❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
Pr✐♥❝✐♣❛✐s ❆①✐♦♠❛s ❞❛ ▼❛t❡♠át✐❝❛
†♣♦r
▼❛❣♥✉♥ ❈és❛r ◆❛s❝✐♠❡♥t♦ ❞♦s ❙❛♥t♦s
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✐❝❛✳
❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
S237p Santos, Magnun César Nascimento dos.
Principais axiomas da matemática / Magnun César Nascimento dos Santos.-- João Pessoa, 2014.
43f.
Orientador: Bruno Henrique Carvalho Ribeiro Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Axiomas. 3. Lema de Zorn. 4. Axioma da escolha.
Pr✐♥❝✐♣❛✐s ❆①✐♦♠❛s ❞❛
▼❛t❡♠át✐❝❛
♣♦r
▼❛❣♥✉♥ ❈és❛r ◆❛s❝✐♠❡♥t♦ ❞♦s ❙❛♥t♦s
❉✐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❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❆❧✈❡s ❞❡ ▼❡❞❡✐r♦s ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ▼❛✉rí❝✐♦ ❈❛r❞♦s♦ ❙❛♥t♦s ✲ ❯❋P❊
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣❡❧♦ ❞♦♠ q✉❡ ♠❡ ❢♦✐ ❞❛❞♦✳ ❆ ❯❋P❇✱ ♣❡❧❛ ❡①❝❡❧ê♥❝✐❛ ❞❡ ❡♥s✐♥♦✳
❆♦ Pr♦❢❡ss♦r ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ♣❡❧❛ ❝♦♥✲ ✜❛♥ç❛ ❡♠ ♠✐♠ ❞❡♣♦s✐t❛❞❛ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ tã♦ ❞❡s❛✜❛❞♦r✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✱ ♦❜r✐❣❛❞♦ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ❡①❡♠♣❧♦ ❛♦ ❧♦♥❣♦ ❞❡st❛ ❥♦r♥❛❞❛✳
❆ ♠✐♥❤❛ ♠ã❡✱ ▼✐r✐❛♠ ▼❛r✐❛✱ q✉❡ s❡♠♣r❡ ❛❝r❡❞✐t♦✉✱ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❜❛st❛♥t❡ ❡ q✉❡ ♥ã♦ ♠❡❞✐✉ ❡s❢♦rç♦ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ❛té ❛q✉✐✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ▼✐t❝❤❡❧ ❡ ▼♦❡♠❛✱ ♠❡✉s ❡t❡r♥♦s ❛♠✐❣♦s✳
❆♦s ❣r❛♥❞❡s ❛♠✐❣♦s ❝♦♥q✉✐st❛❞♦s ❞✉r❛♥t❡ ♦ ❝✉rs♦✱ ❝♦♠ ♦ q✉❛❧ ✜③❡♠♦s ✉♠ ❣r✉♣♦ ❞❡ ❡st✉❞♦ ✭❖s ❈♦♥❣r✉❡♥t❡s✮ q✉❡ ✈✐r♦✉ ✉♠❛ ❢❛♠í❧✐❛✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ❆❧❡ss❛♥❞r♦ ▼✐❣✲ ♥❛❝✱ ❆♥❞ré ❘♦❞r✐❣✉❡s✱ ❲❛s❤✐♥❣t♦♥ ●♦♥ç❛❧✈❡s ❡ ❈②❜❡❧❡ ❱❡r❞❡✱ ❞✉r❛♥t❡ ❡ss❛ ❥♦r♥❛❞❛ ❡✉ ❛♣r❡♥❞✐ ♠✉✐t♦ ❝♦♠ ✈♦❝ês✳
❆ ❆❞r✐❛♥❛ ◆❛s❝✐♠❡♥t♦✱ q✉❡ ♠✉✐t♦ ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥❡st❛ ❥♦r♥❛❞❛✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦ ♣❡❧♦s ♣✉①õ❡s ❞❡ ♦r❡❧❤❛✱ ❡❧❡s t✐✈❡r❛♠ ê①✐t♦ ❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝❤❡❣❛r ❛té ❛q✉✐✳
❆♦s ♠❡✉s ❛❧✉♥♦s✱ q✉❡ ♣♦r ♠✉✐t❛s ✈❡③❡s ♠❡s♠♦ s❡♠ s❛❜❡r ♦ s✐❣♥✐✜❝❛❞♦ ❞❡st❡ ❝✉rs♦✱ ♠❛s ❡❧❡s s❡♠♣r❡ ❡st❛✈❛♠ ❛❧✐ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦✱ ♣❛r❛ q✉❡ ❡st❡ ♦❜❥❡t✐✈♦ ❢♦ss❡ ❛❧❝❛♥ç❛❞♦✳ ❙❛✐❜❛♠ q✉❡ ✈♦❝ês sã♦ ♠✐♥❤❛s ✐♥s♣✐r❛çõ❡s✳
❆♦ ❛♠✐❣♦ ❡ ▼❡str❡ ▲❛ér❝✐♦ ❋r❛♥❝✐s❝♦ ❋❡✐t♦s❛✱ ♣❡❧❛ ✐♥❡st✐♠á✈❡❧ ❛❥✉❞❛ ♥❛ ❢♦r♠❛✲ t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❢❛③❡r ✉♠❛ ❛❜♦r❞❛❣❡♠ s♦❜r❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ s✐st❡♠❛s ❛①✐♦♠át✐❝♦s ♥❛ ▼❛t❡♠át✐❝❛✳ ❊st✉❞❛r❡♠♦s ❛❧❣✉♥s ❛①✐♦♠❛s ❝❧áss✐❝♦s✱ s✉❛s ❡q✉✐✈❛❧ê♥❝✐❛s ❡ ✈❡r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦s ♠❡s♠♦s✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆①✐♦♠❛s✱ ▲❡♠❛ ❞❡ ❩♦r♥✱ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✳
❆❜str❛❝t
❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s s❤♦✇✐♥❣ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ s②st❡♠s ❛①✐♦♠❛t✐❝ ✐♥ ♠❛t❤❡♠❛t✐❝s✳ ❲❡ ✇✐❧❧ st✉❞② s♦♠❡ ❝❧❛ss✐❝ ❛①✐♦♠s✱ t❤❡✐r ❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ ✇❡ ✇✐❧❧ s❡❡ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡♠✳
❑❡②✇♦r❞s✿ ❆①✐♦♠s✱ ❩♦r♥✬s ▲❡♠♠❛✱ ❆①✐♦♠ ♦❢ ❈❤♦✐❝❡
❙✉♠ár✐♦
✶ ❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦ ✶
✶✳✶ ❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❖♣❡r❛çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ ▲❡♠❛ ❞❡ ❩♦r♥ ❡ ❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦ ✶✸ ✷✳✶ ◆♦çõ❡s ❇ás✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❘❡❧❛çõ❡s ❞❡ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❇♦❛ ❖r❞❡♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✹ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✺ ▲❡♠❛ ❞❡ ❩♦r♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻ ❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✿ ❊q✉✐✈❛❧ê♥❝✐❛s ❡ ❆♣❧✐❝❛çõ❡s ✷✺ ✸✳✶ ❊q✉✐✈❛❧ê♥❝✐❛s ❞♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷ ❆♣❧✐❝❛çõ❡s ❞♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
❆ ●ö❞❡❧ ❡ s❡✉s ❚❡♦r❡♠❛s ✸✶
■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛ ❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ♥✉♠❛ ✈✐sã♦ ❛①✐♦♠át✐❝❛✳ ❖ ♦❜✲ ❥❡t✐✈♦ ❞♦ ♠❡s♠♦ é ❛♣r❡s❡♥t❛r ♦s ❛①✐♦♠❛s ❡ s✉❛s ♥❡❝❡ss✐❞❛❞❡s ❞❡ ❡①✐stê♥❝✐❛ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛✱ ❡ ♦s s✐st❡♠❛s ❛①✐♦♠át✐❝♦s✱ ✉♠❛ ✈❡③ q✉❡ ❞❡s❞❡ ❞❡ ❝r✐❛♥ç❛ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♦s ❛①✐♦♠❛s ❞❡ ✉♠❛ ❢♦r♠❛ ❜❡♠ ✐♥t✉✐t✐✈❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡✜♥✐çã♦ ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s✳ P♦rt❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❛①✐♦♠❛s ❡ s✐st❡♠❛s ❛①✐♦♠át✐❝♦s ❡ ❛s ❡q✉✐✈❛❧ê♥❝✐❛s q✉❡ ❡①✐st❡♠ ❡♥tr❡ ♦s ♣r✐♥❝✐♣❛✐s ❛①✐♦♠❛s ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s✱ ❛ s❛❜❡r✱ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ ▲❡♠❛ ❞❡ ❩♦r♥ ❡ ❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦✳
▼✉✐t♦ s❡ ❞✐s❝✉t✐✉ s♦❜r❡ ♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛ ♥♦s ú❧t✐♠♦s ✶✵✵ ❛♥♦s✱ ♣♦✐s ❛té ♦s ✜♥s ❞♦ sé❝✉❧♦ ❳■❳✱ ❛❝❡✐t❛✈❛✲s❡ ❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♠♦ s❡♥❞♦ ✉♠❛ ❝♦❧❡çã♦ ❞❡ q✉❛❧q✉❡r ♦❜❥❡t♦s✱ ♣♦ré♠ ♣❡r❝❡❜❡✉✲s❡ q✉❡ t❛❧ ❞❡✜♥✐çã♦ ❡r❛ ♠✉✐t♦ ✈❛❣❛ ❡ ♥ã♦ s✉✜❝✐✲ ❡♥t❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ✉♠❛ ✈❡③ q✉❡✱ ❡♠ ✶✾✵✶✱ ❘✉ss❡❧ ❢❡③ ✉♠❛ ❝♦♥str✉çã♦ q✉❡ ❛t✉❛❧♠❡♥t❡ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♦ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧✱ ❡①♣❧♦r❛♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ ❛té ❡♥tã♦ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❞❛✈❛ ❡s♣❛ç♦ ❛ ✐❞❡✐❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✳ ❆ ♣❛rt✐r ❞❡ ❡♥tã♦✱ ✈ár✐♦s ♠❛t❡♠át✐❝♦s s❡ ❞❡❜r✉ç❛✲ r❛♠ s♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝r✐❛r ❛①✐♦♠❛s ♣❛r❛ q✉❡ ♣✉❞❡ss❡ s❡r ❡st❛❜❡❧❡❝✐❞❛ ✉♠❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ❜❛s❡❛❞❛s ❡♠ r❡❣r❛s ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ ♦❜❥❡t♦s q✉❡ s❡ ❝❤❛♠❛r✐❛♠ ❝♦♥❥✉♥t♦s✳ ❊♠ ✶✾✵✹✱ ❩❡r♠❡❧♦ ❛♣r❡s❡♥t❛ à ❝♦♠✉♥✐❞❛❞❡ ♠❛t❡♠át✐❝❛ ✉♠❛ ♣♦ssí✈❡❧ ❝♦❧❡çã♦ ❞❡ ❛①✐♦♠❛s ♣❛r❛ ❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s q✉❡✱ ❝♦♠ ♣❡q✉❡♥❛s ❛❧t❡r❛çõ❡s✱ é ❝♦♥❤❡❝✐❞❛ ❛t✉❛❧♠❡♥t❡ ❝♦♠♦ ♦s ❆①✐♦♠❛s ❞❡ ❩❡r♠❡❧♦✲❋r❛❡♥❦❡❧✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡✜♥✐♠♦s ♦ q✉❡ ✈ê♠ ❛ s❡r ❆①✐♦♠❛s ❡ ❙✐st❡♠❛s ❆①✐♦♠át✐✲ ❝♦s✱ ♦s ❛①✐♦♠❛s ❞❡ ❩❡r♠❡❧♦ ✲ ❋r❛❡♥❦❡❧ ❡ s✉❛s ✐♠♣♦rtâ♥❝✐❛s✱ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ♦♣❡r❛çõ❡s ❝♦♠ ❝♦♥❥✉♥t♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ t❡♠♦s ❛s s❡çõ❡s ❝♦♠ ❛s ❞❡✜♥✐çõ❡s ❞❡ r❡❧❛çõ❡s ❞❡ ♦r❞❡♠✱ ❞♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ ❞♦ ▲❡♠❛ ❞❡ ❩♦r♥ ❡ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛♠♦s ❛s ❡q✉✐✈❛❧ê♥❝✐❛s ❞♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛ ❡ ❝✐♥❝♦ ❛♣❧✐❝❛çõ❡s ❞♦ ♠❡s♠♦ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ t❛✐s ❝♦♠♦ ➪❧❣❡❜r❛ ▲✐♥❡❛r✱ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ❡ ❆♥á❧✐s❡✳
❊st❡ t❡①t♦ ❢♦✐ ❝♦♠♣❧❡t❛♠❡♥t❡ ✐♥s♣✐r❛❞♦ ❡♠ ❬✹❪✱ ❬✺❪✱ ❬✻❪ ❡ ❬✶✸❪✳
❈❛♣ít✉❧♦ ✶
❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛r ❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s s♦❜r❡ ✉♠❛ ❜❛s❡ ❛①✐♦♠át✐❝❛ ❡✱ ♣♦rt❛♥t♦✱ ❜❛st❛♥t❡ ❛❜str❛t❛✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❝♦♥str✉✐r ✉♠ s✐st❡♠❛ ❣❡r❛❧ q✉❡ s✐r✈❛ ❞❡ ♠♦❞❡❧♦ ♣❛r❛ ❢✉t✉r❛s ✐♥✈❡st✐❣❛çõ❡s ❛❧é♠ ❞♦ ❝❛♠♣♦ ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s✳ ◆♦ sé❝✉❧♦ ❳■❳✱ ❝r✐♦✉ ✲ s❡ ❛ ❊s❝♦❧❛ ❋♦r♠❛❧✐st❛ ❞❡ ❉❛✈✐❞ ❍✐❧❜❡rt✱ ♦♥❞❡ s✉❛ ✜❧♦s♦✜❛ ❝♦♥❝❡♥tr❛✈❛ ❡♠ ✏❚♦❞❛ ▼❛t❡♠át✐❝❛ é s✉st❡♥t❛❞❛ ♣♦r ✉♠ s✐st❡♠❛ ❛❜str❛t♦✑✱ ❡♠ ♦♣♦s✐çã♦ ❛ t❛❧ ♣❡♥s❛♠❡♥t♦✱ ❡①✐st❡ ❛ ❊s❝♦❧❛ ■♥t✉✐❝✐♦♥✐st❛✱ q✉❡ s❡ ♠❛♥✐s❢❡st❛ ❝♦♥✲ trár✐❛ ❛♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❛①✐♦♠át✐❝♦✳ P♦ré♠✱ ❛♠❜❛s ❛s ❡s❝♦❧❛s ❛♣r❡s❡♥t❛♠ ❞❡✜❝✐ê♥❝✐❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❋♦r♠❛❧✐st❛ ❛♣r❡s❡♥t❛ s✐st❡♠❛s ❛❜str❛t♦s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡ ❛✉t♦❡①✲ ♣❧✐❝❛r✱ t❛❧ ❛✜r♠❛çã♦ é ❥✉st✐✜❝❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❛ ■♥❝♦♠♣❧❡t✉❞❡ ❞❡ ●ö❞❡❧✱ ♦ q✉❛❧ ✈❡r❡♠♦s ♣♦st❡r✐♦r♠❡♥t❡❀ ❥á ❛ ■♥t✉✐❝✐♦♥✐st❛ ♥ã♦ ♣❡r♠✐t❡ ❣❡♥❡r❛❧✐③❛çõ❡s s♦❜r❡ ❛ ❡str✉✲ t✉r❛ ❞❛ ▼❛t❡♠át✐❝❛✳
❊♠ ✶✾✸✶✱ ❑✉rt ●ö❞❡❧ ✭✶✾✵✻ ✲ ✶✾✼✽✮✱ ❡♥tã♦ ❝♦♠ ✈✐♥t❡ ❡ ❝✐♥❝♦ ❛♥♦s✱ ♣✉❜❧✐❝♦✉ ♦ ❛rt✐❣♦✱ ❝✉❥♦ t✐t✉❧♦ ♣♦❞❡ s❡r tr❛❞✉③✐❞♦ ♣♦r ✏❙♦❜r❡ Pr♦♣♦s✐çõ❡s ❋♦r♠❛❧♠❡♥t❡ ■♥❞❡❝✐✲ ❞í✈❡✐s ❞♦s Pr✐♥❝✐♣✐❛ ▼❛t❤❡♠❛t✐❝❛ ❡ ❙✐st❡♠❛s ❘❡❧❛❝✐♦♥❛❞♦s✑✱ ♥♦ q✉❛❧ ❛♣r❡s❡♥t♦✉ ♦s s❡✉s ❞♦✐s ❚❡♦r❡♠❛s ❞❡ ■♥❝♦♠♣❧❡t✉❞❡✱ q✉❡ ❡stã♦ ❡♥tr❡ ♦s t❡♦r❡♠❛s ♠❛✐s ♣r♦❢✉♥❞♦s ❡ ❝♦♠ ❝♦♥s❡q✉ê♥❝✐❛s ♠❛✐s ♠❛r❝❛♥t❡s ❞❡ t♦❞❛ ❛ ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛ ❡ ▲ó❣✐❝❛ ❡♠ ❣❡r❛❧✳ ❚❛✐s t❡♦r❡♠❛s s✉r❣✐r❛♠ ❝♦♠♦ r❡s✉❧t❛❞♦ ❞❛s ✐♥✈❡st✐❣❛çõ❡s ❞❡ ●ö❞❡❧ q✉❛♥❞♦ ❛❜♦r✲ ❞❛✈❛ ❛ q✉❡stã♦ ❞❛ ❝♦♥s✐stê♥❝✐❛ ❞♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛✱ s❡❣✉♥❞♦ ❧✐♥❤❛s ❡st❛❜❡❧❡❝✐❞❛s ♣♦r ❍✐❧❜❡rt✱ ❡ ✈✐❡r❛♠ ❢♦r♥❡❝❡r ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❝♦♠♣❧❡t❛♠❡♥t❡ ♥♦✈❛ s♦❜r❡ ♦ ♣♦ssí✈❡❧ ❛❧❝❛♥❝❡ ❞❡ t♦❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ❢✉♥❞❛♠❡♥t❛çã♦ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦s s✐st❡♠❛s ❧ó❣✐❝♦s ❢♦r♠❛✐s q✉❡ t✐♥❤❛♠ ✈✐♥❞♦ ❛ s❡r ❞❡s❡♥✈♦❧✈✐❞♦s✳
