❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
◆♦çõ❡s ❇ás✐❝❛s ❞❡ ■♥✜♥✐t♦ ❡
◆ú♠❡r♦s ❈❛r❞✐♥❛✐s
†♣♦r
❆❧❡ss❛♥❞r♦ ▼✐❣♥❛❝ ❈❛r♥❡✐r♦ ▲❡ã♦
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❢❡✈❡r❡✐r♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
L436n Leão, Alessandro Mignac Carneiro.
Noções básicas de infinito e números cardinais / Alessandro Mignac Carneiro Leão.-- João Pessoa, 2014. 57p.
Orientador: Bruno Henrique Carvalho Ribeiro Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Cantor. 3. Teoria dos Conjuntos. 4.Números cardinais. 5. Números transfinitos. 6. Aritmética cardinal.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣♦r ♣❡r♠✐t✐r ❡ss❛ ♥♦✈❛ ❝♦♥✲ q✉✐st❛ ♥♦s ♠❡✉s ❡st✉❞♦s✳ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ▼✐❝❤❡❧② ❆❧✈❡s ❈❛r♥❡✐r♦ ▲❡ã♦ ❡ ♠✐♥❤❛ q✉❡r✐❞❛ ✜❧❤❛ ❈❧❛r✐ss❛ ▼✐❣♥❛❝ ❈❛r♥❡✐r♦ ▲❡ã♦✱ ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦ ❡ ❛♣♦✐♦ ✐♥❝♦♥❞✐✲ ❝✐♦♥❛❧✳ ➚ ❝♦♦r❞❡♥❛çã♦ ❡ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✳ ❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉♦✉t♦r ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❝♦♥st❛♥t❡ ❡ ♣❡❧❛s ❝♦♥tr✐❜✉✐çõ❡s✱ s✉❣❡stõ❡s ❡ ❝rít✐❝❛s q✉❡ ♠✉✐t♦ ❝♦♥tr✐❜✉í✲ r❛♠ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ♠♦str❛♠♦s ✉♠ ♣♦✉❝♦ ❛ t❡♦r✐❛ s♦❜r❡ ♦s ❝❤❛♠❛❞♦s ♥ú♠❡r♦s tr❛♥s✜✲ ♥✐t♦s ❡ s✉❛ ❛r✐t♠ét✐❝❛ ❝❛r❞✐♥❛❧✳ P❛r❛ t❛♥t♦✱ tr❛❜❛❧❤❛♠♦s t❛♠❜é♠ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❥✉♥t♦s✱ ❜❡♠ ❝♦♠♦ ❡q✉✐♣♦tê♥❝✐❛✱ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✐♥✜♥✐t♦s✱ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s ❡ ♥ã♦✲❡♥✉♠❡rá✈❡✐s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❈❛♥t♦r✱ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s✱ ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s✱ ◆ú♠❡r♦s ❚r❛♥s✜♥✐t♦s✱ ❆r✐t♠ét✐❝❛ ❈❛r❞✐♥❛❧
■♥ t❤✐s ✇♦r❦✱ ✇❡ s❤♦✇ ❜❛s✐❝ r❡s✉❧ts ❛❜♦✉t t❤❡ s♦✲❝❛❧❧❡❞ tr❛♥s✜♥✐t❡ ♥✉♠❜❡rs ❛♥❞ t❤❡✐r ❝❛r❞✐♥❛❧ ❛r✐t❤♠❡t✐❝✳ ❋♦r t❤❡s❡ ♣✉r♣♦s❡✱ ✇❡ ❛❧s♦ s❤♦✇ s♦♠❡ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ t❤❡ s❡t t❤❡♦r②✱ ❛s ✇❡❧❧ ❛s ❡q✉✐♥✉♠❡r♦s✐t②✱ ✜♥✐t❡ s❡ts✱ ✐♥✜♥✐t❡ s❡ts✱ ❝♦✉♥t❛❜❧❡ s❡ts ❛♥❞ ✉♥✲ ❝♦✉♥t❛❜❧❡ s❡ts✳
❑❡②✇♦r❞s✿ ❈❛♥t♦r✱ ❙❡t ❚❤❡♦r②✱ ❈❛r❞✐♥❛❧ ◆✉♠❜❡rs✱ ❚r❛♥s✜♥✐t❡ ◆✉♠❜❡rs✱ ❈❛r❞✐♥❛❧ ❆r✐t❤♠❡t✐❝✳
❙✉♠ár✐♦
✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ❞❡ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ✶ ✶✳✶ ❆ ▲✐♥❣✉❛❣❡♠ ❞♦s ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ❈♦♥❥✉♥t♦ ❞❡ P❛rt❡s ❡ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ ❋❛♠í❧✐❛ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ❈♦♠♣♦s✐çã♦ ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ■♥✈❡rsã♦ ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷✳✸ ❈♦♥❥✉♥t♦ ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✸ ❇♦❛ ❖r❞❡♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s ❇ás✐❝❛s ❞♦s ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s ✶✻ ✷✳✶ ❈♦♥❥✉♥t♦s ❊q✉✐♣♦t❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✳✶ ❖ P❛r❛❞♦①♦ ❞♦ ❍♦t❡❧ ■♥✜♥✐t♦ ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ ❈♦♥❥✉♥t♦s ❊♥✉♠❡rá✈❡✐s ❡ ◆ã♦✲❊♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸ ◆ú♠❡r♦s ❚r❛♥s✜♥✐t♦s ❡ ❆r✐t♠ét✐❝❛ ❈❛r❞✐♥❛❧ ✷✾ ✸✳✶ ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶✳✶ ❘❡❧❛çã♦ ❞❡ ❖r❞❡♠ ❡♥tr❡ ❈❛r❞✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷ ❆r✐t♠ét✐❝❛ ❈❛r❞✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳✶ ❆❞✐çã♦ ❞❡ ❈❛r❞✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ❈❛r❞✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✸ ❊①♣♦♥❡♥❝✐❛çã♦ ❞❡ ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❖✉tr♦s ❘❡s✉❧t❛❞♦s ❊♥✈♦❧✈❡♥❞♦ ❆r✐t♠ét✐❝❛ ❈❛r❞✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
❆ ❯♠ ❇r❡✈❡ ❈♦♠❡♥tár✐♦ ❙♦❜r❡ ❛ ❍✐♣ót❡s❡ ❞♦ ❈♦♥tí♥✉♦ ✸✾
❖s ❣r❡❣♦s s❡♠♣r❡ ❡✈✐t❛r❛♠ ❧✐❞❛r ❝♦♠ ♦ ✐♥✜♥✐t♦✱ ♣♦✐s ❡ss❡ ❝♦♥❝❡✐t♦ ❧❤❡s tr❛③✐❛ ❞✐✜❝✉❧✲ ❞❛❞❡s q✉❡ ❡❧❡s ♥✉♥❝❛ s♦✉❜❡r❛♠ r❡s♦❧✈❡r✱ ❡ ♣♦r ✐ss♦ ❡❧❡s ♥✉♥❝❛ tr❛t❛r❛♠ ♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳ ◆❡♠ ❡❧❡s ♥❡♠ s❡✉s s✉❝❡ss♦r❡s ❞❛s ❝✐✈✐❧✐③❛çõ❡s ❤❡❧❡♥íst✐❝❛✱ ár❛❜❡ ❡ ❞❛ ❊✉✲ r♦♣❛ ♠❡❞✐❡✈❛❧✳ ❋♦✐ só ♥♦ sé❝✉❧♦ ❳■❳ q✉❡ ♦s ♠❛t❡♠át✐❝♦s ❝♦♠❡ç❛r❛♠ ❛ ❡st✉❞❛r ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❞❡ ♠❛♥❡✐r❛ s✐st❡♠át✐❝❛✳ ❊ ♦ ♣r✐♠❡✐r♦ ❛ ❢❛③❡r ✐ss♦ ❢♦✐ ❇❡r♥❤❛r❞ ❇♦❧③❛♥♦✭✶✼✽✶✲✶✽✹✽ ✮✱ q✉❡ ♥❛s❝❡✉✱ ✈✐✈❡✉ ❡ ♠♦rr❡✉ ❡♠ Pr❛❣❛✳ ❊r❛ s❛❝❡r❞♦t❡ ❝❛tó✲ ❧✐❝♦ q✉❡✱ ❛❧é♠ ❞❡ s❡ ❞❡❞✐❝❛r ❛ ❡st✉❞♦s ❞❡ ❋✐❧♦s♦✜❛✱ ❚❡♦❧♦❣✐❛ ❡ ▼❛t❡♠át✐❝❛✱ t✐♥❤❛ ❣r❛♥❞❡s ♣r❡♦❝✉♣❛çõ❡s ❝♦♠ ♦s ♣r♦❜❧❡♠❛s s♦❝✐❛✐s ❞❡ s✉❛ é♣♦❝❛✳ ❙❡✉ ❛t✐✈✐s♠♦ ❡♠ ❢❛✈♦r ❞❡ r❡❢♦r♠❛s ❡❞✉❝❛❝✐♦♥❛✐s✱ s✉❛ ❝♦♥❞❡♥❛çã♦ ❞♦ ♠✐❧✐t❛r✐s♠♦ ❡ ❞❛ ❣✉❡rr❛✱ s✉❛ ❞❡✲ ❢❡s❛ ❞❡ ❧✐❜❡r❞❛❞❡ ❞❡ ❝♦♥s❝✐ê♥❝✐❛ ❡ ❡♠ ❢❛✈♦r ❞❛ ❞✐♠✐♥✉✐çã♦ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s s♦❝✐❛✐s ❝✉st❛r❛♠✲❧❤❡ sér✐♦s ❡♠❜❛r❛ç♦s ❝♦♠ ♦ ❣♦✈❡r♥♦✳ ❆s ✐❞❡✐❛s ❞❡ ❇♦❧③❛♥♦ ❡♠ ▼❛t❡♠át✐❝❛ ♥ã♦ ❢♦r❛♠ ♠❡♥♦s ❛✈❛♥ç❛❞❛s✳ ➱ ❛té ❛❞♠✐rá✈❡❧ q✉❡✱ ✈✐✈❡♥❞♦ ❡♠ r❡❧❛t✐✈♦ ✐s♦❧❛♠❡♥t♦ ❡♠ Pr❛❣❛✱ ❛❢❛st❛❞♦ ❞♦ ♣r✐♥❝✐♣❛❧ ❝❡♥tr♦ ❝✐❡♥tí✜❝♦ ❞❛ é♣♦❝❛✱ q✉❡ ❡r❛ P❛r✐s✱ ❡❧❡ t❡♥❤❛ t✐❞♦ s❡♥s✐❜✐❧✐❞❛❞❡ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ✈❛♥❣✉❛r❞❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛✳ ■♥❢❡❧✐③♠❡♥t❡✱ s❡✉s tr❛❜❛❧❤♦s ♣❡r♠❛♥❡❝❡r❛♠ ♣r❛t✐❝❛♠❡♥t❡ ❞❡s❝♦♥❤❡❝✐❞♦s ♣♦r ✈ár✐❛s ❞é❝❛❞❛s ❛♣ós ❛ s✉❛ ♠♦rt❡✳ ✭✈❡r ❬✷❪✮
❋✐❣✉r❛ ✶✿ ❇❡r♥❤❛r❞ ❇♦❧③❛♥♦
