (−1)k 0n (k − i)n k k−j (−1)k−j = |Xi1 ∩ Xi2 | + j=0 jn k (−1)k−j = j=0 k j j n ❱➻ ✈➟②✱ k (−1)k−j k!S(n, k) = j=0 k j j n ⇔ S(n, k) = k (−1)k−j k! j=0 k j j n ✷✳✸✳ P❤➙♥ ❤♦↕❝❤✱ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐ ▼ư❝ ♥➔② ❣✐ỵ✐ t❤✐➺✉ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐ ✈➔ ù♥❣ ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ✤➳♠ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ t♦➔♥ →♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ❙è t➜t ❝↔ ❝→❝ ♣❤➙♥ ❤♦↕❝❤ t❤➔♥❤ k t➟♣ ❝õ❛ ♠ët t➟♣ n ♣❤➛♥ tû ✤÷đ❝ ❣å✐ ❧➔ sè ❙tr✐r❧✐♥❣ ❧♦↕✐ ữủ ỵ S(n, k) ú ỵ q ữợ S(n, 0) = ✈ỵ✐ ♠å✐ n ∈ N∗✱ S(0, k) = ✈ỵ✐ ♠å✐ k ∈ N∗✱ S(n, k) = ♥➳✉ k > n ✈➔ S(0, 0) = ✭✐✐✮ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐ ❧➔ sè ❝→❝❤ ♣❤➙♥ ♣❤è✐ n q✉↔ ❜â♥❣ ♣❤➙♥ ❜✐➺t ✈➔♦ k ❤ë♣ ❣✐è♥❣ ♥❤❛✉ ♠➔ ❦❤ỉ♥❣ ❝â ❤ë♣ ♥➔♦ ré♥❣✳ ❱➼ ❞ư ✸✳ ❝→❝ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ t➟♣ {a, b, c, d} t❤➔♥❤ ❜❛ ❦❤è✐ ❧➔✿ {{a}, {b}, {c, d}}✱ {{a}, {b, c}, {d}} ✱ {{a}, {b, d}, {c}} {{b}, {a, c}, {d}}✱ {{b}, {a, d}, {c}}✱ {{c}, {a, b}, {d}} ❉♦ ✤â✱ S ✭✹✱✸✮ ❂ ✻✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❤➺ t❤ù❝ s S(n, k) ỵ S(n + 1, k) = kS(n, k) + S(n, k − 1) ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t➟♣ ❜➜t ❦ý ❝â n + ♣❤➛♥ tû✱ ❝❤➥♥❣ ❤↕♥ t➟♣ A = {x1 , x2 , , xn+1 } ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â S(n + 1, k) ♣❤➙♥ ❤♦↕❝❤ t➟♣ A t❤➔♥❤ k t➟♣✳ ▼➦t ❦❤→❝ t❛ ❝â t❤➸ ❝❤✐❛ t➟♣ A t➜t ❝↔ ❝→❝ ♣❤➙♥ ❤♦↕❝❤ tr➯♥ t❤➔♥❤ ❤❛✐ t➟♣ ❝♦♥ P1 ✈➔ P2 rí✐ ♥❤❛✉ ♥❤÷ s❛✉✿ P1 ỗ tt A t k t➟♣ tr♦♥❣ ✤â ♠ët t➟♣ ❧➔ {xn+1}✱ ❝á♥ P2 ỗ tt A t k t➟♣ tr♦♥❣ ✤â ❦❤æ♥❣ t➟♣ ♥➔♦ ❧➔ {xn+1 }✳ ❑❤✐ ✤â ♠é✐ ♣❤➙♥ ❤♦↕❝❤ t❤✉ë❝ P1 s➩ ❝❤✐❛ t➟♣ {x1, x2, , xn} t❤➔♥❤ (k − 1) t➟♣ ✈➔ ❝â S(n, k − 1) ❝→❝❤ ❝❤✐❛ ♥❤÷ t❤➳✳ ❱➻ ✈➟②✱ |P1| = S(n, k − 1)✳ ◆➳✉ {xn+1} ❦❤æ♥❣ ❧➔ ♠ët t➟♣ t❤➻ xn+1 ❝➛♥ ❝❤ù❛ tr♦♥❣ ♠ët t➟♣ ✈ỵ✐ ➼t ♥❤➜t ♠ët ♣❤➛♥ tû ❦❤→❝ ♥ú❛ ❝õ❛ A✳ ❱➻ ❝â S(n, k) ❝→❝❤ ♣❤➙♥ ❤♦↕❝❤ t➟♣ {x1, x2, , xn} t❤➔♥❤ k t➟♣ ✈➔ xn+1 ❝â t❤➸ t❤✉ë❝ ♠ët t➟♣ ❜➜t ❦ý tr♦♥❣ sè ❝→❝ t➟♣ ✤â✱ ♥➯♥ t❛ ❝â t➜t ❝↔ ❧➔ kS(n, k) ❝→❝❤ ♣❤➙♥ ❤♦↕❝❤ t➟♣ A t❤➔♥❤ k t➟♣ s❛♦ ❝❤♦ {xn+1} ❦❤æ♥❣ ❧➔ ♠ët t➟♣ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ❦➨♦ t❤❡♦ [P ] = kS(n, k)✳ ❱➻ P = P1 ∪ P2 ✈ỵ✐ P1 ∩ P2 = ∅ ♥➯♥ t❤❡♦ q✉② t➢❝ ❝ë♥❣ S(n + 1, k) = |P | = |P1 | + |P2 | ❤❛② S(n + 1, k) = kS(n, k) + S(n, k − 1) ú ỵ r ợ n 1, S(n, 0) = 0, S(n, 1) = 1, S(n, n) = ✈➔ S(n, k) = ♥➳✉ k > n✳ ❱➻ ✈➟② t❛ ♥❤➟♥ ✤÷đ❝ t❛♠ ❣✐→❝ s❛✉ ✤➙② ❝❤♦ ❝→❝ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐✿ ✷✽ S(1, 1) = S(2, 1) = S(2, 2) = ✳✳ S(3, 1) = S(3, 2) = S(3, 3) = ỵ sỷ A ✈➔ B ❧➔ ❤❛✐ t➟♣ ❤ú✉ ❤↕♥ ✈ỵ✐ |A| = n ✈➔ |B| = k✳ ❙è t♦➔♥ →♥❤ f : A → B ❧➔ k!S(n, k)✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû B = {b1, , bk } ✈➔ f : A → B ❧➔ ♠ët t♦➔♥ →♥❤✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ f tr➯♥ t➟♣ A ♥❤÷ s❛✉✿ a1f a2 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (a1) = f (a2)✳ ∼ ∼ ❉➵ t❤➜② r➡♥❣ q✉❛♥ ❤➺ f ❧➔ ♠ët q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ A✳ ❱➻ t❤➳ ợ tữỡ ữỡ f t t ♠ët ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ A✳ ❱➻ f ❧➔ t♦➔♥ →♥❤✱ ∼ ♥➯♥ ♣❤➙♥ ❤♦↕❝❤ ♥➔② ❝â ✤ó♥❣ k ❦❤è✐✱ tù❝ ❧➔ t❛ ❝â t❤➸ ①❡♠ ♣❤➙♥ ❤♦↕❝❤ ✤â ❧➔ t➟♣ A = {A1, , Ak } ✈ỵ✐ ❝→❝ Ai, i = 1, , m ❧➔ ❝→❝ ❦❤è✐ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤✳ ❍ì♥ t❤➳ ♥ú❛✱ →♥❤ ①↕ f ❝↔♠ s✐♥❤ r❛ →♥❤ ①↕ f = A → B : Ai → f (Ai) = f (ai) ✈ỵ✐ ∈ Ai✳ ❉➵ t❤➜② r➡♥❣ f ❧➔ ♠ët s♦♥❣ →♥❤ ❣✐ú❛ A ✈➔ B ✳ ◆❣÷đ❝ ❧↕✐✱ ♠ët ♣❤➙♥ ❤♦↕❝❤ A ❝õ❛ A t❤➔♥❤ k ❦❤è✐ ❝ị♥❣ ✈ỵ✐ ♠ët s♦♥❣ →♥❤ f : A → B ①→❝ ✤à♥❤ ✤ó♥❣ ♠ët t♦➔♥ →♥❤ f : A → B : → f (ai) = f ([ai]) ✈ỵ✐ [ai] ❧➔ ❦❤è✐ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ A ❝❤ù❛ ai✳ ❚â♠ ❧↕✐✱ ♠ët t♦➔♥ →♥❤ f : A → B ❝â t❤➸ ❝♦✐ ❧➔ ♠ët ❝→❝❤ t❤ü❝ ❤✐➺♥ ❝õ❛ ♠ët ❤➔♥❤ ✤ë♥❣ H t r t ỗ H1 ✈➔ H2 ♥❤÷ s❛✉✿ ●✐❛✐ ✤♦↕♥ H1✿ ❚↕♦ r❛ ♠ët A A ỗ k ố ♥❣❤➽❛ ❝õ❛ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ❤❛✐✱ t❛ ❝â S(n, k) ❝→❝❤ t❤ü❝ ❤✐➺♥ ❣✐❛✐ ✤♦↕♥ H1✳ ●✐❛✐ ✤♦↕♥ H2✿ ❚↕♦ r❛ ♠ët ❤➔♠ s♦♥❣ →♥❤ f : A → B ✳ ◆❤÷ t❛ ✤➣ t➼♥❤ ð ❜➔✐ t♦→♥ ✤➳♠ t➜t ❝→ ❝→❝ ✤ì♥ →♥❤✱ t❛ ❝â k! ❝→❝❤ t❤ü❝ ❤✐➺♥ H2 ỵ t õ số ❝→❝ t♦➔♥ →♥❤ f : A → B ✈ỵ✐ |A| = n ✈➔ |B| = k ❜➡♥❣ k!S(n, k) ❚❛ ❝â t❤➸ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❦❤→❝ ♥❤÷ s❛✉✿ ●å✐ Fnk ❧➔ t➟♣ ❝→❝ →♥❤ ①↕ tø A ✤➳♥ B ❚❛ ❝â t❤➸ ❣✐↔ sû A = {1, 2, , n}✳ ❉➵ t❤➜②✱ ♠é✐ t♦➔♥ →♥❤ f ∈ Fn,k ❝❤♦ ♣❤➙♥ ❤♦↕❝❤ ❞✉② ♥❤➜t A = {1, 2, , n} = f −1 (1) ∪ f −1 (2) ∪ ∪ f −1 (k) ◆❣÷đ❝ ❧↕✐✱ ♠é✐ ♣❤➙♥ ❤♦↕❝❤ A = {1, 2, , n} = A1 ∪ A2 ∪ ∪ Ak t÷ì♥❣ ù♥❣ ✤ó♥❣ k! t♦➔♥ →♥❤✿ ✷✾ ✭✶✮ ❚➟♣ A1✱ ❝❤å♥ i1 ∈ B ✤➸ t➟♣ A1 = f −1(i1)✳ ❈â ✤ó♥❣ k ❝→❝❤ ❝❤å♥ sè ♥❣✉②➯♥ i1 ∈ B ✤➸ A1 = f −1(i1)✳ ✭✷✮ ❚÷ì♥❣ tü✱ ❝â ✤ó♥❣ k − ❝→❝❤ ❝❤å♥ i2 ∈ B \ {i1} ✤➸ A2 = f −1(i2)✳ ··· ✭❦✮ ❈â ❞✉② ♥❤➜t ✶ ❝→❝❤ ❝❤å♥ sè ik ∈ B ✤➸ Ak = f −1(ik )✳ ◆❤÷ ✈➟②✱ ♠é✐ ♣❤➙♥ ❤♦↕❝❤ t➟♣ A ❝❤♦ k! t♦➔♥ →♥❤✳ ❱➟② sè ❝→❝ t♦➔♥ →♥❤ tr♦♥❣ t➟♣ Fn,k ❜➡♥❣ k!S(n, k)✳ ❑➳t q✉↔ s❛✉ ❝❤♦ t❛ ❝æ♥❣ t❤ù❝ ①→❝ ✤à♥❤ số tr ỵ ợ số ♥❣✉②➯♥ ❞÷ì♥❣ n, k t❤ä❛ ♠➣♥ ≤ k ≤ n ❝â k S(n, k) = [ (−1)k−i k! i=0 k i in ] ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t➟♣ A = {a1, a2, , an} ✈➔ B = {1, 2, , k} ỵ số t tø A ❧➯♥ B ❜➡♥❣ k!S(n, k)✳ ▼➦t ❦❤→❝✱ ❜✐➸✉ ❞✐➵♥ →♥❤ ①↕ f : A → B ❜ð✐ a1 a2 an f (a1 ) f (a2 ) f (an ) t❛ ❝â ♥❣❛② {f (a1 ), f (a2 ), , f (an )} = {1, 2, , k} ❱✐➳t ❞➣② sè f (a1), , f (an) ♥❤÷ ♠ët ❝❤➾♥❤ ❤đ♣ ❧➦♣ ❝❤➟♣ n ❝õ❛ k sè✳ ◆❤÷ tữỡ ự ộ f ợ ú ởt ủ ❧➦♣ ❝❤➟♣ n ❝õ❛ k sè✳ ❙è ❝❤➾♥❤ ❤ñ♣ ❧➦♣ ú kn ỵ X t tt ❝↔ ❝→❝ →♥❤ ①↕ tø A ✈➔♦ B ✈➔ ❝❤♦ ộ i ỵ Xi t X ỗ tt tứ A B \ {i}✳ ❚❛ ❝â |X| = kn✱ |Xi| = (k − 1)n ✈➔ sj=1 Xij = (k − s)n✳ ❚➟♣ t➜t ❝↔ ❝→❝ t♦➔♥ →♥❤ tø A ❧➯♥ B ✤ó♥❣ X \ ki=1 Xi ỵ ũ trứ ỵ t ữủ k k!S(n, k) = k n − = |X| − | ❚❛ ❝â (k −1)n + k k k (k −2)n − +(−1)k (k −k)n k i=1 Xi | k i k!S(n, k) = (−1) i=0 k i k n (−1)k−1 (k − i) = i=0 ✸✵ k i in q ợ số ữỡ n, k tọ ♠➣♥ ≤ k ≤ n ❝â k (−1)k−i ak = i=0 k i bi ❚❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ❝ư t❤➸ ♥❤÷ s❛✉✿ ✣➦t ck = (−1)k ak õ ck = ki=0(1)i ki bi ỵ ✤➥♥❣ t❤ù❝ ♥➔② ♥❤÷ s❛✉ ck = (1 − b)k ✈➔ ❤✐➸✉ ❧➔ s❛✉ ❦❤✐ ❦❤❛✐ tr✐➸♥ t❤❛② bi ❜ð✐ bi ợ ỵ tự ú ợ tr x õ t t ỗ t tự ♥❤÷ s❛✉✿ (c + x)k = (−b + + x)k ✳ ❈❤♦ x = −1 t❛ ❝â (−1)k bk = (c − 1)k ❤❛② bk = (1 − c)k = ki=0(−1)i ki ci✳ ❱➟② kn = ki=0 ki i!S(n, i) ỵ kn = n i=0 S(n, i)Akn ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû A ✈➔ B ❧➔ ❝→❝ t➟♣ ✈ỵ✐ |A| = n ✈➔ |B| = k ✈➔ F ❧➔ t➟♣ t➜t ❝↔ ❝→❝ →♥❤ ①↕ f : A B ỵ Fi = {f F ||f (A)| = i} , i = 1, , k✳ ❑❤✐ ✤â Fi ∩ Fj = ∅ ♥➳✉ i = j ✈➔ F = F1 ∪ F2 ∪ ∪ Fk ✳ ❚❤❡♦ ♥❣✉②➯♥ ỵ |F | = |F1 ∪ Fk | = |F1| + + |Fk |✳ ▼é✐ f ∈ Fi ❝â t❤➸ ❝♦✐ ❧➔ ♠ët ❝→❝❤ t❤ü❝ ❤✐➺♥ ❝õ❛ ❤➔♥❤ ✤ë♥❣ H ✧t↕♦ r❛ tở Fi ỗ H1 ✈➔ H2 ♥❤÷ s❛✉✿ ●✐❛✐ ✤♦↕♥ H1✿ ❚↕♦ r❛ ❝→❝ t➟♣ ❝♦♥ K ⊆ B ❧ü❝ ❧÷đ♥❣ i✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ tê ❤ñ♣✱ t❛ ❝â ki ❝→❝❤ t❤ü❝ ❤✐➺♥ H1✳ ●✐❛✐ ✤♦↕♥ H2✿ ❚↕♦ r❛ ♠ët t♦➔♥ →♥❤ f : A → K ✳ ❚❤❡♦ ❜➔✐ t♦→♥ ✤➳♠ ❝→❝ t♦➔♥ →♥❤ tø ♠ët t➟♣ ❤ú✉ ❤↕♥ ❧➯♥ ♠ët t➟♣ ❤ú✉ ❤↕♥✱ t❛ ❝â i!S(n, i) ❝→❝❤ t❤ü❝ ❤✐➺♥ H2 ỵ |Fi | = n i k!S(n, i) = S(n, i)Aim ❉♦ ✤â |F | = n i i=1 S(n, i)Am ✸✶ ❚❤❡♦ ❜➔✐ t♦→♥ ✤➳♠ t➜t ❝↔ ❝→❝ →♥❤ ①↕ tø ♠ët t➟♣ ❤ú✉ ❤↕♥ ✈➔♦ ♠ët t➟♣ ❤ú✉ ❤↕♥✱ |F | = kn ú ỵ S(n, 0) = 0, S(n, i) = ♥➳✉ i > n, Aik = ♥➳✉ i > k ✱ ♥➯♥ t❛ ♥❤➟♥ ✤÷đ❝ n n k = |F | = S(n, 0)A0k n S(n, i)Aik + S(n, i)Aik = i=1 i=1 ỵ t ữủ ự ú ỵ r ữớ t❛ ❝ơ♥❣ ✤➣ ✤÷❛ r❛ ♠ët sè ❜✐➸✉ ❞✐➵♥ ❦❤→❝ ❝❤♦ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ✷ ♥❤÷ s❛✉✿ ✭✶✮ S(n, k) = k!1 ni + +i