✶✳✶ ❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦
❉❡✜♥✐çã♦ ✶✳✶✳✶ ✭❆①✐♦♠❛s✮✿ ❙ã♦ ✈❡r❞❛❞❡s ✐♥q✉❡st✐♦♥á✈❡✐s ✉♥✐✈❡rs❛❧♠❡♥t❡ ✈á❧✐❞❛s✱ ♠✉✐t❛s ✈❡③❡s ✉t✐❧✐③❛❞❛s ❝♦♠♦ ♣r✐♥❝í♣✐♦s ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ t❡♦r✐❛ ♦✉ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ✉♠❛ ❛r❣✉♠❡♥t❛çã♦✳
❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦ ❈❛♣ít✉❧♦ ✶
❆ ♣❛❧❛✈r❛ ❛①✐♦♠❛ ❞❡r✐✈❛ ❞❛ ❣r❡❣❛ ❛①✐♦s✱ ❝✉❥♦ s✐❣♥✐✜❝❛❞♦ é ❞✐❣♥♦ ♦✉ ✈á❧✐❞♦✳ ❊♠ ♠✉✐t♦s ❝♦♥t❡①t♦s✱ ❛①✐♦♠❛ é s✐♥ô♥✐♠♦ ❞❡ ♣♦st✉❧❛❞♦✱ ❧❡✐ ♦✉ ♣r✐♥❝í♣✐♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✷ ✭❆①✐♦♠át✐❝♦✮✿ ➱ ❛❧❣♦ ❡✈✐❞❡♥t❡✱ ✐♥q✉❡st✐♦♥á✈❡❧✱ ✐♥❝♦♥t❡stá✈❡❧✱ é r❡❧❛t✐✈♦ ❛♦s ❛①✐♦♠❛s✳
❉❡✜♥✐çã♦ ✶✳✶✳✸ ✭❙✐st❡♠❛ ❆①✐♦♠át✐❝♦✮✿ ➱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛①✐♦♠❛s q✉❡ ❞❡✜♥❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ t❡♦r✐❛ ❡ q✉❡ ❝♦♥st✐t✉❡♠ ❛s ✈❡r❞❛❞❡s ♠❛✐s s✐♠♣❧❡s ❛ ♣❛rt✐r ❞❛s q✉❛✐s s❡ ❞❡♠♦♥str❛♠ ♦s ♥♦✈♦s r❡s✉❧t❛❞♦s ❞❡ss❛ t❡♦r✐❛✳
❖s s✐st❡♠❛s ❛①✐♦♠át✐❝♦s tê♠ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡ ♥❛s ❝✐ê♥❝✐❛s ❡①❛t❛s✱ ♥♦♠❡❛❞❛✲ ♠❡♥t❡ ♥❛ ▼❛t❡♠át✐❝❛ ❡ ♥❛ ❋ís✐❝❛✱ s❡♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❡♠♦♥str❛❞♦s ♥❛s ♠ú❧t✐♣❧❛s t❡♦r✐❛s ❞❡ss❛s ❝✐ê♥❝✐❛s ✉s✉❛❧♠❡♥t❡ ❞❡s✐❣♥❛❞♦s ♣♦r t❡♦r❡♠❛s ♦✉ ❧❡✐s✳ ❙✉❛s ❛♣❧✐❝❛çõ❡s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❛ ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ t❛✐s ❝♦♠♦ ❧ó❣✐❝❛✱ ♠❛t❡♠át✐❝❛✱ ❡♥❣❡♥❤❛r✐❛✱ ❞❡♥tr❡ ♦✉tr❛s✳ ❖ s✐st❡♠❛ ❛①✐♦♠át✐❝♦ ❡①✐st❡♥t❡ ♥❛ ❧ó❣✐❝❛ é ✉♠❛ ❢♦r♠❛ ❞❡ t❡♦r✐❛ ❞❡❞✉t✐✈❛✱ q✉❡ ❢♦✐ ❝♦♥str✉í❞❛ ♣♦r ♠❡✐♦s ❞❡ t❡r♠♦s ✐♥✐❝✐❛✐s✱ ♦♥❞❡ s❡✉ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ s❡ ❞❡✉ ❛tr❛✈és ❞❡ r❡❣r❛s ❞❡ ❞❡✜♥✐çã♦✳ ❏á ♥❛ ▼❛t❡♠át✐❝❛✱ ♦ s✐st❡♠❛ ❛①✐♦♠át✐❝♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛①✐♦♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉s❛❞♦s ♣❛r❛ ❛ ❞❡r✐✈❛çã♦ ❞❡ ❚❡♦r❡♠❛s✳ ❊♥tr❡ ❛s ❞✐✈❡rs❛s ❛①✐♦♠át✐❝❛s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❞❛ ❋ís✐❝❛ ❣❛♥❤❛r❛♠ ♥♦✲ t♦r✐❡❞❛❞❡ ♦s Pr✐♥❝í♣✐♦s ❞❡ ❊✉❝❧✐❞❡s ♥❛ ●❡♦♠❡tr✐❛ ❈❧áss✐❝❛✱ ♦s ❆①✐♦♠❛s ❞❡ P❡❛♥♦ ♥❛ ❆r✐t♠ét✐❝❛✱ ❛s ▲❡✐s ❞❡ ◆❡✇t♦♥ ♥❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ❡ ♦s P♦st✉❧❛❞♦s ❞❡ ❊✐♥st❡✐♥ ♥❛ ❚❡♦r✐❛ ❞❛ ❘❡❧❛t✐✈✐❞❛❞❡✳
◆♦ â♠❜✐t♦ ❞❛ ▼❛t❡♠át✐❝❛✱ t❡♠♦s ❛❧❣✉♥s s✐st❡♠❛s ❛①✐♦♠át✐❝♦s q✉❡ ♣♦ss✐❜✐❧✐t❛r❛♠ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❛♦ s❡✉ r❡s♣❡✐t♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦s✱
✭■✮ ❆①✐♦♠❛s ❞❡ P❡❛♥♦ ✭❆P✮✿ ❆P1 : ❩❡r♦ é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧;
AP2 : ❩❡r♦ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧;
AP3]: ❖ s✉❝❡ss♦r ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧;
AP4 : ❉♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s q✉❡ t✐✈❡r❡♠ ♦ ♠❡s♠♦ s✉❝❡ss♦r sã♦ ✐❣✉❛✐s;
AP5 : ❙❡ ③❡r♦ ♣♦ss✉✐r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ P✱ ❡ s❡ ❞♦ ❢❛t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❛❧✲
q✉❡r ♥ ♣♦ss✉✐r P✱ ✐st♦ ❛❝❛rr❡t❛ q✉❡ ♥✬✭s✉❝❡ss♦r ❞❡ ♥✮ t❛♠❜é♠ ❛ ♣♦ss✉✐✱ ❡♥tã♦ t♦❞♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ P.