❇♦❧③❛♥♦ ♣r♦❞✉③✐✉ ✈ár✐♦s tr❛❜❛❧❤♦s ♠❛t❡♠át✐❝♦s ✐♠♣♦rt❛♥t❡s✱ ♠❛s ❛q✉✐ ✈❛♠♦s ♥♦s ❧✐♠✐t❛r ❛♣❡♥❛s ❛ ♠❡♥❝✐♦♥❛r s❡✉ ♣✐♦♥❡✐r✐s♠♦ ♥♦ tr❛t❛♠❡♥t♦ ❞❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳
❞♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳ ❉❡♣♦✐s ❞❡ ❇♦❧③❛♥♦✱ ❞❡✈❡♠♦s ♠❡♥❝✐♦♥❛r ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ ✭✶✽✸✶✲✶✾✶✻ ✮✱ ✉♠ ❣r❛♥❞❡ ♥♦♠❡ ❞❛ ▼❛t❡♠át✐❝❛ ❞♦ sé❝✉❧♦ ❳■❳✳ ❊❧❡ ❢♦✐ ♠❛✐s ❧♦♥❣❡ q✉❡ ❇♦❧③❛♥♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦ ♥❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
❋✐❣✉r❛ ✷✿ ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞
▼❛s ❢♦✐ ●❡♦r❣ ❈❛♥t♦r ✭✶✽✹✺✲✶✾✶✽ ✮ q✉❡♠ ♠❛✐s ❛✈❛♥ç♦✉ ♥♦ ❡st✉❞♦ ❞♦s ❝♦♥❥✉♥✲ t♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ ❞❡st❡ ♥♦ss♦ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ♦s q✉❛✐s ❥✉❧❣❛♠♦s ♣r❡❧✐♠✐♥❛r❡s ❡♥✈♦❧✈❡♥❞♦ ❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s q✉❡ s❡rã♦ ♠✉✐t♦ út❡✐s ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳
❋✐❣✉r❛ ✸✿ ●❡♦r❣ ❈❛♥t♦r
❊♠ ✶✽✼✷ ❈❛♥t♦r ❡st❛✈❛ ✐♥✐❝✐❛♥❞♦ s✉❛ ❝❛rr❡✐r❛ ♣r♦✜ss✐♦♥❛❧ ❡ s❡ ♦❝✉♣❛✈❛ ❞♦ ❡st✉❞♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❛s ❢✉♥çõ❡s ♣♦r ♠❡✐♦ ❞❡ sér✐❡s tr✐❣♦♥♦♠étr✐❝❛s✳ ◆❡ss❛ ♦❝✉♣❛çã♦ ❡❧❡ ❢♦✐ ❧❡✈❛❞♦ ❛ ✐♥✈❡st✐❣❛r ♦s ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ t❛✐s ❢✉♥çõ❡s✱ ♦s ♠❛✐s s✐♠♣❧❡s ❞♦s q✉❛✐s sã♦ ♦s ❝♦♥❥✉♥t♦s ❝♦♠ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s✳
❞❡ ❡q✉✐♣♦tê♥❝✐❛ ❞❡ ❝♦♥❥✉♥t♦s✱ q✉❡ s❡rá ❛❜♦r❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✷ ❞❡st❡ ♥♦ss♦ tr❛❜❛❧❤♦✳ ◆♦ ❝❛s♦ ❞❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ s❡r❡♠ ❡q✉✐♣♦t❡♥t❡s ❝♦rr❡s♣♦♥❞❡ ❛ t❡r❡♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ❊ ♥♦ ❝❛s♦ ❞♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s❄ ❇❡♠✱ ♥❡ss❡ ❝❛s♦ ♥ã♦ ❢❛③ s❡♥t✐❞♦ ❢❛❧❛r ❡♠ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✱ ♣♦✐s t❛✐s ❝♦♥❥✉♥t♦s s❡♠♣r❡ t❡♠ ✐♥✜♥✐t♦s ❡❧❡♠❡♥t♦s✳ ▼❛s ❝♦♠♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ é ✈á❧✐❞♦ t❛♥t♦ ♣❛r❛ ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❝♦♠♦ ♣❛r❛ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✱ ❡ ❝♦♠♦ t❛❧ ❝♦♥❝❡✐t♦ ❝♦rr❡s♣♦♥❞❡ ❡①❛t❛♠❡♥t❡ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ♥♦ ❝❛s♦ ❞❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ é ❡ss❡ ❝♦♥❝❡✐t♦ q✉❡ ❡st❡♥❞❡✱ ♣❛r❛ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✱ ❛ ♥♦çã♦ ❞❡ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✳
❆ss✐♠✱ ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ❞✐③✲s❡ q✉❡ ❞♦✐s ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r A ❡ B sã♦ ❡q✉✐✲ ♣♦t❡♥t❡s s❡ ❡❧❡s t✐✈❡r❡♠ ❛ ♠❡s♠❛ ❝❛r❞✐♥❛❧✐❞❛❞❡✳ ❆ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ ❡st❡ ❝♦♥❥✉♥t♦ ♣♦ss✉✐✳ ❊ss❛ ❞❡✜♥✐çã♦✱ ♥♦ ❝❛s♦ ❞♦s ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ ♥ã♦ tr❛③ ♥❛❞❛ ❞❡ ♥♦✈♦✳ ▼❛s✱ ❝♦♠♦ ✈❡r❡♠♦s✱ ❡st❡♥❞❡✱ ♣❛r❛ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✱ ❛ ♥♦çã♦ ❞❡ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦✳ ❚❛✐s ♥ú♠❡r♦s sã♦ ♦s ❝❤❛♠❛❞♦s ♥ú♠❡r♦s tr❛♥s✜♥✐t♦s✱ ♦ q✉❛❧ ❛❜♦r❞❛r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦✳
❈❤❛♠❛✲s❡ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ t♦❞♦ ❝♦♥❥✉♥t♦ ❡q✉✐♣♦t❡♥t❡ ❛ N✳ ❆ss✐♠✱ ♦ ❝♦♥✲
❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣❛r❡s ♣♦s✐t✐✈♦s é ❡♥✉♠❡rá✈❡❧✱ ♣♦✐s✱ ❝♦♠♦ ♠♦str❛r❡♠♦s ♥♦ ❝❛♣í✲ t✉❧♦ ✷✱ ❡❧❡ é ❡q✉✐♣♦t❡♥t❡ ❛ N✳ ◆ã♦ ❞❡✐①❛ ❞❡ s❡r s✉♣r❡❡♥❞❡♥t❡✱ ♣❛r❛ q✉❡♠ ❛❞q✉✐r❡
❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ ❝♦♥st❛t❛r q✉❡ ❡①✐st❡♠ s✉❜❝♦♥❥✉♥t♦s ♣ró♣r✐♦s ❞❡ N q✉❡ sã♦ ❡q✉✐♣♦t❡♥t❡s ❛ N✳ ◆ã♦ ❛♣❡♥❛s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ♣♦s✐t✐✈♦s✱ ♠❛s
t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s í♠♣❛r❡s✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s q✉❛❞r❛❞♦s ❞♦s ✐♥t❡✐r♦s
{1,4,9,16,25,36, ...} ❡ ♠✉✐t♦s ♦✉tr♦s ♠❛✐s✱ t♦❞♦s ❡q✉✐♣♦t❡♥t❡s ❛ N✳ ❊ss❡ ❢❡♥ô♠❡♥♦
é ✉♠❛ ♣❡❝✉❧✐❛r✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❡ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r t❛✐s ❝♦♥❥✉♥t♦s✳
❙❡rá q✉❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s sã♦ ❡♥✉♠❡rá✈❡✐s❄ ❖✉ s❡❥❛✱ ❡q✉✐♣♦t❡♥t❡s ❛
N❄ ❱❡r❡♠♦s q✉❡ ♥ã♦ é ❛ss✐♠✳ ❊st❛❜❡❧❡❝❡r❡♠♦s✱ ♥♦ ❝❛♣ít✉❧♦ ✷✱ ❛ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❞♦s
♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ✉♠ r❡s✉❧t❛❞♦ ❥á ❡♠ s✐ s✉r♣r❡❡♥❞❡♥t❡✳ ❱❡r✐✜❝❛r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✷ ❞❡st❡ tr❛❜❛❧❤♦✱ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s R é ♥ã♦ ❡♥✉♠❡rá✈❡❧✳ ❊✱ ❞✐❛♥t❡
❞❡st❡ r❡s✉❧t❛❞♦✱ ❈❛♥t♦r ♠♦str♦✉ q✉❡ ❡①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s t✐♣♦s ❞❡ ✐♥✜♥✐t♦✿ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
◆♦ t♦❝❛♥t❡ ❛♦s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s✱ q✉❡ s❡rá ❛❜♦r❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✷✱ t❡♠♦s q✉❡ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ♥♦çã♦ ❞❡ ❢✉♥çã♦ ❜✐❥❡t✐✈❛ ❡♥tr❡ ❝♦♥❥✉♥t♦s é ❛ ❛❜♦r❞❛❣❡♠ ❛❞❡q✉❛❞❛ ♣❛r❛ ❝♦♠♣❛r❛r ♦ t❛♠❛♥❤♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳ ❊st❛ ❛❜♦r❞❛❣❡♠ t❛♠❜é♠ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❈❛♥t♦r ❡✱ s✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡✱ ❝♦♥❞✉③ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞✐❢❡r❡♥t❡s t❛♠❛♥❤♦s ❞❡ ✐♥✜♥✐t♦s✳ ▼♦t✐✈❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❧❡✈❛♥t❛♥❞♦ ❛s s❡❣✉✐♥t❡s q✉❡stõ❡s✿ ◗✉❡ ❝♦♥❥✉♥t♦ s❡r✐❛ ♠❛✐♦r✱ ♦ ❝♦♥❥✉♥t♦ N ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♦✉ ♦ ❝♦♥❥✉♥t♦ Q ❞♦s ♥ú♠❡r♦s