i , i n, , i ✭✷✮ S(n + 1, k) = k n j=0 n j k S(j, k − 1)✳ ✷✳✹✳ ✣➳♠ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ✈➔ q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ❜➢❝ ❝➛✉ ▼ư❝ ♥➔② t➻♠ ❤✐➸✉ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✱ sè q✉❛♥ ❤➺ t❤ù tü✱ sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ tr➯♥ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ❚❤❡♦ ỵ số q tữỡ ữỡ tr ởt t➟♣ A ❝❤➼♥❤ ❧➔ sè ♣❤➙♥ ❤♦↕❝❤ tr➯♥ A ✈➔ õ số ữủ ợ t r ❚❡♠♣❧❡ ❇❡❧❧ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✸✵✳ ❙è ❇❡❧❧ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ tê ❤đ♣✱ ①→❝ s✉➜t✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✳ ❙è ❇❡❧❧ t❤ù n ❧➔ sè ❝→❝❤ ♣❤➙♥ ❤♦↕❝❤ t➟♣ n ♣❤➛♥ tû ✭❤❛② ♥â✐ ❝→❝❤ ❦❤→❝ ❧➔ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ t➟♣ n ♣❤➛♥ tû✮✳ ●å✐ sè Bn ❧➔ sè ❇❡❧❧ t❤ù n✳ ❱ỵ✐ n = 3✱ ✈➔ t➟♣ S = {1, 2, 3}✱ t❛ ❝â B3 = ✈➻ ❝â ❝→❝❤ ♣❤➙♥ ❤♦↕❝❤✿ {{1}, {2}, {3}}✱ {{1, 2}, {3}}✱ {{1, 3}, {2}}✱ {{1}, {2, 3}} {{1, 2, 3}}✳ ▼ët ✈➔✐ sè ❇❡❧❧ ✤➛✉ t✐➯♥✿ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ú ỵ Bn = S(n, 1) + S(n, 2) + + S(n, n) ❚❛ ❝â ổ tự tr ỗ s số ỵ ✷✳✹✳✸✳ ❙è ❇❡❧❧ t❤ä❛ ♠➣♥ n Bn+1 = k=0 ✸✷ n k Bk ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ♠ët ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ t➟♣ S = {1, 2, , n + 1}, A1 , , Am ❚❛ ❝â t❤➸ ❣✐↔ sû n + ∈ A1 ✈➔ |A1| = k + 1, ≤ k ≤ n ❑❤✐ ✤â A2, , Am ❧➟♣ t❤➔♥❤ ♠ët ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ t➟♣ n − k ♣❤➛♥ tû ❝á♥ ❧↕✐ ❝õ❛ S ✱ tù❝ ❧➔ t➟♣ S \ A1 ✳ ❈â Bn−k ♣❤➙♥ ❤♦↕❝❤ t➟♣ ♥➔②✳ ❱➻ ✈➟② tê♥❣ sè ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ S ♠➔ n n + tr♦♥❣ ♠ët t➟♣ k + ♣❤➛♥ tû ❧➔ Bn−k ❉♦ ✤â k n n k Bn+1 = k=0 ❤❛② n n n−k Bn+1 = ❱➟② k=0 n Bn+1 = k=0 n k Bn−k , Bn−k Bk ●❤✐ ❝❤ó ✷✳✹✳✹ ✭❚❛♠ ❣✐→❝ ❇❡❧❧✮✳ ❈→❝ sè ❇❡❧❧ ❝â t❤➸ ❞➵ ❞➔♥❣ t➼♥❤ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ t❛♠ ❣✐→❝ ❇❡❧❧✱ ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❆✐t❦❡♥ ❤♦➦❝ t❛♠ ❣✐→❝ P❡✐r❝❡✿ ❇➢t ✤➛✉ ✈ỵ✐ sè ♠ët✳ ✣➦t sè ♥➔② tr➯♥ ❞á♥❣ t❤ù ♥❤➜t✳ ❚↕♦ ♠ët ❞á♥❣ ♠ỵ✐ ❜➡♥❣ ✈✐➺❝ ❧➜② ♣❤➛❤ tû ❝ü❝ ♣❤↔✐ ❝õ❛ ❞á♥❣ ♥❣❛② tr➯♥ ♥â ❧➔♠ ♣❤➛♥ tû ✤➛✉ t✐➯♥ ❜➯♥ tr→✐ ❝õ❛ ỏ ợ ữủt t số t t ❞á♥❣ ♠ỵ✐ ❜➡♥❣ ❝→❝❤ ❧➜② tê♥❣ ♣❤➛♥ tû ❜➯♥ tr→✐ ♥â ✈ỵ✐ ♣❤➛♥ tû ✤ù♥❣ ❝ị♥❣ ❝ët ♣❤➛♥ tû ➜② ỏ trữợ õ tử ữợ ❦❤✐ sè ♣❤➛♥ tû ❝õ❛ ❞á♥❣ ♠ỵ✐ ♥❤✐➲✉ ❤ì♥ sè ♣❤➛♥ tû ❝õ❛ ❞á♥❣ tr➯♥ ♠ët ♣❤➛♥ tû✳ ❙è ♥➡♠ ♣❤➼❛ tr→✐ ♠é✐ ❞á♥❣ ❧➔ sè ❇❡❧❧ ❝❤♦ ♠é✐ ❞á♥❣✳ ữ ỏ tự t ỗ số ỏ t✐➳♣ t❤❡♦ ✭t❤ù ❤❛✐✮ ✤÷đ❝ t↕♦ r❛ ❜➡♥❣ ❝→❝❤ ❧➜② ♣❤➛♥ tû ✤➛✉ t✐➯♥ ❜➯♥ ♣❤↔✐ ❝õ❛ ❞á♥❣ tr➯♥ ✤➦t ✈➔♦ ✈à tr➼ ✤➛✉ t✐➯♥ ❜➯♥ tr→✐✳ ❚❛ ❝â✿ 1 x ●✐→ trà ❝õ❛ x ❧➔ tê♥❣ ❝õ❛ ❤❛✐ tỷ ởt tữợ õ ú ỏ ✈➔ ❞á♥❣ ✸✸ tr➯♥ ✭❝ô♥❣ ❧➔ ✶✮ ❜➡♥❣ ✷✳ 1 y ●✐→ trà y ❜➡♥❣ ❣✐→ trà ✤➛✉ t✐➯♥ t➼♥❤ tø ❜➯♥ ♣❤↔✐ ❝õ❛ ❞á♥❣ tr➯♥ ✭❜➡♥❣ ✷✮✱ ✈➔ t✐➳♣ t❤❡♦✿ 1 2 x ❇➡♥❣ ❝→❝❤ ➜② t❛ ❝â ✺ ❞á♥❣ ✤➛✉ ❝õ❛ t❛♠ ❣✐→❝ ❧➔✿ 1 15 10 15 20 27 37 52 ❉á♥❣ t❤ù ♥➠♠ ✤÷đ❝ t➼♥❤ ♥❤÷ s❛✉✿ ▲➜② ✶✺ tø ❞á♥❣ t❤ù t÷ ✶✺ ✰ ✺ ❂ ✷✵❀ ✷✵ ✰ ✼ ❂ ✷✼❀ ✷✼ ✰ ✶✵ ❂ ✸✼❀ ✸✼ ✰ ✶✺ ❂ ✺✷✳ ❙è ✤ù♥❣ ð ❞á♥❣ t❤ù n ✈➔ ❝ët t❤ù k ❧➔ sè ❝→❝ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ t➟♣ {1, , n} s❛♦ ❝❤♦ n ❧➔ ❦❤ỉ♥❣ ❝ị♥❣ ♠ët ❧ỵ♣ ✈ỵ✐ ❜➜t ❦ý sè ♥➔♦ tr♦♥❣ ❝→❝ ♣❤➛♥ tû k, k + 1, , n − ❈❤➥♥❣ ❤↕♥ ❝â ✼ ♣❤➙♥ ❤♦➦❝ ❝õ❛ {1, , 4} s❛♦ ❝❤♦ ✹ ❦❤ỉ♥❣ ❝ị♥❣ ❧ỵ♣ ✈ỵ✐ ❝→❝ ♣❤➛♥ tû ✷✱ ✸✱ ✈➔ ❝â ✶✵ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ {1, , 4} s❛♦ ❝❤♦ ✹ ❦❤ỉ♥❣ ❝ị♥❣ ❧ỵ♣ ✈ỵ✐ ♣❤➛♥ tû ✸✳ ❍✐➺✉ ❝õ❛ ❤❛✐ sè tr➯♥ ✭❜➡♥❣ ✸✮ ❧➔ sè ❝→❝ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ {1, , 4} s❛♦ ❝❤♦ ✹ ❝ị♥❣ ❧ỵ♣ ✈ỵ✐ ✷ ữ ổ ũ ợ ợ ố õ r➡♥❣ ❝â ✸ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ {1, , 3} s❛♦ ❝❤♦ ✸ ❦❤ỉ♥❣ ❝ị♥❣ ❧ỵ♣ ✈ỵ✐ ✷✳ ❚✐➳♣ t❤❡♦ t❛ t➻♠ ❤✐➸✉ sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❜➢❝ ❝➛✉ ❧✐➯♥ ❤➺ ✈ỵ✐ sè q✉❛♥ ❤➺ t❤ù tü tr➯♥ ♠ët t➟♣ ❤ú✉ ❤↕♥✳ ❈❤♦ A ❧➔ t➟♣ ❝â n tỷ ỵ 2X t t➟♣ ❝♦♥ ❝õ❛ X ✳ ◆❤➢❝ ❧↕✐ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤÷đ❝ ❣å✐ ❧➔ t❤ù tü ♥➳✉ ♣❤↔♥ ①↕ ✈➔ ❜➢❝ ❝➛✉❀ ✤÷ì❝ ❣å✐ ❧➔ t❤ù tü ✭tø♥❣ ♣❤➛♥✮ ♥➳✉ ♣❤↔♥ ①↕✱ ✤è✐ ①ù♥❣ ✈➔ ỵ T (A) ỵ t q ổ tr A P (A) ỵ ❤✐➺✉ t➟♣ q✉❛♥ ❤➺ t❤ù tü tr➯♥ A✳ ✣➦t Tn = |T (A)| sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ tr➯♥ A✱ Pn = |P (A)| sè q✉❛♥ ❤➺ t❤ù tü tr➯♥ A✱ Qn sè q✉❛♥ ❤➺ tü❛ t❤ù tü tr➯♥ A✳ ✸✹ ◆➳✉ n = 0✱ ✤➦t T0 = P0 = Q0 = ỵ P (A) t ❤♦↕❝❤ ❝õ❛ A✳ ●✐↔ sû P ∈ P (A) ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ A✱ ♣❤➛♥ tû ❝õ❛ P ❝❤ù❛ x A ỵ [x] s rr ❬✺❪ ✤➣ ✤÷❛ r❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ sè q✉❛♥ ❤➺ tü❛ t❤ù tü ✈➔ q✉❛♥ ❤➺ t❤ù tü tr➯♥ t➟♣ n ♣❤➛♥ tû n Qn = S(n, k)Pk k=1 ❚r♦♥❣ ♠ư❝ ♥➔② t❛ ✤÷❛ r❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ ✈➔ sè q✉❛♥ ❤➺ t❤ù tü t❤❡♦ ❜➔✐ ❜→♦ ❬✻❪✳ ❚ø ❝→❝ ❦➳t q✉↔ ✤➣ ❜✐➳t ❬✸✱ ✹✱ ✺❪ t❛ ①→❝ ✤à♥❤ ✤÷đ❝ sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ tr➯♥ t➟♣ n ♣❤➛♥ tû ✈ỵ✐ n ≤ 14✳ ✣➦t Ta(A) ❧➔ t➟♣ ❝→❝ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ❜➢❝ ❝➛✉ ✈➔ ♣❤↔♥ ✤è✐ ①ù♥❣ tr➯♥ A✳ ●✐↔ sû X ⊆ A ✈➔ P ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ t➟♣ A \ X ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ Ta∗ (X ∪ P) = {ϕ ∈ Ta (X ∪ P)|∀z ∈ X ∪ P, zϕz ⇔ z ∈ P} ỗ t s f : T (A) → Ta∗ (X ∪ P) ( X∈2A P∈P (A\X) ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ ϕ ∈ T (A) t❛ ✤à♥❤ ♥❣❤➽❛ t➟♣ Aϕ = {x ∈ A|(x, x) ∈ ϕ} ❱ỵ✐ ∀x, y ∈ Aϕ✱ t❛ ✤à♥❤ ♥❣❤➽❛ x ∼ y ⇔ (x, y) ∈ ϕ ✈➔ (y, x) ∈ ϕ ❱ỵ✐ ♠å✐ x ∈ Aϕ t❛ ❝â (x, x) ∈ ϕ ♥➯♥ x ∼ x✳ ❙✉② r❛ ∼ ❝â t➼♥❤ ❝❤➜t ♣❤↔♥ ①↕✳ ❱ỵ✐ ♠å✐ x, y ∈ Aϕ ❣✐↔ sû x ∼ y✳ ❑➨♦ t❤❡♦ (x, y) ∈ ϕ ✈➔ (y, x) ∈ ϕ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ y ∼ x✳ s✉② r❛ ∼ ❝â t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣✳ ❱ỵ✐ ♠å✐ x, y, z ∈ Aϕ✱ ❣✐↔ sû x ∼ y ✈➔ y ∼ z ✳ ❑➨♦ t❤❡♦ (x, y) ∈ ϕ, (y, z) ∈ ϕ, (z, y) ∈ ϕ, (y, x) ∈ ϕ✳❱➻ ϕ ❝â t➼♥❤ ❝❤➜t ❜➢❝ ❝➛✉ ♥➯♥ (x, z) ∈ ϕ ✈➔ (z, x) ∈ ϕ✳ ❙✉② r❛ ∼ ❝â t➼♥❤ ❝❤➜t ❜➢❝ ❝➛✉✳ ✈➟② ∼ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ tr➯♥ Aϕ✳ ✣➦t X = A\Aϕ ✈➔ P = Aϕ/∼✳ ❚❛ ❝â P ❧➔ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ Aϕ = A\X ❤❛② P ∈ P (A \ X)✳ ❱ỵ✐ ♠é✐ ϕ ∈ T (A)✱ t❛ ✤à♥❤ ♥❣❤➽❛ σ ∈ Ta∗(X ∪ P) ♥❤÷ s❛✉✳ ●✐↔ sû (x, y) ∈ ϕ ✸✺ ◆➳✉ x, y ∈ X t❛ ✤➦t (x, y) ∈ σ✱ ◆➳✉ x ∈ A \ X = Aϕ, y ∈ X t❛ ✤➦t ([x], y) ∈ ϕ✱ ◆➳✉ x ∈ X, y ∈ A \ X ✱ ✤➦t (x, [y]) ∈ σ✱ ◆➳✉ x, y ∈ A \ X ✱ ✤➦t ([x], [y]) ∈ σ✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ X ✱ t❛ ❝â zσz ⇔ z ∈ P ❘ã r➔♥❣ σ ❝â t➼♥❤ ❝❤➜t ❜➢❝ ❝➛✉✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ σ ♣❤↔♥ ①ù♥❣✳ ❱ỵ✐ ♠å✐ ∀z, w ∈ X ∪ P ❣✐↔ sû zσw ✈➔ wσz ✳ ◆➳✉ z, w ∈ X t❛ ❝â zϕw ✈➔ wϕz ✳ ❑➨♦ t❤❡♦ zσz ❤❛② z ∈ A = A \ X ổ ỵ ữỡ tỹ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ t❛ ❝â z = [x] ∈ P, w = [y] ∈ P✳ ❱➻ zσz ✈➔ wσz ♥➯♥ xϕy ✈➔ yϕx✳ ❑➨♦ t❤❡♦ [x] = [y]✳ ❱➟② σ ⊆ Ta∗ (X ∪ P)✳ ✣➦t f (ϕ) = σ ✳ ❚❛ ❝â f ❧➔ →♥❤ ①↕✳ ●✐↔ sû ϕ, ϕ ∈ T (A), f (ϕ) = σ = f (ϕ = σ ✈ỵ✐ (x, y) ∈ ϕ✳ ✣➦t X = A \ Aϕ✱ X = A \ Aϕ ✳ ●✐↔ sû x, y ∈ X ✱ t❛ ❝â x ∈ X, y ∈ Aϕ ✱ t❛ ❝â (x; [y]) ∈ σ = σ ✳ ❑➨♦ t❤❡♦ x ∈ X , y ∈ Aϕ ✈➔ xϕ y ❚✐➳♣ tö❝ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ t❛ ❝â (x, y) ∈ ϕ ✳ ❙✉② r❛ ϕ ⊆ ϕ ✳ ❚÷ì♥❣ tü t❛ ❝â ϕ = ϕ ✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ ♠✐♥❤ f ỡ ợ tở t ỗ t↕✐ X ⊆ A✱ P ❧➔ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ A \ X s❛♦ ❝❤♦ σ ∈ Ta∗(X ∪ P)✳ ❱ỵ✐ ♠é✐ (z, w) ∈ σ✱ ♥➳✉ z, w ∈ X ❧➜② (z, w) ∈ ϕ✳ ◆➳✉ z = [x] ∈ P, w ∈ X ❧➜② (u, w) ∈ ϕ, ∀u ∈ [x], ◆➳✉ z ∈ X, w = [y] ∈ P✱ ❧➜② (z, θ) ∈ ϕ, ∀θ ∈ [y], ◆➳✉ z = [x], w = [y] ∈ P✱ ❧➜② (u, θ) ∈ ϕ, ∀u ∈ [x], ∀θ ∈ [y]✳ ❉➵ t❤➜② ϕ ∈ T (A) ✈➔ f (ϕ) = σ✳ ❉♦ ✤â f ❧➔ t♦➔♥ →♥❤✳ ❱➟② f ❧➔ s♦♥❣ ỗ t s f : Ta∗ (X ∪ P) → P (X ∪ P) ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ σ ∈ Ta∗(X ∪ P)✱ ✤➦t f (σ) = σ∪ X∪P tr♦♥❣ ✤â {(z, z)|z ∈ X ∪ P}✳ ❚❛ ❝â f ❧➔ s♦♥❣ →♥❤✳ ❱➼ ❞ö s❛✉ ♠✐♥❤ ❤å❛ ❝→❝ ❦➳t q✉↔ ð tr➯♥✳ ❱➼ ❞ö ✷✳✹✳✼✳ ❳➨t A = {a, b, c, d, e, f } ✈➔ q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ X∪P = ϕ = {(a, a), (b, b), (d, d), (e, e), (f, f ), (a, b), (b, a), (a, c), (b, c)} {(d, e), (e, d), (d, c), (e, e), (d, f ), (e, f )} tr➯♥ A✳ ❚❛ ❝â Aϕ = {a, b, c, d, e, f }✱ X = {c}✳ ◗✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣ ∼ ❧➔ {[a, a], [b, b], [d, d], [e, e], [f, f ], [a, b], [b, a], [d, e], [e, d]}, ✸✻ P = Aϕ/∼ = {[a] = [b] = {a, b}, [d] = [e] = {d, e}, [f ] = {f }}, X ∪ P = {c, {a, b}, {d, e}, {f }}, σ = {({a, b}, {a, b}), ({d, e}, {d, e}), ({f }, {f })} {({a, b}, c), ({d, e}, c), ({d, e}, {f })} ◗✉❛♥ ❤➺ σ ❧➔ ♣❤↔♥ ✤è✐ ①ù♥❣ ✈➔ ❜➢❝ ❝➛✉ ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔♥ ①↕✳ t❛ ❝â σ∪ ❧➔ q✉❛♥ ❤➺ t❤ù tü tr➯♥ X ∪ P✳ ❚ø ✤â t❛ õ t q ỵ sû n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❑❤✐ ✤â n Nk (n)Pk , Tn = k=1 tr♦♥❣ ✤â Nk (n) = k s=0 n S(n − s, k − s) s ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✳✷ ✈➔ ❇ê ✤➲ ✷✳✶✳✸✱ t❛ ❝â Tn = |T (A)| = | P (X ∪ P))| ( X∈2A P∈P (A\X) ❱➻ X ∩ P = ∅ ♥➯♥ t❛ ❝â n Tn = k=0 n k n−k S(n − k, m)Pk+m m=0 ❘ót ♥❤➙♥ tû Pk ✱ t❛ ❝â n Tn = k ( k=1 s=0 n )Pk = S(n − s, k − s) s n Nk (N )Pk k=1 ❍➺ q✉↔ ✷✳✹✳✾✳ ❈❤♦ n ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❛ ❝â n−1 ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â n s=0 Pn = n (Tn − Nk (n)Pk ) k=1 n S(n − s, n − s) = s ❚ø ✤â t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✸✼ n s=0 n = 2n s X∪P ❱➼ ❞ö ✷✳✹✳✶✵✳ ❚r♦♥❣ ✈➼ ❞ö ♥➔② t❛ t➼♥❤ T4✳ ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ✷ t❛ ❝â N1(4) = 1, N3(4) = 24 ✈➔ N4(4) = 16✳ ❚❤❡♦ ❬✶❪✱ P1 = 1, P2 = 3, P4 = 219✳ ❚ø ✤â t❛ ❝â T4 = P1 + 11P2 + 24P3 + 16P4 = + 33 + 456 + 3504 = 3994 ❇↔♥❣ s❛✉ ❝❤♦ t❛ sè Pn ✈➔ Tn ✈ỵ✐ n ≤ 14✳ ❚➼♥❤ tỵ✐ t❤í✐ ✤✐➸♠ ❜➔✐ ❜→♦ T14 ❧➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❞➣② tr➯♥✳ ❚❤❡♦ ❤✐➸✉ ❜✐➳t ❝õ❛ t→❝ ❣✐↔ ✤➳♥ ♥❛② ♥❣÷í✐ t❛ t➻♠ ✤÷đ❝ Tn✱ n ≤ 18✳ P1 = P2 = P3 = 19 P4 = 219 P5 = 4231 P6 = 130023 P7 = 6129859 P8 = 431723379 P9 = 44511042511 P10 = 6611065248783 P11 = 13962816771058999 