❊st❡ ú❧t✐♠♦ ❛①✐♦♠❛ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞♦ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✲ P✳■✳❋✳ ♦✉ ❛①✐♦♠❛ ❞❛ ✐♥❞✉çã♦✳
❊st❡ ❛①✐♦♠❛ é ❛ ❜❛s❡ ❞❡ ✉♠ ❡✜❝✐❡♥t❡ ♠ét♦❞♦ ❞❡ ❞❡♠♦♥str❛çã♦ ❞❡ ♣r♦♣♦s✐çõ❡s r❡✲ ❢❡r❡♥t❡s ❛ ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭❞❡♠♦♥str❛çõ❡s ♣♦r ✐♥❞✉çã♦✱ ♦✉ ♣♦r r❡❝♦rrê♥❝✐❛✮✳ ❊♥✉♥✲ ❝✐❛❞♦ s♦❜ ❛ ❢♦r♠❛ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❡♠ ✈❡③ ❞❡ ❝♦♥❥✉♥t♦s✱ ❡❧❡ s❡ ❢♦r♠✉❧❛ ❛ss✐♠✿
❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦ ❈❛♣ít✉❧♦ ✶
❙❡❥❛ P(n)✉♠❛ ♣r♦♣r✐❡❞❛❞❡ r❡❧❛t✐✈❛ ❛♦ ♥ú♠❡r♦ ♥❛t✉r❛❧ ♥✳ ❙✉♣♦♥❤❛♠♦s q✉❡✿ ✭✐✮ P(1) é ✈á❧✐❞❛❀
✭✐✐✮ P❛r❛ t♦❞♦ n ∈ N ✱ ❛ ✈❛❧✐❞❡③ ❞❡ P(n) ✐♠♣❧✐❝❛ ❛ ✈❛❧✐❞❡③ ❞❡ P(n′)✱ ♦♥❞❡ n′ é ♦ s✉❝❡ss♦r ❞❡ n❀ ❊♥tã♦ P(n) é ✈á❧✐❞❛ ♣❛r❛ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ♥ú♠❡r♦ ♥❛t✉r❛❧
n✳
❊①❡♠♣❧♦✿ ❉❡♠♦♥str❛r q✉❡ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s í♠♣❛r❡s é
✐❣✉❛❧ ❛n2✳
❘❡s♦❧✉çã♦✿ ■♥❞✐❝❛r❡♠♦s ♣♦r Sn ❛ s♦♠❛ ♣r♦❝✉r❛❞❛
Sn = 1 + 3 + 5 +...+ (2n−1),
❡♥tã♦ ♣❡❧♦ P✳■✳❋✳ t❡♠♦s✿
✭✐✮ P❛r❛ n = 1✱ ❛ ❤✐♣ót❡s❡ é ✈á❧✐❞❛✱ ♣♦✐s S1 = 12 = 1✳
✭✐✐✮ ❙❡✱ ♣❛r❛ n ∈ N✱ P(n) ❢♦r ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ q✉❡r❡♠♦s ♣r♦✈❛r q✉❡ P(n + 1) t❛♠❜é♠ ♦ é✱ ♦✉ s❡❥❛✱ q✉❡✿
1 + 3 + 5 +...+ (2n−1) + (2n+ 1) = (n+ 1)2. (1)
▼❛s✱ ♥♦t❡ q✉❡✱ ♣♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡
1 + 3 + 5 +...+ (2n−1) =n2,
❧♦❣♦ ❡♠(1)✱ t❡♠♦s q✉❡
1 + 3 + 5 +...+ (2n−1) + (2n+ 1) =n2+ 2n+ 1 = (n+ 1)2.