r❛❝✐♦♥❛✐s❄ ❖ q✉❡ s❡ ♣♦❞❡ ❞✐③❡r q✉❛♥t♦ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ♦ ✐♥t❡r✈❛❧♦
❡♥tr❡ ❡❧❡s ❝♦♠ r❡s♣❡✐t♦ à q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s❄
❈❛♥t♦r ♣r♦✈♦✉ ♦✉tr♦ ❢❛t♦ ♥ã♦ ♠❡♥♦s ♣❡rt✉❜❛❞♦r✿ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✱ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❝♦♥str✉✐r ♦✉tr♦ ❝♦♥❥✉♥t♦ ♠❛✐♦r ❛✐♥❞❛✱ ♠❛✐♦r ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❡❧❡ ❝♦♥té♠ ♦ ♣r✐♠❡✐r♦ ❝♦♥❥✉♥t♦ ❝♦♠♦ ♣❛rt❡ ♣ró♣r✐❛ ❡ ♥ã♦ é ❡q✉✐♣♦t❡♥t❡ ❛ ❡ss❛ s✉❛ ♣❛rt❡ ♣ró♣r✐❛✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s tê♠ ❝❛r❞✐♥❛❧✐❞❛❞❡s ❞✐❢❡r❡♥t❡s✳ ■ss♦ ♣❡r♠✐t❡ ♦r❞❡♥❛r ❛s ❝❛r❞✐♥❛❧✐❞❛❞❡s ❞♦s ❝♦♥❥✉♥t♦s ❝r✐❛♥❞♦ ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ♥ú♠❡r♦s tr❛♥s✜♥✐t♦s✳ ❯♠ t❛❧ ♥ú♠❡r♦ ♥❛❞❛ ♠❛✐s é q✉❡ ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✳ ❆❜♦r❞❛r❡♠♦s ❡ss❡ ❛ss✉♥t♦ ♥♦ ❝❛♣ít✉❧♦ ✸ ❞❡st❡ tr❛❜❛❧❤♦✳
❆✐♥❞❛✱ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❞❡st❛❝❛♠♦s ❛ ❝❤❛♠❛❞❛ ❆r✐t♠ét✐❝❛ ❈❛r❞✐♥❛❧✳ ❏á ❡①✐st❡ ✉♠❛ ❛r✐t♠ét✐❝❛ ❝❛r❞✐♥❛❧ ♣❛r❛ ♦s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s ✜♥✐t♦s✳ P♦r ❡①❡♠♣❧♦✱ s❡x❡ysã♦ ❞♦✐s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s ✜♥✐t♦s✱ t❡♠♦s q✉❡ ❛ s♦♠❛x+y❡ ♦ ♣r♦❞✉t♦xyt❡♠ s❡✉s s✐❣♥✐✜❝❛❞♦s tr❛❞✐❝✐♦♥❛✐s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛❧é♠ ❞❡ ❛❜♦r❞❛r ❛ ❛r✐t♠ét✐❝❛ ❞♦s ❝❛r❞✐♥❛✐s ✜♥✐t♦s✱ ❣❡♥❡r❛❧✐③❛♠♦s ❡st❡s ❝♦♥❝❡✐t♦s ❞❡ ♠♦❞♦ ❛ ❝♦❜r✐r ♦s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s tr❛♥s✜♥✐t♦s t❛♠❜é♠✳ ❖✉ s❡❥❛✱ ✉♠❛ ❛r✐t♠ét✐❝❛ q✉❡ s❡ ❛♣❧✐❝❛ ❛ t♦❞♦s ♦s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s✱ ✜♥✐t♦s ❡ ✐♥✜♥✐t♦s✱ q✉❡ ♣r❡s❡r✈❡ ♦s s✐❣♥✐✜❝❛❞♦s ❡ ♣r♦♣r✐❡❞❛❞❡s tr❛❞✐❝✐♦♥❛✐s ❞❛ ❛r✐t♠ét✐❝❛ ❞♦s ♥ú♠❡r♦s ❝❛r❞✐♥❛✐s ✜♥✐t♦s✳
◆♦ ❛♣ê♥❞✐❝❡ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❜r❡✈❡✱ ❛ ❍✐♣ót❡s❡ ❞♦ ❈♦♥tí♥✉♦✳ ❆ ❤✐♣ót❡s❡ ❞♦ ❝♦♥tí♥✉♦ é ✉♠❛ ❝♦♥❥❡❝t✉r❛ ♣r♦♣♦st❛ ♣♦r ●❡♦r❣ ❈❛♥t♦r✳ ❊st❛ ❝♦♥❥❡❝t✉r❛ ❝♦♥s✐st❡ ♥♦ s❡❣✉✐♥t❡✿
◆ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ♠❛✐s ❡❧❡♠❡♥t♦s ❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ♠❡♥♦s ❡❧❡♠❡♥t♦s ❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
❆q✉✐ ♠❛✐s ❡❧❡♠❡♥t♦s ❡ ♠❡♥♦s ❡❧❡♠❡♥t♦s t❡♠ ✉♠ s❡♥t✐❞♦ ♠✉✐t♦ ♣r❡❝✐s♦✳ ❊st❛ ❤✐♣ót❡s❡ ❢♦✐ ♦ ♥ú♠❡r♦ ✉♠ ❞♦s ✷✸ Pr♦❜❧❡♠❛s ❞❡ ❍✐❧❜❡rt ❛♣r❡s❡♥t❛❞♦s ♥❛ ❝♦♥❢❡rê♥❝✐❛ ❞♦ ❈♦♥❣r❡ss♦ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ❞❡ ✶✾✵✵✱ ♦ q✉❡ ❧❡✈♦✉ ❛ q✉❡ ❢♦ss❡ ❡st✉❞❛❞❛ ♣r♦❢✉♥❞❛♠❡♥t❡ ❞✉r❛♥t❡ ♦ sé❝✉❧♦ ❳❳✳
❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ❞❡ ❚❡♦r✐❛
❞♦s ❈♦♥❥✉♥t♦s
❆ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ❞❡ ❢♦r♠❛ r✐❣♦r♦s❛ ❡ ♠♦❞❡r♥❛ ♥♦ ✜♥❛❧ ♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡ ♣♦r ●❡♦r❣ ❈❛♥t♦r ✭✶✽✹✺✲✶✾✶✽ ✮ ♣❛r❛ ❛❜♦r❞❛r ❝❡rt❛s q✉❡stõ❡s s✉t✐s ❞❛ t❡♦r✐❛ ❞❛s ❢✉♥çõ❡s✳ ❆s ✐❞❡✐❛s r❡✈♦❧✉❝✐♦♥ár✐❛s ❞❡ ❈❛♥t♦r✱ ❞❡ ✐♥í❝✐♦ ✐♥❝♦♠♣r❡❡♥❞✐❞❛s ♣♦r s❡r❡♠ ❞❡♠❛s✐❛❞♦ ❛❜str❛t❛s ♣❛r❛ ❛ é♣♦❝❛✱ ❢♦r❛♠ r❛♣✐❞❛♠❡♥t❡ s❡ ✐♠♣♦♥❞♦ ❝♦♠♦ ❡❧❡♠❡♥t♦ ✉♥✐✜❝❛❞♦r ❞❡ ✈ár✐♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛✱ ❛ ♣♦♥t♦ ❞❡ s❡ t♦r♥❛r❡♠ ♦ ♠❡✐♦ ♣❡❧♦ q✉❛❧ é ❢♦r♠❛❧✐③❛❞❛ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛ ❝♦♥t❡♠♣♦râ♥❡❛✳
❆ t❡♦r✐❛ ❝♦♥tr✐❜✉✐✉ ❞❡❝✐s✐✈❛♠❡♥t❡ ♣❛r❛ q✉❡ s❡ ♣❛ss❛ss❡ ❛ ❡♥❝❛r❛r s♦❜ ♦✉tr❛ ♣❡rs✲ ♣❡❝t✐✈❛ ♦s ♣r♦❜❧❡♠❛s ❞❛ ♠❛t❡♠át✐❝❛✱ ❞❡s❞❡ ♦s q✉❡ s✉r❣❡♠ ♥♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ❞✐s❝✐♣❧✐♥❛ ❛té ♦s q✉❡ sã♦ tí♣✐❝♦s ❞❡ r❛♠♦s ❡s♣❡❝✐❛❧✐③❛❞♦s ❞❛ á❧❣❡❜r❛✱ ❞❛ ❛♥á❧✐s❡ ❡ ❞❛ ❣❡♦♠❡tr✐❛✳
❆s ❛♣❧✐❝❛çõ❡s ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s à s♦❧✉çã♦ ❞❡ q✉❡stõ❡s r❡❧❛t✐✈❛s à ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ ✈ár✐♦s t✐♣♦s ❞❡ ❝♦♥❥✉♥t♦s ❡ ❛ q✉❡stõ❡s r❡❧❛t✐✈❛s às s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♦♣❡r❛tór✐❛s ❛❜r✐r❛♠ ♥♦✈♦s r✉♠♦s ♣❛r❛ ♦s ♠❛t❡♠át✐❝♦s✱ r❡ss❛❧t❛♥❞♦✱ ❡♥tr❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✱ ❛ ❡①t❡♥sã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ♠❡❞✐❞❛ ❡ ❞❡ ✐♥t❡❣r❛❧✱ ❛ ✐♥tr♦❞✉çã♦ ❞❛s ♥♦çõ❡s ❞❡ ❡s♣❛ç♦ ❛❜str❛t♦✱ ❞❡✜♥✐❞♦ ❝♦♠♦ ❝♦♥❥✉♥t♦s ❞❡ ❡❧❡♠❡♥t♦s ❝♦♠ ❞❛❞❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❡ ❜❡♠ ❛ss✐♠ ♥♦tá✈❡✐s ✐♥♦✈❛çõ❡s ♥♦ ❝❛♠♣♦ ❞❛ ✐♥t❡❣r❛çã♦ ❡ ♥♦ ❞♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s✱ ❡①❛♠✐♥❛❞❛s à ❧✉③ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❝♦♥❥✉♥t♦s✳
◆❡st❡ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♣r♦❝✉r❛♠♦s ❛♣r❡s❡♥t❛r✱ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❝❧❛r❛ ❡ ♦❜❥❡t✐✈❛✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ❥✉❧❣❛♠♦s ✐♠♣♦rt❛♥t❡s ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✱ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s s❡♠ ♦ r✐❣♦r ♠❛t❡♠át✐❝♦ ❛♣r♦♣r✐❛❞♦✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❢❛③❡r ❝♦♠ q✉❡ ♦ ❧❡✐t♦r ✈❡♥❤❛ ❛ ❝♦♠♣r❡❡♥❞❡r✱ ❛❧é♠ ❞❡st❡s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s✱ ♦✉tr♦s ✐♠♣♦rt❛♥t❡s ❝♦♥❝❡✐t♦s ♣❛r❛ q✉❡✱ ✜♥❛❧♠❡♥t❡✱ ✈❡♥❤❛♠♦s ❛ ❛❜♦r❞❛r ❛s ❝❤❛♠❛❞❛s ◆♦✲ çõ❡s ❇ás✐❝❛s ❞❡ ■♥✜♥✐t♦ ❡ ◆ú♠❡r♦s ❈❛r❞✐♥❛✐s✳
✶✳✶ ❆ ▲✐♥❣✉❛❣❡♠ ❞♦s ❈♦♥❥✉♥t♦s
❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✿ ❬✶❪✱ ❬✸❪✱ ❬✼❪✱ ❬✶✶❪✳