P12 = 414864951055853499 P13 = 171850728381587059351 P14 = 98484324257128207032183 T2 = T2 = 13 T3 = 171 T4 = 3994 T5 = 154303 T6 = 9415189 T7 = 878222530 T8 = 122207703623 T9 = 24890747921947 T10 = 7307450299510288 T11 = 3053521546333103057 T12 = 1797003559223770324237 T13 = 1476062693867019126073312 T14 = 1679239558149570229156802997 ✸✽ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ◗✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✲ ❚➻♠ ❤✐➸✉ ❤➺ t❤è♥❣ ❦✐➳♥ t❤ù❝ ✈➲ q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✱ ỵ tt tờ ủ số t➟♣ ❝♦♥ ❝õ❛ ♠ët t➟♣ ✈➔ ù♥❣ ❞ö♥❣✱ t➻♠ ❤✐➸✉ ✤➳♠ sè q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ ♣❤↔♥ ①↕ ✭r❡❢❧❡①✐✈❡✮✱ ✤è✐ ①ù♥❣ ✭s②♠♠❡tr✐❝✮✱ ❜➜t ✤è✐ ①ù♥❣ ✭❛s②♠♠❡tr✐❝✮✱ ♣❤↔♥ ✤è✐ ①ù♥❣ ✭❛♥t✐s②♠♠❡tr✐❝✮✱ ❜➜t ♣❤↔♥ ①↕ ✭✐rr❡❢❧❡①✐✈❡✮✳ ✲ ❚➻♠ ❤✐➸✉ ❜➔✐ t♦→♥ ✤➳♠ sè →♥❤ ①↕ ✈➔ ✤➳♠ ❝→❝ →♥❤ ①↕ ✤➦❝ ❜✐➺t ✭✤ì♥ →♥❤✱ s♦♥❣ →♥❤✱ t♦➔♥ →♥❤✮✳ ✲ ❚➻♠ ❤✐➸✉ sè ❙t✐r❧✐♥❣ ❧♦↕✐ ✷ ✈➔ ❜➔✐ t♦→♥ ♣❤➙♥ ❤♦↕❝❤ t➟♣ ❤ñ♣✳ ✲ ❚➻♠ ❤✐➸✉ ✤➳♠ sè q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✱ sè ❇❡❧❧✱ ✤➳♠ sè q✉❛♥ ❤➺ ❜➢❝ ❝➛✉ ❧✐➯♥ ❤➺ ✈ỵ✐ sè q✉❛♥ ❤➺ t❤ù tü✳ ✸✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ◆❣✉②➯♥ ❆♥ ✈➔ ◆❣✉②➵♥ ❱➠♥ ❍♦➔♥❣ ✭✷✵✶✻✮✱ ❚➟♣ ❤ñ♣ ✈➔ ❧♦❣✐❝ ❚♦→♥✱ ◆❳❇ ✣↕✐ ❤å❝ ❚❤→✐ ổ ỵ tt tờ ủ ỗ t ố ❚✐➳♥❣ ❆♥❤ ❬✸❪ ●✳ ❇r✐♥❦♠❛♥♥ ❛♥❞ ❇✳ ❉✳ ▼❝❑❛② ✭✷✵✵✷✮✱ P♦s❡ts ♦♥ ✉♣ t♦ ✶✻ ♣♦✐♥ts✱ ❖r❞❡r✱ ✶✾ ✭✷✮✱ ✶✹✼✲✶✼✾✳ ❬✹❪ ▼✳ ❊r♥❡ ❛♥❞ ❑✳ ❙t❡❣❡ ✭✶✾✾✶✮✱ ❈♦✉♥t✐♥❣ ❢✐♥✐t❡ ♣♦s❡ts ❛♥❞ t♦♣♦❧♦❣✐❡s✱ ❖❞❡r ✽✭✸✮✱ ✷✹✼✲✷✻✺✳ ❬✺❪ ❏✳❲✳ ❊✈❛♥s✱ ❋✳ ❍❛r❛r② ❛♥❞ ▼✳ ❙✳ ▲②♥♥ ✭✶✾✻✼✮✱ ❖♥ t❤❡ ❝♦♠♣✉t❡r ❡♥✉✲ ♠❡r❛t✐♦♥ ♦❢ ❢✐♥✐t❡ t♦♣♦❧♦❣✐❡s✱ ❈♦♦♠✳ ❆❈▼ ✶✵✱ ✷✾✺✲✷✾✽✳ ❬✻❪ ❏✳ ❑❧❛s❦❛ ✭✶✾✾✼✮✱ ✧❚r❛♥s✐t✐✈✐t② ❛♥❞ ♣❛rt✐❛❧ ♦r❞❡r✧✱ ▼❛t❤✳ ❇♦❤❡♠✳ ✶✷✷✭✶✮✱ ✼✺✲✽✷✳ ❬✼❪ ❑✳ ❍✳ ❘♦s❡♥ ✭✷✵✵✼✮✱ ✧❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✧✱ ✻t❤ ❊❞✐t✐♦♥✱ ▼❝●r❛✇ ❍✐❧❧✳ ... 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: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Trần... ❝æ♥❣ t❤ù❝ ①→❝ ✤à♥❤ số tr ỵ ợ số ♥❣✉②➯♥ ❞÷ì♥❣ n, k t❤ä❛ ♠➣♥ ≤ k ≤ n ❝â k S(n, k) = [ (−1)k−i k! i=0 k i in ] ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t t➟♣ A = {a1, a2, , an} ✈➔ B = {1, 2, , k} ỵ số t tø A ❧➯♥ B... q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ✤➦❝ ❜✐➺t ♥❤÷ sè q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐ ♣❤↔♥ ①↕✱ số q ổ ố ự rữợ t t❛ ✤÷❛ r❛ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚❤❡♦ ỵ tt tờ ủ số t n! k tû ❝õ❛ t➟♣ n ❧➔ nk = ❚❛ ✤÷❛ r❛ ♠ët ❝❤ù♥❣ ♠✐♥❤ ❝➜♣