P♦rt❛♥t♦✱ t❡♠♦s q✉❡P(n)é ✈❡r❞❛❞❡✐r♦ ✐♠♣❧✐❝❛ q✉❡P(n+1)t❛♠❜é♠ é ✈❡r❞❛❞❡✐r♦✳
▲♦❣♦✱ ❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s í♠♣❛r❡s é ✐❣✉❛❧ ❛ n2✳
✭■■✮ ❆①✐♦♠❛s ❞❡ ❊✉❝❧✐❞❡s✭❆❊✮✿
❆❊1 : P♦❞❡✲s❡ tr❛ç❛r ✉♠❛ ú♥✐❝❛ r❡t❛ ❧✐❣❛♥❞♦ q✉❛✐sq✉❡r ❞♦✐s ♣♦♥t♦s;
AE2 : P♦❞❡✲s❡ ❝♦♥t✐♥✉❛r ✭❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛✮ q✉❛❧q✉❡r r❡t❛ ✜♥✐t❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❡♠
❧✐♥❤❛ r❡t❛;
AE3 : P♦❞❡✲s❡ tr❛ç❛r ✉♠ ❝✐r❝✉❧♦ ❝♦♠ q✉❛❧q✉❡r ❝❡♥tr♦ ❡ ❝♦♠ q✉❛❧q✉❡r r❛✐♦;
AE4 : ❚♦❞♦s ♦s â♥❣✉❧♦s r❡t♦s sã♦ ✐❣✉❛✐s;
AE5 : P♦r ✉♠ ♣♦♥t♦ ❢♦r❛ ❞❡ ✉♠❛ r❡t❛ ♣♦❞❡✲s❡ tr❛ç❛r ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛r❛❧❡❧❛ ❛
r❡t❛ ❞❛❞❛.
❚❛✐s s✐st❡♠❛s sã♦ ❞❡ ❢✉♥❞❛♠❡♥t❛✐s ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛✱ ♣♦✐s r❡s✲ ♣❡❝t✐✈❛♠❡♥t❡✱ ❡❧❡s s❡r✈✐r❛♠ ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ❛r✐t♠ét✐❝❛ ❡❧❡♠❡♥t❛r ❡ ❞❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ ❊✉❝❧✐❞✐❛♥❛✳
❆①✐♦♠❛s ❡ ❙✐st❡♠❛ ❆①✐♦♠át✐❝♦ ❈❛♣ít✉❧♦ ✶
❉❡✜♥✐çã♦ ✶✳✶✳✹ ✭❚❡♦r❡♠❛✮✿ ➱ ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡ ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞❛ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❧ó❣✐❝❛ ❛ ♣❛rt✐r ❞❡ ✉♠ ❛①✐♦♠❛ ♦✉ ❞❡ ♦✉tr♦s t❡♦r❡♠❛s q✉❡ t❡♥❤❛♠ s✐❞♦ ♣r❡✲ ✈✐❛♠❡♥t❡ ❞❡♠♦♥str❛❞♦s✳
❖ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ ❛✜r♠❛çã♦ ❞❡ ✐♠♣♦rtâ♥❝✐❛✳ ❊①✐st❡♠ ❛✜r✲ ♠❛çõ❡s ❞❡ ♠❡♥♦r ♦r❞❡♠✱ ❝♦♠♦ ❧❡♠❛ ✭✉♠❛ ❛✜r♠❛çã♦ q✉❡ ♣❡rt❡♥❝❡ ❛ ✉♠ t❡♦r❡♠❛ ♠❛✐♦r✮✱ ♦ ❝♦r♦❧ár✐♦ ✭❛✜r♠❛çã♦ q✉❡ s❡❣✉❡ ❞❡ ❢♦r♠❛ ✐♠❡❞✐❛t❛ ❛♦ t❡♦r❡♠❛✮ ♦✉ ❛ ♣r♦✲ ♣♦s✐çã♦ ✭✉♠ r❡s✉❧t❛❞♦ q✉❡ ♥ã♦ s❡ ❡♥❝♦♥tr❛ ❛ss♦❝✐❛❞♦ ❛ ♥❡♥❤✉♠ t❡♦r❡♠❛ ❡s♣❡❝í✜❝♦✮✳ ❈♦♥✈é♠ ❞❡st❛❝❛r q✉❡✱ ❡♥q✉❛♥t♦ ❛ ❛✜r♠❛çã♦ ♥ã♦ ❢♦r ❞❡♠♦♥str❛❞❛✱ ♥ã♦ ♣❛ss❛ ❡♥tã♦ ❞❡ ✉♠❛ ❤✐♣ót❡s❡ ♦✉ ❞❡ ✉♠❛ ❝♦♥❥❡❝t✉r❛✳
❯♠ ❞♦s t❡♦r❡♠❛s ♠❛✐s ♣♦♣✉❧❛r❡s é ♦ q✉❡ ❝♦♥❤❡❝❡♠♦s ♣❡❧♦ ♥♦♠❡ ❞❡ ❚❡♦r❡♠❛ ❞❡ ❚❛❧❡s✱ s❡❣✉♥❞♦ ♦ q✉❛❧ ♥♦s ❞✐③ q✉❡✱ s❡ ❞✉❛s r❡t❛s sã♦ tr❛♥s✈❡rs❛✐s ❛ ✉♠ ❢❡✐①❡ ❞❡ r❡t❛s ♣❛r❛❧❡❧❛s✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ ❞♦✐s s❡❣♠❡♥t♦s q✉❛✐sq✉❡r ❞❡ ✉♠❛ ❞❡❧❛s é ✐❣✉❛❧ à r❛③ã♦ ❡♥tr❡ ♦s s❡❣♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❛ ♦✉tr❛✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ t❡♦r❡♠❛ t❡♠♦s q✉❡✱ s❡ tr❛ç❛r♠♦s ♥✉♠ tr✐â♥❣✉❧♦ ✉♠❛ ❧✐♥❤❛ q✉❡ s❡❥❛ ♣❛r❛❧❡❧❛ ❛ ❛❧❣✉♥s ❞❡ s❡✉s ❧❛❞♦s ✐♥t❡rs❡❝t❛♥❞♦ ♦s ♦✉tr♦s ❞♦✐s✱ ♦❜t❡♠♦s ❞♦✐s tr✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s ✭✐st♦ é✱ ❞✉❛s ✜❣✉r❛s ❝♦♠ â♥❣✉❧♦s ✐♥t❡r♥♦s ✐❞ê♥t✐❝♦s ❡ ❧❛❞♦s ♣r♦♣♦r❝✐♦♥❛✐s✮✳