❖s t❡r♠♦s ❝♦♥❥✉♥t♦ ❡ ❡❧❡♠❡♥t♦ ❡ ❛ r❡❧❛çã♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ♣❡rt❡♥❝❡r ❛ ✉♠ ❝♦♥❥✉♥t♦ sã♦ ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s❀ ♦✉ s❡❥❛✱ ♥ã♦ s❡rã♦ ❞❡✜♥✐❞♦s✳ ❯s❛♠♦s ♦ t❡r♠♦ ❝♦❧❡çã♦ ❝♦♠♦ s✐♥ô♥✐♠♦ ❞❡ ❝♦♥❥✉♥t♦✳ ❆ ❛✜r♠❛çã♦ q✉❡ ✉♠ ❡❧❡♠❡♥t♦ x ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ A é s✐♠❜♦❧✐③❛❞❛ ♣♦r x∈A ❡ ❛ s✉❛ ♥❡❣❛çã♦ é s✐♠❜♦❧✐③❛❞❛ ♣♦r x6∈A✳
❉❡✜♥✐çã♦ ✶✳✶✳✶ ❉♦✐s ❝♦♥❥✉♥t♦s sã♦ ❝♦♥s✐❞❡r❛❞♦s ✐❣✉❛✐s✱ s❡ ❡❧❡s tê♠ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s q✉❡ A = B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ A é ❡❧❡♠❡♥t♦ ❞❡ B ❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ B é ❡❧❡♠❡♥t♦ ❞❡ A✳
❆ ❝♦♥❞✐çã♦ ❞❡ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦A ♣❡rt❡♥❝❡ ❛ ✉♠ ❝♦♥❥✉♥t♦B✱ ❡st❛❜❡❧❡❝❡ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ A ❡ B✱ ❝❤❛♠❛❞❛ r❡❧❛çã♦ ❞❡ ✐♥❝❧✉sã♦✳ ◗✉❛♥❞♦ ❡①✐st✐r ✉♠❛ t❛❧ r❡❧❛çã♦ ❡♥tr❡ A❡B ❡s❝r❡✈❡r❡♠♦sA⊂B ♦✉B ⊃A✱ q✉❡ s❡ ❧êA❡stá ❝♦♥t✐❞♦ ❡♠ B ♦✉A é s✉❜❝♦♥❥✉♥t♦ ❞❡ B✱ ♦✉ ❛✐♥❞❛✱ B ❝♦♥té♠ A✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✶ ❆ r❡❧❛çã♦ ❞❡ ✐♥❝❧✉sã♦ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✶✮ A⊂A✱ ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ A❀
✭✷✮ A=B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A⊂B ❡ B ⊂A❀ ✭✸✮ ❙❡ A ⊂B ❡ B ⊂C✱ ❡♥tã♦ A⊂C✳
❆ ♥❡❣❛çã♦ ❞❡ A ⊂ B✱ ♦✉ s❡❥❛✱ ♦ ❢❛t♦ ❞❡ A ♥ã♦ s❡r s✉❜❝♦♥❥✉♥t♦ ❞❡ B✱ é s✐♠✲ ❜♦❧✐③❛❞❛ ♣♦r A 6⊂ B ❡ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❡ A q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛B✳ ❙❡A⊂B ❡A6=B✱ ❞✐r❡♠♦s q✉❡ Aé s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡B✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡♠♦s A$B✳ ◆♦ q✉❡ s❡ s❡❣✉❡✱ ❛❞♠✐t✐r❡♠♦s ♦ ❧❡✐t♦r ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s N={0,1,2,3,4· · · } ❡ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✿ Z={· · · −3,−2,−1,0,1,2,3,· · · }✳
❖ ❝♦♥❥✉♥t♦ Q✱ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ é ❢♦r♠❛❞♦ ♣❡❧❛s ❢r❛çõ❡s p
q✱ ♦♥❞❡ p ❡ q ♣❡rt❡♥❝❡♠ ❛ Z✱ s❡♥❞♦ q6= 0✳ ❊♠ sí♠❜♦❧♦s✱ t❡♠♦s q✉❡
Q={p/q;p∈Z, q ∈Z, q6= 0}✳
▲ê✲s❡✿ ✧Q é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢r❛çõ❡s p/q t❛✐s q✉❡ p♣❡rt❡♥❝❡ ❛ Z✱q ♣❡rt❡♥❝❡ ❛ Z❡
q é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✧✳
❆ ♠❛✐♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ♠❛t❡♠át✐❝❛ ♥ã♦ sã♦ ❞❡✜♥✐❞♦s ❡s♣❡❝✐✜❝❛♥❞♦✲ s❡✱ ✉♠ ❛ ✉♠✱ ♦s s❡✉s ❡❧❡♠❡♥t♦s✳ ❖ ♠ét♦❞♦ ♠❛✐s ❢r❡q✉❡♥t❡ ❞❡ ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ é ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠✉♠ ❡ ❡①❝❧✉s✐✈❛ ❞♦s s❡✉s ❡❧❡♠❡♥t♦s✳ ▼❛✐s ♣r❡❝✐✲ s❛♠❡♥t❡✱ ♣❛rt❡✲s❡ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ P✳ ❊❧❛ ❞❡✜♥❡ ✉♠ ❝♦♥❥✉♥t♦ X✱ ❛ss✐♠✿ s❡ ✉♠ ❡❧❡♠❡♥t♦ x ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P✱ ❡♥tã♦ x ∈ X❀ s❡ ♥ã♦ ❣♦③❛ ❞❡ P✱ ❡♥tã♦ x 6∈ X✳ ❊s❝r❡✈❡✲s❡
X ={x; ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡P}✳
▲ê✲s❡✿ ✧X é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s xt❛✐s q✉❡ x ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P✧✳
▼✉✐t❛s ✈❡③❡s ❛ ♣r♦♣r✐❡❞❛❞❡P s❡ r❡❢❡r❡ ❛ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❢✉♥❞❛♠❡♥t❛❧ A✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡✲s❡
X ={x∈A;x ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P}✳
P♦r ❡①❡♠♣❧♦✱ s❡❥❛N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ❝♦♥s✐❞❡r❡♠♦s ❛ s❡❣✉✐♥t❡
♣r♦♣r✐❡❞❛❞❡✱ q✉❡ s❡ r❡❢❡r❡ ❛ ✉♠ ❡❧❡♠❡♥t♦ ❣❡♥ér✐❝♦ x∈N✿ ✧x é ♠❛✐♦r ❞♦ q✉❡ 5✧✳
❆ ♣r♦♣r✐❡❞❛❞❡ P✱ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ s❡r ♠❛✐♦r ❞♦ q✉❡ ✺✱ ❞❡✜♥❡ ♦ ❝♦♥❥✉♥t♦ X ={6,7,8,9· · · }✱ ♦✉ s❡❥❛✱
X ={x∈N;x >5}✳
▲ê✲s❡✿ ✧X é ♦ ❝♦♥❥✉♥t♦ ❞♦s x ♣❡rt❡♥❝❡♥t❡s ❛ N t❛✐s q✉❡ x é ♠❛✐♦r ❞♦ q✉❡ ✺✧✳
➚s ✈❡③❡s✱ ♦❝♦rr❡ q✉❡ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡A❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P✳ ◆❡st❡ ❝❛s♦✱ ♦ ❝♦♥❥✉♥t♦{x∈A❀ x❣♦③❛ ❞❡ P} ♥ã♦ ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ❛❧❣✉♠✳ ■st♦ é ♦ q✉❡ s❡ ❝❤❛♠❛ ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦✳ P❛r❛ r❡♣r❡s❡♥tá✲❧♦✱ ✉s❛r❡♠♦s ♦ sí♠❜♦❧♦ ∅✳
P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ é ❞❡✜♥✐❞♦ ❛ss✐♠✿ ◗✉❛❧q✉❡r q✉❡ s❡❥❛ x✱ t❡♠✲s❡x6∈ ∅✳
❆❧❣✉♥s ❡①❡♠♣❧♦s✿ • {x∈A;x6=x}=∅❀
• {x∈N; 1< x <2}=∅❀
• {x;x6=x}=∅✳
❉❡✜♥✐çã♦ ✶✳✶✳✷ ❆✜r♠❛♠♦s q✉❡ ∅ ⊂A✱ ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ❝♦♥❥✉♥t♦ A✳ ❊st❛ ❛✜r♠❛çã♦ ♣❛r❡❝❡ ❡str❛♥❤❛ à ♣r✐♠❡✐r❛ ✈✐st❛✱ ♠❛s ✈❡❥❛♠♦s ❝♦♠♦ é ♥❛t✉r❛❧ ❛ ❢❛❧s✐❞❛❞❡ ❞❡ s✉❛ ♥❡❣❛çã♦ ✭✐st♦ é✱ s✉❛ ✈❡r❛❝✐❞❛❞❡✮✳ ❆ ❛✜r♠❛çã♦ ∅ 6⊂ A✱ ♣❛r❛ ❛❧❣✉♠ ❝♦♥❥✉♥t♦ A✱ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ x∈ ∅ t❛❧ q✉❡ x 6∈A ❡ ✐st♦ é ❝❧❛r❛♠❡♥t❡ ❢❛❧s♦✱ ✈✐st♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ∅ ♥ã♦ ♣♦ss✉✐ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦✳
✶✳✶✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❈♦♥❥✉♥t♦s
❉❛❞❛ ✉♠❛ ❝♦❧❡çã♦ q✉❛❧q✉❡r ❞❡ ❝♦♥❥✉♥t♦s✱ ❛❞♠✐t✐r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ t❛❧ q✉❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s ♣❡rt❡♥❝❡ ❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝♦♥❥✉♥t♦s ❞❛ ❝♦❧❡çã♦✳ ❚❛❧ ❝♦♥❥✉♥t♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ✉♥✐ã♦ ❞♦s ❝♦♥❥✉♥t♦s ❞❛ ❝♦❧❡çã♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✸ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ❛ ✉♥✐ã♦ ❞❡ A ❡ B é ♦ ❝♦♥❥✉♥t♦
A∪B ={x;x∈A ♦✉ x∈B}✳
❆s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r ❞❡❝♦rr❡♠ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛s ❞❡✜♥✐çõ❡s✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✷ P❛r❛ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s A✱ B ❡ C✱ t❡♠♦s q✉❡✿ ✭✶✮ A∪ ∅=A ❡ A∪A=A❀
✭✷✮ A⊂A∪B ❡ B ⊂A∪B❀ ✭✸✮ A∪B =B∪A❀
✭✹✮ (A∪B)∪C =A∪(B ∪C)✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✸ ❉❛❞♦s ❝♦♥❥✉♥t♦s A, A′, B ❡ B′✱ ❝♦♠ A ⊂ B ❡ A′ ⊂ B′✱ ❡♥tã♦
A∪A′ ⊂B∪B′✳
Pr♦✈❛✳ ❙❡ A ∪A′ = ∅✱ ❛ ❛ss❡rçã♦ é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ A ∪A′ 6= ∅✳ ❙❡ x ∈ A∪A′✱ t❡♠♦s q✉❡ x ∈ A ♦✉ x ∈ A′✱ ❡ ❝♦♠♦ A ⊂ B ❡ A′ ⊂ B′✱ s❡❣✉❡✲s❡ q✉❡ x∈B ♦✉x∈B′✳ ■st♦ ♣♦st♦✱ x∈B∪B′✳ P♦rt❛♥t♦✱ ♣r♦✈❛♠♦s q✉❡ A∪A′ ⊂B∪B′✳
❈♦r♦❧ár✐♦ ✶✳✶✳✶ A∪B =A s❡✱ ❡ s♦♠❡♥t❡ s❡✱ B ⊂A✳
Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ A ∪B = A✳ ❈♦♠♦ B ⊂ A∪B✱ s❡❣✉❡✲s❡ q✉❡ B ⊂ A✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ B ⊂ A✳ ❈♦♠♦ A ⊂ A✱ s❡❣✉❡✲s❡ ❞❛ ♣r♦♣♦s✐çã♦ q✉❡ A∪B ⊂A∪A=A✳ ▲♦❣♦✱A∪B ⊂A✳ ❈♦♠♦ A⊂A∪B✱ s❡❣✉❡✲s❡ q✉❡ A∪B =A✳
❉❡✜♥✐çã♦ ✶✳✶✳✹ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ❛ ✐♥t❡rs❡çã♦ ❞❡ A ❡ B é ♦ ❝♦♥❥✉♥t♦ A∩B ={x;x∈A ❡ x∈B}✳ ◗✉❛♥❞♦ A∩B =∅✱ ❞✐③❡♠♦s q✉❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B sã♦ ❞✐s❥✉♥t♦s✳
❆s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r ❞❡❝♦rr❡♠ ❞❛s ❞❡✜♥✐çõ❡s✿
Pr♦♣♦s✐çã♦ ✶✳✶✳✹ P❛r❛ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s A, B ❡ C✱ t❡♠♦s q✉❡✿ ✭✶✮ A∩B =∅ ❡ A∩A=A❀
✭✷✮ A∩B ❡ A∩B ⊂B❀ ✭✸✮ A∩B =B∩A❀
✭✹✮ (A∩B)∩C =A∩(B ∩C)✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✺ ❉❛❞♦s ❝♦♥❥✉♥t♦s A, B ❡ C q✉❛✐sq✉❡r✱ t❡♠♦s q✉❡ A∩(B∪C) =
(A∩B)∪(A∩C)✳
Pr♦✈❛✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s ❛ ✐♥❝❧✉sã♦ A∩(B ∪C) ⊂ (A∩B)∪(A∩C)✳ ❙❡ A∩(B ∪C) = ∅✱ ♥❛❞❛ t❡♠♦s ❛ ♣r♦✈❛r✳ ❙✉♣♦♥❤❛ q✉❡ A∩(B ∪C) 6= ∅✳ ❙❡❥❛ x ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❡ A∩(B∪C)✳ ▲♦❣♦✱ t❡♠♦s q✉❡ x ∈A ❡ x∈ B∪C✳ ❙❡❥❛ x∈B✱ ❡♥tã♦ x∈A∩B✳ ❙❡ x∈C✱ ❡♥tã♦x∈A∩C✳ ❊♠ q✉❛❧q✉❡r ❝❛s♦✱ t❡♠♦s q✉❡ x∈(A∩B)∪(A∩C)✳ ❆❣♦r❛✱ ♣r♦✈❡♠♦s ❛ ✐♥❝❧✉sã♦(A∩B)∪(A∩C)⊂A∩(B∪C)✳ ❙❡ ♦ ❝♦♥❥✉♥t♦ ❞❛ ❡sq✉❡r❞❛ ❢♦r ✈❛③✐♦✱ ❛ ✐♥❝❧✉sã♦ é ✈❡r✐✜❝❛❞❛✳ ❙✉♣♦♥❤❛ q✉❡ t❛❧ ❝♦♥❥✉♥t♦ é ♥ã♦ ✈❛③✐♦✱ ❡ s❡❥❛ x ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❡❧❡✳ ▲♦❣♦✱ x ∈ A∩B ♦✉ x ∈ A∩C✳ ❊♠ q✉❛❧q✉❡r ❝❛s♦✱x∈A ❡ t❡♠♦s q✉❡x∈B ♦✉x∈C✳ ■st♦ ♣♦st♦✱x∈A∩(B∪C)✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✻ ❉❛❞♦s ❝♦♥❥✉♥t♦s A✱ B ❡C✱ q✉❛✐sq✉❡r✱ t❡♠♦s q✉❡A∪(B∩C) =
(A∪B)∩(A∪C)✳
❉❡✜♥✐çã♦ ✶✳✶✳✺ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦sA ❡B✱ ❛ ❞✐❢❡r❡♥ç❛A♠❡♥♦s B✱ é ♦ ❝♦♥❥✉♥t♦ A−B = {x;x ∈A ❡ x 6∈ B}✳ ◗✉❛♥❞♦ B ⊂A✱ ❛ ❞✐❢❡r❡♥ç❛ A−B é ❞❡♥♦t❛❞❛ ♣♦r
∁A(B) ❡ é ❝❤❛♠❛❞❛ ❞❡ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ B ❡♠ A✳
P♦r ❡①❡♠♣❧♦✱ s❡ A={a, b, c} ❡ B ={b, c, d}✱ ❡♥tã♦ A−B ={a}✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✼ P❛r❛ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s A ❡ B✱ t❡♠♦s q✉❡✿ ✭✶✮ A− ∅=A ❡ A−A=∅❀
✭✷✮ ❙❡ A∩B =∅✱ ❡♥tã♦ A−B =A ❡ B−A=B❀
✭✸✮ ∁A(∅) = A ❡ ∁A(A) =∅✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✽ ❙❡❥❛♠ B ❡ B′ s✉❜❝♦♥❥✉♥t♦s ❞❡ A✳ ❙❡ B ⊂ B′✱ ❡♥tã♦ ∁
A(B′) ⊂
∁A(B)✳
Pr♦✈❛✳ ❙✉♣♦♥❤❛ q✉❡ B ⊂ B′✳ ❙❡ ∁
A(B′) = ∅✱ ♥❛❞❛ t❡♠♦s ❛ ♣r♦✈❛r✳ ❙❡∁A(B′)6=∅ ❡ s❡❥❛ x ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❡ ∁A(B′)✳ ■st♦ ♣♦st♦✱ x6∈ B′✳ ❙❡❣✉❡✲s❡ q✉❡ x 6∈ B ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❝♦♠♦B ⊂B′✱ t❡rí❛♠♦s x∈B′✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱x∈∁
A(B)✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✾ ❙❡❥❛♠ B ❡ B′ s✉❜❝♦♥❥✉♥t♦s ❞❡ A✳ ❚❡♠♦s q✉❡ ∁
A(B ∪ B′) =
∁A(B)∩∁A(B′)✳
Pr♦✈❛✳ ❆ ♣r♦♣♦s✐çã♦ ❞❡❝♦rr❡ ❞❛ s❡❣✉✐♥t❡ ❝❛❞❡✐❛ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s✿
x∈∁A(B∪B′)⇔x6∈B∪B′ ⇔x6∈B ❡x6∈B′ ⇔x∈∁
A(B)∩∁A(B′)✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ x ❞❡ A✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✶✵ ❙❡❥❛♠ B ❡ B′ s✉❜❝♦♥❥✉♥t♦s ❞❡ A✳ ❚❡♠♦s q✉❡ ∁
A(B ∩B′) =
∁A(B)∪∁A(B′)✳
✶✳✶✳✷ ❈♦♥❥✉♥t♦ ❞❡ P❛rt❡s ❡ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦
❉❡✜♥✐çã♦ ✶✳✶✳✻ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ A q✉❛❧q✉❡r✱ ❛❞♠✐t✐r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ℘(A)✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ t♦❞♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ A✱ ❝❤❛♠❛❞♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛rt❡s ♦✉ ❝♦♥❥✉♥t♦ ♣♦tê♥❝✐❛ ❞❡ A✳
❉❡✜♥✐çã♦ ✶✳✶✳✼ ❯♠ ♣❛r ♦r❞❡♥❛❞♦(a, b)❞❡ ❡❧❡♠❡♥t♦s ❞❡Aé ♦ ❡❧❡♠❡♥t♦ ❞❡℘(℘(A)) ❞❛❞♦ ♣♦r {{a},{a, b}}✳ ◆ã♦ é ❞✐❢í❝✐❧ ❝♦♥✈❡♥❝❡r✲s❡ q✉❡ (a, b) = (a′, b′) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=a′ ❡ b =b′✳
❉❡✜♥✐çã♦ ✶✳✶✳✽ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡ A ❡ B é ♦ ❝♦♥❥✉♥t♦ A×B ❞❡ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (a, b) ❞❡ ❡❧❡♠❡♥t♦s ❞❡ A∪B t❛✐s q✉❡ a∈A ❡ b ∈B✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ ❡s❝r❡✈❡♠♦s A×B ={(a, b);a∈A ❡ b ∈B}✳
P♦r ❡①❡♠♣❧♦✱ s❡A={a, b}❡B ={c, d}✱ t❡♠♦s q✉❡A×B ={(a, c),(a, d),(b, c),(b, d)}
❡ B×A={(c, a),(c, b),(d, a),(d, b)}✳
◆♦t❡ q✉❡✱ ❡♠ ❣❡r❛❧✱A×B 6=B×A✳ ❚❡♠♦s t❛♠❜é♠ q✉❡A×B =∅s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A=∅♦✉ B =∅✳
✶✳✶✳✸ ❋❛♠í❧✐❛ ❞❡ ❈♦♥❥✉♥t♦s
❉❡✜♥✐çã♦ ✶✳✶✳✾ ❙❡❥❛ I ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ q✉❛❧q✉❡r✳ ❯♠❛ ❢❛♠í❧✐❛ ✐♥❞❡①❛❞❛ ♣♦r I é ✉♠❛ ❝♦❧❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s Ai ❝♦♠ i∈I✳ ❯♠❛ t❛❧ ❢❛♠í❧✐❛ s❡rá ❞❡♥♦t❛❞❛ ♣♦r (Ai)i∈I✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✵ ❆ ✉♥✐ã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ é[
i∈I
Ai ={x;x∈Ai ♣❛r❛
❛❧❣✉♠ i∈I} ❡ ❛ s✉❛ ✐♥t❡rs❡çã♦ é \ i∈I
Ai ={x;x∈Ai ♣❛r❛ t♦❞♦ i∈I}✳
❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ j ∈I✱ t❡♠♦s q✉❡ Aj ⊂
[
i∈I
Ai ❡
\
i∈I
Ai ⊂Aj✳
✶✳✷ ❋✉♥çõ❡s
❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❛ ♣❛rt✐r ❞❛s s❡❣✉✐♥t❡s r❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✿ ❬✸❪✱ ❬✶✶❪✱ ❬✶✷❪✳
❉❡✜♥✐çã♦ ✶✳✷✳✶ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦sX ❡Y✱ ✉♠❛ r❡❧❛çã♦ ❞❡X ❡♠ Y ✭♦✉ ❡♥tr❡ X ❡ Y✱ ♥❡ss❛ ♦r❞❡♠✮✱ é ✉♠ s✉❜❝♦♥❥✉♥t♦ R ❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ X×Y✱ ✐st♦ é✱ R é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞♦ t✐♣♦ (x, y)✱ ❝♦♠ x∈X ❡ y ∈Y✳ ❙❡ R é ✉♠❛ r❡❧❛çã♦ ❞❡ X ❡♠ X✱ ❞✐r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ q✉❡ R é ✉♠❛ r❡❧❛çã♦ ❡♠ X✳
◆♦ ❡①❡♠♣❧♦ q✉❡ s❡ s❡❣✉❡✱ ❞❛❞♦sX ={1,2,3} ❡Y ={2,3,4,5}✱ ♦ ❝♦♥❥✉♥t♦ R=
{(x, y)∈X×Y;x≥y} é ❛ r❡❧❛çã♦ ❞❡ X ❡♠ Y ❞❛❞❛ ♣♦rR ={(2,2),(3,2),(3,3)}❀
❞❡ ❢❛t♦✱ ❡ss❡s sã♦ ♦s ú♥✐❝♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (x, y) ❝♦♠ x ∈ {1,2,3}, y ∈ {2,3,4,5} ❡ t❛✐s q✉❡ x≥y✳ ❙❡ R é ✉♠❛ r❡❧❛çã♦ ❞❡ X ❡♠ Y✱ ❡♥tã♦ R ⊂X×Y ♣♦r ❞❡✜♥✐çã♦✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❡s❝♦❧❤✐❞♦ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (x, y) ∈ X×Y✱ ♣♦❞❡ ♦❝♦rr❡r q✉❡
(x, y) ∈ R ♦✉ (x, y) 6∈ R ✭✐st♦ é✱ q✉❡ x ❡ y s❡❥❛♠ r❡❧❛❝✐♦♥❛❞♦s ♦✉ ♥ã♦ ♣♦r R✮✳ ◆♦
♣r✐♠❡✐r♦ ❝❛s♦✱ ✈❛♠♦s ❞❡♥♦t❛r ♣♦r xRy✳ ❚❡♠♦s q✉❡✿ xRy ⇔(x, y)∈R✳
❉❡✜♥✐çã♦ ✶✳✷✳✷ ❯♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ X 6= ∅ é ✉♠❛ s❡♥t❡♥ç❛ ❛❜❡rt❛ xRy ♥♦ ❝♦♥❥✉♥t♦ X×X✳
❙ã♦ ❡①❡♠♣❧♦s ❞❡ r❡❧❛çõ❡s ❜✐♥ár✐❛s ❛ ✐❣✉❛❧❞❛❞❡ x = y ❡♥tr❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ X ❡ ❛ r❡❧❛çã♦ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ x≤y ❡♠ Z✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ❯♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛xRy ❡♠ ✉♠ ❝♦♥❥✉♥t♦X 6=∅ s❡rá ❝❤❛♠❛❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ s❡ ♣♦ss✉✐r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭✐✮ xRx é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ x∈X ✭Pr♦♣r✐❡❞❛❞❡ ❘❡✢❡①✐✈❛✮❀
✭✐✐✮ ❙❡ xRy é ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ yRx é ✈❡r❞❛❞❡✐r❛ ✭Pr♦♣r✐❡❞❛❞❡ ❙✐♠étr✐❝❛✮❀ ✭✐✐✐✮ ❙❡ xRy ❡ yRz sã♦ ✈❡r❞❛❞❡✐r❛s✱ ❡♥tã♦xRz é ✈❡r❞❛❞❡✐r❛ ✭Pr♦♣r✐❡❞❛❞❡ ❚r❛♥✲
s✐t✐✈❛✮✳
❉❡✜♥✐çã♦ ✶✳✷✳✸ ❉❛❞❛ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ≡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ X✱ ❞❡✜✲ ♥✐♠♦s ❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ a ∈ X ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦
[a] = {x ∈ X;x ≡ a} ❡ ♦ ❡❧❡♠❡♥t♦ a s❡rá ❝❤❛♠❛❞♦ ❞❡ r❡♣r❡s❡♥t❛♥t❡ ❞❛ ❝❧❛ss❡
[a]✳
P♦r ❡①❡♠♣❧♦✱ s❡ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ é ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ X✱ t❡♠♦s q✉❡ [a] ={a}✱ ♣❛r❛ t♦❞♦ a∈X✳
❉❡✜♥✐çã♦ ✶✳✷✳✹ ❙❡❥❛♠ ❞❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s X ❡ Y✳ ❯♠❛ ❢✉♥çã♦ f ❞❡ X ❡♠ Y é ✉♠❛ r❡❣r❛ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ x∈X ✉♠ ú♥✐❝♦ y∈Y✳ ❖s ❝♦♥❥✉♥t♦s X ❡ Y sã♦ ❝❤❛♠❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✳ ❉❡♥♦♠✐✲ ♥❛r❡♠♦s X =D(f), Y =CD(f) ❡f(X)❝♦♠♦ s❡♥❞♦ ❛ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ f✳ ❚❡♠♦s q✉❡ f(X)⊂Y✳
❆s três ❞❡✜♥✐çõ❡s ❛ s❡❣✉✐r ❡①♣❧✐❝✐t❛♠ ❛❧❣✉♥s t✐♣♦s ❡①tr❡♠❛♠❡♥t❡ út❡✐s ❞❡ ❢✉♥çõ❡s✳
❉❡✜♥✐çã♦ ✶✳✷✳✺ ❉❛❞♦s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s X ❡ Y✱ ❡ ✜①❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ c∈Y✱ ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ c ❞❡ X ❡♠ Y é ❛ ❢✉♥çã♦ f :X →Y t❛❧ q✉❡ f(x) =c ♣❛r❛ t♦❞♦
x∈X✳
◆♦ ❝❛s♦ ❡①tr❡♠♦ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ❡ ✐❣✉❛❧ ❛ c✱ ❞❡✜♥✐❞❛ ❛❝✐♠❛✱ t♦❞♦ x ∈ X ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠ ♠❡s♠♦ y∈Y✱ ❛ s❛❜❡r✱ y=c✳ ❈♦♥t✉❞♦✱ ❛s ❝♦♥❞✐çõ❡s ✐♠♣♦st❛s ♥❛ ❞❡✜♥✐çã♦ sã♦ ♣❧❡♥❛♠❡♥t❡ s❛t✐s❢❡✐t❛s✱ ✐st♦ é✱ t♦❞♦x∈X ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠ ú♥✐❝♦
y∈Y✳
❉❡✜♥✐çã♦ ✶✳✷✳✻ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ X✱ ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X✱ ❞❡♥♦t❛❞❛ ♣♦r IdX :X →X✱ é ❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦r Id(x) =x ♣❛r❛ t♦❞♦ x∈X✳
❉❡✜♥✐çã♦ ✶✳✷✳✼ ❉✉❛s ❢✉♥çõ❡s f :X →Y ❡ g :W →Z sã♦ ✐❣✉❛✐s s❡X =W, Y =
Z ❡ f(x) =g(x) ♣❛r❛ t♦❞♦ x∈X✳
❙❡ ❞✉❛s ❢✉♥çõ❡s f : X → Y ❡ g : W → Z ❢♦r❡♠ ✐❣✉❛✐s✱ ❡s❝r❡✈❡♠♦s f = g✳ ❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ s✐❣♥✐✜❝❛ ❛ ✐❣✉❛❧❞❛❞❡ ❞♦s ❞♦♠í♥✐♦s✱ X =W✱ ❡ ❞♦s ❝♦♥tr❛❞♦♠í♥✐♦s✱ Y =Z✱ ❛ss✐♠ ❝♦♠♦ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❢✉♥çã♦ f(x) =g(x) ♣❛r❛ t♦❞♦x∈X✳ ❙❡ ❢✉♥çõ❡s f ❡ g ♥ã♦ ❢♦r❡♠ ✐❣✉❛✐s✱ ❡s❝r❡✈❡♠♦s f 6=g ❡ ❞✐r❡♠♦s q✉❡ f ❡g sã♦ ❢✉♥çõ❡s ❞✐❢❡r❡♥t❡s ♦✉ ❞✐st✐♥t❛s✳
✶✳✷✳✶ ❈♦♠♣♦s✐çã♦ ❞❡ ❋✉♥çõ❡s
❉❛❞❛s ❞✉❛s ❢✉♥çõ❡s f :X →Y ❡ g :Y →Z✱ t❡♠♦s✱ ❡♠ ú❧t✐♠❛ ❛♥á❧✐s❡✱ r❡❣r❛s ❜❡♠ ❞❡✜♥✐❞❛s ♣❛rt✐♥❞♦ ❞❡ x ∈ X ✈✐❛ f✱ ♦❜t❡r y = f(x) ∈ Y ❡✱ ✈✐❛ g✱ ♦❜t❡r g(z) ∈ Z✳ P❛r❡❝❡ ♠✉✐t♦ r❛③♦á✈❡❧ q✉❡ ♣♦ss❛♠♦s ❢♦r♠❛r ✉♠❛ ❢✉♥çã♦ q✉❡ ♥♦s ♣❡r♠✐t❛ s❛✐r ❞❡ X ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ Z✳ ❊st❡ é ❞❡ ❢❛t♦ ♦ ❝❛s♦✱ ❡ ❛ ❢✉♥çã♦ r❡s✉❧t❛♥t❡ é ❞❡♥♦♠✐♥❛❞❛ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ f ❡ g✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ s❡❣✉✐♥t❡✿
❉❡✜♥✐çã♦ ✶✳✷✳✽ ❉❛❞❛s ❛s ❢✉♥çõ❡s f :X →Y ❡ g :Y → Z✱ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ f ❡ g✱ ♥❡ss❛ ♦r❞❡♠✱ é ❛ ❢✉♥çã♦ g ◦f : X → Z ❞❡✜♥✐❞❛✱ ♣❛r❛ ❝❛❞❛ x ∈ X✱ ♣♦r
(g◦f)(x) =g(f(x))✳ ❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ❜❛st❛ q✉❡ t❡♥❤❛♠♦s f(X)⊂Y ♣❛r❛ q✉❡
❛ ❢✉♥çã♦ g◦f ❢❛ç❛ s❡♥t✐❞♦✳
❆♣❡s❛r ❞❡ ♥ã♦ s❡r ❝♦♠✉t❛t✐✈❛✱ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s é ❛ss♦❝✐❛t✐✈❛✱ ❝♦♥❢♦r♠❡ s❡❣✉❡✿
Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ❉❛❞❛s ❢✉♥çõ❡s f :X →Y✱ g :Y →Z ❡ h:Z →W✱ t❡♠♦s q✉❡
h◦(g◦f) = (h◦g)◦f✳
Pr♦✈❛✳ ❱❡❥❛ ♣r✐♠❡✐r♦ q✉❡ ❛s ❛♠❜❛sh◦(g◦f)❡ (h◦g)◦f sã♦ ❢✉♥çõ❡s ❞❡X ❡♠ W✳ P♦rt❛♥t♦✱ ♣❛r❛ s❡r❡♠ ✐❣✉❛✐s✱ é s✉✜❝✐❡♥t❡ q✉❡ ❛ss♦❝✐❡♠ ❝❛❞❛ x ∈ A ❡♠ ✉♠ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ W✳ P❛r❛ ✈❡r ✐st♦✱ ❜❛st❛ ♥♦t❛r q✉❡ (h◦(g ◦f))(x) = h((g◦f)(x)) =
h((g(f(x))) = (h◦g)(f(x)) = ((h◦g)◦f)(x)✳
❆ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣♦✐s✱ s❡ t✐✈❡r♠♦s ❢✉♥çõ❡s f✱ g ❡ h ❡ ♣✉❞❡r♠♦s ❝♦♠♣ô✲❧❛s ✭♥❡ss❛ ♦r❞❡♠✮✱ ♣♦❞❡♠♦s ❞❡♥♦t❛r ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ♣♦rh◦g◦f s✐♠♣❧❡s♠❡♥t❡✱ ♥ã♦ ♥♦s ♣r❡♦❝✉♣❛♥❞♦ ❝♦♠ q✉❛❧ ❝♦♠♣♦s✐çã♦ ❡❢❡t✉❛r ♣r✐♠❡✐r♦✳
❉❡✜♥✐çã♦ ✶✳✷✳✾ ❯♠❛ ❢✉♥çã♦ f :X →Y é ❞✐t❛✿
✭❛✮ ■♥❥❡t♦r❛✱ ✐♥❥❡t✐✈❛ ♦✉ ❛✐♥❞❛ ✉♠❛ ✐♥❥❡çã♦✱ s❡ ♣❛r❛ q✉❛✐sq✉❡r x1, x2 ∈X t❛✐s
q✉❡ x1 6=x2 ⇒f(x1)6=f(x2)❀
✭❜✮ ❙♦❜r❡❥❡t♦r❛✱ s♦❜r❡❥❡t✐✈❛ ♦✉ ❛✐♥❞❛ ✉♠❛ s♦❜r❡❥❡çã♦✱ s❡ s✉❛ ✐♠❛❣❡♠ ❢♦r t♦❞♦ ♦ ❝♦♥❥✉♥t♦ Y✱ ✐st♦ é✱ f(X) =Y❀
✭❝✮ ❇✐❥❡t♦r❛✱ ❜✐❥❡t✐✈❛ ♦✉ ❛✐♥❞❛ ✉♠❛ ❜✐❥❡çã♦ s❡ ❢♦r ❛♦ ♠❡s♠♦ t❡♠♣♦ ✐♥❥❡t✐✈❛ ❡ s♦❜r❡❥❡t✐✈❛✳
❚❡♦r❡♠❛ ✶✳✷✳✶ ❙❡ X ⊂ R é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ f : X → X é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡ f(f(x)) = x ♣❛r❛ t♦❞♦ x✱ ❡♥tã♦ f é ❜✐❥❡t✐✈❛✳
Pr♦✈❛✳ ❙❡❥❛♠ x1 ❡ x2 ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ f(x1) = f(x2)✳ P❛r❛ ♠♦str❛r♠♦s
q✉❡ f é ✐♥❥❡t✐✈❛ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡ x1 = x2✳ P❛r❛ t❛♥t♦✱ ♦❜s❡r✈❡ q✉❡ f(x1) =
f(x2) ⇒ f(f(x1)) = f(f(x2))✳ ▲♦❣♦✱ x1 = x2 ♣♦r ❤✐♣ót❡s❡✳ ❆ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡
f é ✐♠❡❞✐❛t❛✳ ❋✐①❛❞♦ y ∈ X ❡ t♦♠❛♥❞♦ f(y) ∈X✱ t❡♠♦s f(f(y)) = y✳ ■st♦ ♣♦st♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ y∈f(X)✳
❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❡♥s✐♥❛ ❝♦♠♦ s❡ ❝♦♠♣♦rt❛♠ ❢✉♥çõ❡s ✐♥❥❡t✐✈❛s✱ s♦❜r❡❥❡t✐✈❛s ❡ ❜✐❥❡t✐✈❛s ❡♠ r❡❧❛çã♦ à ❝♦♠♣♦s✐çã♦✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✸ ❙❡❥❛♠ f :X →Y ❡ g :Y →Z ❞✉❛s ❢✉♥çõ❡s ❞❛❞❛s✳ ❊♥tã♦✿ ✭❛✮ g◦f ✐♥❥❡t✐✈❛ ⇒f ✐♥❥❡t✐✈❛✱ ♠❛s ❛ r❡❝í♣r♦❝❛ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳ ✭❜✮ g◦f s♦❜r❡❥❡t✐✈❛ ⇒g s♦❜r❡❥❡t✐✈❛✱ ♠❛s ❛ r❡❝í♣r♦❝❛ ♥❡♠ s❡♠♣r❡ é ✈❡r❞❛❞❡✐r❛✳
✭❝✮ g, f ✐♥❥❡t✐✈❛s ⇒g◦f ✐♥❥❡t✐✈❛✳ ✭❞✮ g, f s♦❜r❡❥❡t✐✈❛s ⇒g◦f s♦❜r❡❥❡t✐✈❛✳ ✭❡✮ g, f ❜✐❥❡t✐✈❛s ⇒g◦f ❜✐❥❡t✐✈❛✳
Pr♦✈❛✳
✭❛✮ P❛r❛ x1 ❡ x2 ❡♠ X✱ t❡♠♦s q✉❡ f(x1) = f(x2) ⇒ g(f(x1)) = g(f(x2)) ⇒
(g ◦f)(x1) = (g ◦ f)(x2) ⇒ x1 = x2✱ ♦♥❞❡ ♥❛ ú❧t✐♠❛ ♣❛ss❛❣❡♠ ✉s❛♠♦s ♦ ❢❛t♦
❞❡ g◦f s❡r ✐♥❥❡t✐✈❛✳ ❚❡♠♦s ❛❣♦r❛ q✉❡ ❞❛r ✉♠ ❡①❡♠♣❧♦ ♥♦ q✉❛❧ f s❡❥❛ ✐♥❥❡t✐✈❛ ♠❛s g◦f ♥ã♦ ♦ s❡❥❛✳ P❛r❛ t❛♥t♦✱ ❜❛st❛ t♦♠❛r♠♦sX =Y =Z =R, f(x) = x❡g(x) =x2✳
✭❜✮ ❉❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡z ∈Z✱ ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡ g◦f ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ x ∈ X t❛❧ q✉❡ z = (g ◦f)(x)✳ ▼❛s ❛í✱ z = g(f(x)) ❡ g t❛♠✲ ❜é♠ é s♦❜r❡❥❡t✐✈❛✳ P❛r❛ ♦ ❡①❡♠♣❧♦ ♥❡❝❡ssár✐♦ à s❡❣✉♥❞❛ ♣❛rt❡✱ t♦♠❡♠♦s ♥♦✈❛♠❡♥t❡
X =Y =Z =R, g(x) = x❡ f(x) =x2✳
✭❝✮ ❯s❛♥❞♦ s✉❝❡ss✐✈❛♠❡♥t❡ ❛s ✐♥❥❡t✐✈✐❞❛❞❡s ❞❡ g ❡ ❞❡ f✱ t❡♠♦s ♣❛r❛ x1 ❡ x2 ❡♠
X q✉❡(g◦f)(x1) = (g◦f)(x2)⇒g(f(x1)) =g(f(x2))⇒f(x1) =f(x2)⇒x1 =x2✱
❡ g◦f t❛♠❜é♠ é ✐♥❥❡t✐✈❛✳
✭❞✮ ❉❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ z ∈Z✱ ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡ g ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ y∈Y t❛❧ q✉❡z=g(y)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡f ❛ss❡❣✉r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ x ∈ X t❛❧ q✉❡ f(x) = y✳ ❊♥tã♦✱ t❡♠♦s q✉❡ (g ◦f)(x) = g(f(x)) = g(y) = z✱ ❞♦♥❞❡ g◦f t❛♠❜é♠ é s♦❜r❡❥❡t✐✈❛✳
✭❡✮ ❙❡❣✉❡ ❞♦s ✐t❡♥s ✭❝✮ ❡ ✭❞✮ q✉❡g ❡f ❜✐❥❡t♦r❛s⇒g ❡f sã♦ ✐♥❥❡t✐✈❛s ❡ s♦❜r❡❥❡t✐✈❛s ⇒ g◦f ✐♥❥❡t✐✈❛ ❡ s♦❜r❡❥❡t✐✈❛ ⇒ g◦f ❜✐❥❡t✐✈❛✳
✶✳✷✳✷ ■♥✈❡rsã♦ ❞❡ ❋✉♥çõ❡s
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❢✉♥çã♦ f :X → Y ❜✐❥❡t✐✈❛✳ ❚❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ X ❡ Y ❡stã♦ ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✱ ♦✉ s❡❥❛✱ ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡X ❝♦rr❡s♣♦♥❞❡ ✉♠ ❡ só ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Y ✈✐❛ f✱ ❡ ✈✐❝❡✲✈❡rs❛✳ ◗✉❛♥❞♦ t❛❧ ♦❝♦rr❡r✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ♦✉tr❛ ❢✉♥çã♦ g :Y →X✱ s✐♠♣❧❡s♠❡♥t❡ ❡①✐❣✐♥❞♦ q✉❡ f(x) =y⇔g(y) = x✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✵ ❉✐r❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ g :Y → X é ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❞❡ f s❡ g◦f =IdX ❡ g é ✉♠❛ ✐♥✈❡rs❛ à ❞✐r❡✐t❛ ❞❡ f s❡ f ◦g =IdY✳
❯♠❛ ♣❡r❣✉♥t❛ ♥❛t✉r❛❧ ❛ ❡st❛ ❛❧t✉r❛ é ♣♦r q✉❡ ♥ã♦ ♣♦❞❡♠♦s ✉s❛r ❛ ❞❡❝❧❛r❛çã♦ ❛❝✐♠❛ ♣❛r❛ ❞❡✜♥✐r ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛✳ ❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ✐♥t✉✐t✐✈♦✱ s❡ f ♥ã♦ ❢♦ss❡ s♦❜r❡❥❡t✐✈❛✱ ❡①✐st✐r✐❛ ✉♠ ❡❧❡♠❡♥t♦ y ❞❡ Y q✉❡ ♥ã♦ s❡r✐❛ ✐♠❛❣❡♠ ♣♦r
f ❞❡ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡ X✳ ❆ss✐♠✱ ♥ã♦ t❡rí❛♠♦s ✉♠❛ ♠❛♥❡✐r❛ ♥❛t✉r❛❧ ❞❡ ❞❡✜♥✐r g(y) ❛ ♣❛rt✐r ❞❡ f✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ f ♥ã♦ ❢♦ss❡ ✐♥❥❡t✐✈❛✱ ❡①✐st✐r✐❛♠ ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s x1 ❡x2 ❡♠ X ❝♦♠ ✉♠❛ ♠❡s♠❛ ✐♠❛❣❡♠ y∈Y ✈✐❛f✳ ◗✉❛♥❞♦ t❡♥táss❡♠♦s
❞❡✜♥✐r g ♣♦r ♠❡✐♦ ❞❡ f✱ t❛♠❜é♠ ♥ã♦ ❤❛✈❡r✐❛ ♠❛♥❡✐r❛ ♥❛t✉r❛❧ ❞❡ ❞❡❝✐❞✐r♠♦s q✉❡♠✱ ❞❡♥tr❡ x1 ❡ x2✱ ❞❡✈❡r✐❛ s❡r ✐❣✉❛❧ ❛ g(y)✳
❱♦❧t❛♥❞♦ ❛♦ ❝❛s♦ ❡♠ q✉❡ f é ❜✐❥❡t✐✈❛✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ g✱ ❞❡✜♥✐❞❛ ❝♦♠♦ ❛❝✐♠❛✱ é ❞❡ ❢❛t♦ ✉♠❛ ❢✉♥çã♦✱ ❡ ❛❞❡♠❛✐s t❛❧ q✉❡ (g◦f)(x) = x ♣❛r❛ t♦❞♦ x ∈ X ❡
(f ◦g)(y) =y ♣❛r❛ t♦❞♦ y∈ Y✳ ❉❡ ♦✉tr♦ ♠♦❞♦✱ t❡♠♦s g◦f =IdX ❡ f ◦g =IdY✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ f : X → Y ❡ g : Y → X sã♦ ❢✉♥çõ❡s t❛✐s q✉❡ g◦f = IdX ❡ f◦g =IdY✱ ❡♥tã♦f ❞❡✈❡ s❡r✱ ❞❡ ❢❛t♦ ✉♠❛ ❜✐❥❡çã♦✱ ❡ g é ❛ ú♥✐❝❛ ❢✉♥çã♦ q✉❡ s❛t✐s❢❛③ t❛✐s ✐❣✉❛❧❞❛❞❡s ❞❡ ❝♦♠♣♦s✐çã♦✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✶ ❙❡❥❛ f :X →Y ✉♠❛ ❜✐❥❡çã♦ ❞❛❞❛✳ ❆ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ f é ❛ ❢✉♥çã♦ g :Y →X t❛❧ q✉❡✱ ♣❛r❛ x∈X, y ∈Y✱ t❡♠♦s q✉❡ g(y) = x⇔y =f(x)✳
❉❛q✉✐ ❡♠ ❞✐❛♥t❡✱ ❞❡♥♦t❛r❡♠♦s ❛ ✐♥✈❡rs❛ ❞❡ ✉♠❛ ❜✐❥❡çã♦ f : X → Y ♣♦r f−1 :
Y →X✳ ❖❜s❡r✈❡ q✉❡ ♦ ❡①♣♦❡♥t❡ −1♥❛ ♥♦t❛çã♦ ❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ♥ã♦ t❡♠ ♥❡♥❤✉♠ s✐❣♥✐✜❝❛❞♦ ❛r✐t♠ét✐❝♦✳ ❊❧❡ s✐♠♣❧❡s♠❡♥t❡ ❝❤❛♠❛ ❛t❡♥çã♦ ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ f−1 ❢❛③
♦ ❝❛♠✐♥❤♦ ✐♥✈❡rs♦ ❞❡ f✱ ✐st♦ é✱ ❛♣❧✐❝❛ Y ❡♠ X ❡♠ ✈❡③ ❞❡ X ❡♠ Y✱ r❡✈❡rt❡♥❞♦ ❛s s❡t❛s ❞❛s ❛ss♦❝✐❛çõ❡s ❢❡✐t❛s ♣♦r f✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✹ ❯♠❛ ❢✉♥çã♦ é s♦❜r❡❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ ❛❞♠✐t❡ ✐♥✈❡rs❛ à ❞✐r❡✐t❛✳
Pr♦✈❛✳ ❙❡❥❛ f :X →Y ✉♠❛ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛✳ ❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ y∈Y é ♣♦ssí✈❡❧ ❡s❝♦❧❤❡r ♣❡❧♦ ♠❡♥♦s ✉♠x∈X t❛❧ q✉❡y=f(x)✳ ❋✐①❡ ✉♠ t❛❧x♣❛r❛ ❝❛❞❛y✳ ❉❡✜♥❛ g : Y → X t❛❧ q✉❡ g(y) = x ✭♥♦t❡ q✉❡ ❡♠ ❣❡r❛❧ t♦❞❛ ❢✉♥çã♦ g ♥ã♦ é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✱ ❡❧❛ ♦ s❡rá s❡ f é ✐♥❥❡t✐✈❛✮✳ ❙❡❣✉❡✲s❡ ❡♥tã♦ q✉❡✱ ♣❛r❛ t♦❞♦ y ∈ Y✱
f◦g(y) =f(g(y)) =f(x) =y✳ ■st♦ ♣♦st♦✱f ◦g =IdY ❡✱ ♣♦rt❛♥t♦✱g é ✉♠❛ ✐♥✈❡rs❛
à ❞✐r❡✐t❛ ❞❡ f✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ f ◦g = IdY ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ g :Y →X✳ ❈♦♠♦ IdY é s♦❜r❡❥❡t✐✈❛✱ s❡❣✉❡✲s❡ q✉❡f é t❛♠❜é♠ s♦❜r❡❥❡t✐✈❛✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✺ ❯♠❛ ❢✉♥çã♦ é ✐♥❥❡t✐✈❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ ❛❞♠✐t❡ ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛✳
Pr♦✈❛✳ ❙❡❥❛ f :X →Y ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t✐✈❛✳ ❊♥tã♦✱ ❝❛❞❛y∈ f(X) ❞❡t❡r♠✐♥❛ ✉♠ ú♥✐❝♦ x∈X t❛❧ q✉❡ y=f(x)✳ ❉❡✜♥❛ g :Y →X ❝♦♠♦ ❛ s❡❣✉✐r✿
g(y) =
x, s❡ y∈f(X)
a, s❡ y6∈f(X)
♦♥❞❡a é ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ✜①❛❞♦ ❞❡X✳ ◆♦t❡ q✉❡ ❡♠ ❣❡r❛❧g ♥ã♦ é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✱ ❡❧❛ ♦ s❡rá s❡f ❢♦r s♦❜r❡❥❡t✐✈❛✳ ❙❡❣✉❡✲s❡ ❡♥tã♦ q✉❡g◦f(x) = g(f(x)) =
g(y) = x✱ ♣❛r❛ t♦❞♦x ∈X✳ ■st♦ ♣♦st♦✱ g◦f =IdX ❡✱ ♣♦rt❛♥t♦✱ g é ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❞❡ f✳ ❙✉♣♦♥❤❛ r❡❝✐♣r♦❝❛♠❡♥t❡ q✉❡ ❡①✐st❡ g :Y →X t❛❧ q✉❡ g◦f =IdX✳ ❈♦♠♦ IdX é ✐♥❥❡t✐✈❛✱ s❡❣✉❡✲s❡ q✉❡ f é ✐♥❥❡t✐✈❛✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✻ ❙❡ ✉♠❛ ❢✉♥çã♦ ❛❞♠✐t❡ ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❡ ✉♠❛ ✐♥✈❡rs❛ à ❞✐r❡✐t❛✱ ❡♥tã♦ ❡st❛s sã♦ ✐❣✉❛✐s✳
Pr♦✈❛✳ ❙❡❥❛♠g1, g2 :Y →X✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✉♠❛ ✐♥✈❡rs❛ à ❞✐r❡✐t❛ ❡ ✉♠❛ ✐♥✈❡rs❛
à ❡sq✉❡r❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f :X →Y✳ ❙❡❣✉❡✲s❡ q✉❡ g1 =IdX ◦g1 = (g2◦f)◦g1 =
g2◦(f◦g1) =g2 ◦IdY =g2✳
▲❡♠❛ ✶✳✷✳✶ ❙❡❥❛♠A❡B ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s✳ ❊①✐st❡ ✉♠❛ ✉♠❛ ❢✉♥çã♦ s♦❜r❡❥❡t✐✈❛ f :A→B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t✐✈❛ g :B →A✳
Pr♦✈❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ ❡①✐st❡ ✉♠❛ s♦❜r❡❥❡çã♦ f : A → B✱ s❡❣✉❡✲s❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ x ∈ B✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ y ∈ A t❛❧ q✉❡ f(y) = x✳ ❊♥tã♦✱ ❞❡✜♥✐♠♦s g : B → A✱ t❛❧ q✉❡ g(x) = y✳ ◆♦t❡ q✉❡ ✐ss♦ é✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱ ✉♠❛ ✐♥❥❡çã♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ ❡①✐st❡ g :B →A ✐♥❥❡t✐✈❛✱ ✜①❛♥❞♦ ✉♠a∈B q✉❛❧q✉❡r✱ ❞❡✜♥❡✲s❡ f :A→B t❛❧ q✉❡ f(x) = g−1(x)✱ s❡ x∈g(B) ❡ f(x) = a✱ s❡ x6∈g(B)✳ ◆♦t❡ q✉❡ f
é ❡✈✐❞❡♥t❡♠❡♥t❡ s♦❜r❡❥❡t✐✈❛✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✼ ❯♠❛ ❢✉♥çã♦ ❛❞♠✐t❡ ✐♥✈❡rs❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ é ❜✐❥❡t✐✈❛✳
Pr♦✈❛✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t✐✈❛✳ ▲♦❣♦✱ f ❛❞♠✐t❡ ✉♠❛ ✐♥✈❡rs❛ à ❡sq✉❡r❞❛ ❡ ✉♠❛ ✐♥✈❡rs❛ à ❞✐r❡✐t❛✳ ▲♦❣♦✱ ❡st❛s sã♦ ✐❣✉❛✐s✱ ❞❡✜♥✐♥❞♦ ✉♠❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ♣❛r❛ f✳ ❆ r❡❝í♣r♦❝❛ s❡❣✉❡ ❞♦ ❡①♣❧✐❝✐t❛❞♦ ❛❝✐♠❛✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✽ ❙❡ f : X → Y ❡ g :Y → Z sã♦ ❢✉♥çõ❡s ❜✐❥❡t✐✈❛s✱ ❡♥tã♦ g◦f : X →Z é ❜✐❥❡t✐✈❛ ❡ (g◦f)−1 =f−1◦g−1✳
Pr♦✈❛✳ ❏á s❛❜❡♠♦s q✉❡g◦f é ❜✐❥❡t✐✈❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦(g◦f)−1❡f−1◦g−1 sã♦
❛♠❜❛s ❢✉♥çõ❡s ❞❡Z❡♠X✱ ❛ ✜♠ ❞❡ ✈❡r✐✜❝❛r q✉❡(g◦f)−1 =f−1◦g−1é s✉✜❝✐❡♥t❡✱ ♣❡❧❛
✉♥✐❝✐❞❛❞❡ ❞❛ ✐♥✈❡rs❛✱ ♥♦t❛r q✉❡(f−1◦g−1)◦(g◦f) =Id
X ❡(g◦f)◦(f−1◦g−1) = IdZ✳
✶✳✷✳✸ ❈♦♥❥✉♥t♦ ❞❡ ❋✉♥çõ❡s
❉❡✜♥✐çã♦ ✶✳✷✳✶✷ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦sX ❡Y✳ ❉❡♥♦t❛r❡♠♦s ♣♦rF(X, Y) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ❞❡ X ❡♠ Y✳