❖✉tr♦ t❡♦r❡♠❛ ✐❣✉❛❧♠❡♥t❡ ♣♦♣✉❧❛r é ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ ♦ q✉❛❧ ❞❡❢❡♥❞❡ q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❝♦♥str✉í❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ é ✐❣✉❛❧ à s♦♠❛ ❞❛s ár❡❛s ❞♦s q✉❛✲ ❞r❛❞♦s ❝♦♥str✉í❞♦s s♦❜r❡ ♦s ❝❛t❡t♦s✳ ❈♦♠ ✐ss♦ t❡♠♦s q✉❡✱ ♥✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ♦ q✉❛❞r❛❞♦ ❞❛ ❤✐♣♦t❡♥✉s❛ ✭♦✉ s❡❥❛✱ ♦ ❧❛❞♦ ❞❡ ♠❛✐♦r ❝♦♠♣r✐♠❡♥t♦ ❡ q✉❡ s❡ ♦♣õ❡ ❛♦ â♥❣✉❧♦ r❡t♦✮ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛❞r❛❞♦s ❞♦s ❝❛t❡t♦s ✭✐st♦ é✱ ♦s ❞♦✐s ❧❛❞♦s ♠❡♥♦r❡s ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✮✳
❈❛♥t♦r✱ ❡♠ s✉❛ ♦❜r❛✱ ❝♦♥❝❡♥tr❛✈❛✲s❡ ♥♦ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ❞❡ ❝♦♥❥✉♥t♦s✱ ❡st❡ ❝♦♥t✐✲ ♥✉❛✈❛ ❛ s❡r ❞❡✜♥✐❞♦ ❛ ✉♠ ♥í✈❡❧ ♠❡r❛♠❡♥t❡ ✐♥t✉✐t✐✈♦✳ P❛r❛ ❡❧❡✱ ✉♠ ❝♦♥❥✉♥t♦ ♣♦❞❡r✐❛ s❡r ✐❞❡♥t✐✜❝❛❞♦✱ q✉❡r ❞❡s✐❣♥❛♥❞♦ ♦s s❡✉s ❡❧❡♠❡♥t♦s ✭❡①t❡♥sã♦✮✱ q✉❡r ✐♥❞✐❝❛♥❞♦ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♦s ❝❛r❛❝t❡r✐③❛ss❡ ✭❝♦♠♣r❡❡♥sã♦✮✱ ❡ ❡st❛ ❝♦♥❝❡♣çã♦ ✐♥❢♦r♠❛❧ ❝♦♥❞✉✲ ③✐❛ ❛♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ✈ár✐♦s ♣❛r❛❞♦①♦s✱ ❝♦♠♦ ♦ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧ ✲ q✉❡ ✈❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡✳ ❈❛♥t♦r ❡st❛✈❛ ❜❡♠ ❝♦♥s❝✐❡♥t❡ ❞❡st❛s ❞✐✜❝✉❧❞❛❞❡s✱ r❡❝♦♥❤❡❝❡♥❞♦ q✉❡ ♥ã♦ ♣♦❞❡r✐❛ ❡①✐st✐r ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✱ ❡ ❛❞♠✐t✐❛ q✉❡ ❤á ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❞❡t❡r♠✐♥❛♠ ❝♦♥❥✉♥t♦s ❡ ♦✉tr❛s ♥ã♦✱ s❡♠ q✉❡ t✐✈❡ss❡ ❛♣♦♥t❛❞♦ ✉♠ ❝r✐tér✐♦ ❜❡♠ ❞❡✜♥✐❞♦ ♣❛r❛ ❞❡❝✐❞✐r s♦❜r❡ ✐ss♦✳
❚♦r♥❛✈❛✲s❡ ❡✈✐❞❡♥t❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ tr❛t❛r ❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❞❡ ✉♠❛ ❢♦r♠❛ r✐❣♦r♦s❛ ❡ ❛ ❚❡♦r✐❛ ❞❡ ❩❡r♠❡❧♦✲❋r❛❡♥❦❡❧ ✈❡♠ ♣r❡❝✐s❛♠❡♥t❡ ❞❛r r❡s♣♦st❛ ❛ ❡st❡ ♣r♦✲ ❜❧❡♠❛✱ r❡❣✉❧❛♠❡♥t❛♥❞♦ ❛①✐♦♠❛t✐❝❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❥✉♥t♦✳ ❊ss❡ s✐st❡♠❛ ❛①✐✲ ♦♠át✐❝♦ ♣r❡t❡♥❞❡ ✐♥❝♦r♣♦r❛r ♦ ❝♦♥❝❡✐t♦ ✐♥t✉✐t✐✈♦ ❞❡ ❝♦♥❥✉♥t♦s q✉❡ ♦s ♠❛t❡♠át✐❝♦s✱ ❞❡ ❢❛t♦✱ ✉s❛♠✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❖s ❛①✐♦♠❛s ❞❡ ❩❡r♠❡❧♦✲❋r❛❡♥❦❡❧ ♣♦❞❡♠ s❡r ❡①♣r❡ss♦s ❝♦♠♦ ❢ór♠✉❧❛s ▲ó❣✐❝❛s ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s✳ ❘❡♣r❡s❡♥t❛✲s❡ ✉s✉❛❧♠❡♥t❡ ♣♦r ❩❋❈ ♦ ❝♦♥❥✉♥t♦ ❞❡st❡s ❛①✐♦✲ ♠❛s✱ s❡♥❞♦ q✉❡ ❛ ❧❡tr❛ ❈ ✭❞♦ ✐♥❣❧ês ❝❤♦✐❝❡✮ ♥❡st❛ ❛❜r❡✈✐❛t✉r❛ s❡ r❡❢❡r❡ à ✐♥❝❧✉sã♦ ❞♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛❀ ♥❛t✉r❛❧♠❡♥t❡✱ ❩❋ r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ❡ss❡s ❛①✐♦♠❛s ❡①❝❡t♦ ❡st❡ ú❧t✐♠♦✳ ◗✉❛♥❞♦ ✐♥t❡r♣r❡t❛❞♦s ♥✉♠❛ ❡str✉t✉r❛ ❞❡st❛ ▲ó❣✐❝❛✱ ♦s ❛①✐♦♠❛s tr❛❞✉③❡♠ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ✉♥✐✈❡rs♦ ❝♦rr❡s♣♦♥❞❡♥t❡✱ ❛q✉✐ ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ✉♥✐✲ ✈❡rs♦ ❞❡ ❝♦♥❥✉♥t♦s✳ ●❡♥❡r✐❝❛♠❡♥t❡✱ ✉♠ ❝♦♥❥✉♥t♦ s❡rá ❡♥tã♦ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡st❛ ❧✐♥❣✉❛❣❡♠ q✉❡ s❛t✐s❢❛ç❛ ♦s ❛①✐♦♠❛s ❞❡ ❩❋❈✳
✶✳✷ ❖s ❆①✐♦♠❛s ❞❡ ❩❋❈
❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❣❛r❛♥t✐r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❡st❡❥❛ s❡♠♣r❡ ✉♥✐✈♦❝❛♠❡♥t❡ ❞❡✲ t❡r♠✐♥❛❞♦✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ❛❧❣✉♥s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❡♠ ♥♦ss❛ t❡♦r✐❛✳
❆①✐♦♠❛ ❞❛ ❈♦♠♣r❡❡♥sã♦ ♦✉ ❆①✐♦♠❛ ❞❡ ❡s♣❡❝✐✜❝❛çã♦ ❞✐③ q✉❡ s❡ ✉♠ ❝♦♥✲ ❥✉♥t♦ A ❡①✐st❡ ❡ ❝♦♥s❡❣✉✐♠♦s ❞❡s❝r❡✈❡r ✭❛tr❛✈és ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡✮ ❡❧❡♠❡♥t♦s
❞❡st❡ ❝♦♥❥✉♥t♦✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ B✱ s✉❜❝♦♥❥✉♥t♦ ❞❡ A✱ q✉❡ ❝♦♥té♠ ❡ss❡s
❡❧❡♠❡♥t♦s✳
❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡ ❆ é ✉♠ ❝♦♥❥✉♥t♦ ❡ ① é ✉♠ ❡❧❡♠❡♥t♦ q✉❡ ♣❡rt❡♥❝❡ ❛ ❡ss❡ ❝♦♥✲ ❥✉♥t♦✱ ❡♥tã♦ ❡s❝r❡✈❡♠♦sx∈A ❡ ❞✐③❡♠♦s q✉❡ ① ♣❡rt❡♥❝❡ ❛ ❆✳ ❙❡ ❆ é ✉♠ ❝♦♥❥✉♥t♦ ❡
① ✉♠ ❡❧❡♠❡♥t♦ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ ❡ss❡ ❝♦♥❥✉♥t♦✱ ❡♥tã♦ ❡s❝r❡✈❡♠♦sx6∈A ❡ ❞✐③❡♠♦s
q✉❡ ① ♥ã♦ ♣❡rt❡♥❝❡ ❛ ❆✳
❖ ❝♦♥❝❡✐t♦ ❞❡ ♣❡rt✐♥ê♥❝✐❛ ❞❛❞♦ ♣♦r x ∈A ♦✉ x 6∈A é ✉♠ ❝♦♥❝❡✐t♦ ♣r✐♠✐t✐✈♦ ❡✱
❛✐♥❞❛✱ ♦ ♣r✐♥❝✐♣❛❧ ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✳
✶ ✲ ❆①✐♦♠❛ ❞❛ ❊①t❡♥sã♦✿
❙❡❥❛♠ A ❡ B ❝♦♥❥✉♥t♦s✱ t❡♠♦s q✉❡ ❡st❡s ❝♦♥❥✉♥t♦s sã♦ ✐❣✉❛✐s✱ s❡ ❡ s♦♠❡♥t❡ s❡✱
t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ A ❢♦r❡♠ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s ❞❡B✳
❊ss❡ ❛①✐♦♠❛✱ ♠✉✐t♦ ❝♦♥❤❡❝✐❞♦✱ ❛✜r♠❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧❛ s✉❛ ❡①t❡♥sã♦✱ ♣❡❧♦ s❡✉ t❛♠❛♥❤♦✱ ✐st♦ é✱ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦s s❡✉s ♠❡♠❜r♦s✳ ❚❛❧ ❛①✐♦♠❛ r❡✢❡t❡ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ❞♦✐s ❝♦♥❥✉♥t♦s sã♦ ✐❣✉❛✐s s❡ t❡♠ ❛ ♠❡s♠❛ ❡①t❡♥sã♦✱ ✐st♦ é✱ s❡ ♣♦ss✉❡♠ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❊①❡♠♣❧♦✿
❆✿ é ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ x2−5x+ 6 = 0❀
❇✿ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞♦✐s ♣r✐♠❡✐r♦s ♥❛t✉r❛✐s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡ ♣r✐♠♦s✳
❈♦♠♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s sã♦ ❡①❛t❛♠❡♥t❡ ✷ ❡ ✸✱ ❡♥tã♦ ♦s ❝♦♥❥✉♥✲ t♦sA ❡ B ❝♦✐♥❝✐❞❡♠✱ ♦✉ s❡❥❛ A=B✳
❆ ♣❛rt✐r ❞♦ ❛①✐♦♠❛ ❞❛ ❡①t❡♥sã♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♦♣❡r❛çã♦ ❞❡ ✐♥❝❧✉sã♦ ❡♥tr❡ ❝♦♥❥✉♥t♦s ❝♦♠♦ s❡❣✉❡✿
❉❡✜♥✐çã♦ ✶✳✷✳✷ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ ❞✐③❡♠♦s q✉❡ A ❡stá ❝♦♥t✐❞♦ ❡♠ B✱
r❡♣r❡s❡♥t❛❞♦ ♣♦rA⊆B✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡At❛♠❜é♠ é ✉♠ ❡❧❡♠❡♥t♦
❞❡ B✳
❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ t❡♠♦s q✉❡✿ A⊆B ⇐⇒ ∀x✱ s❡ x∈A✱ ❡♥tã♦ x∈B✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ A ={3,4,5,6,7} ❡ B ={3,4,5}✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ B ⊂ A✱
✉♠❛ ✈❡③ q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ B ❡stã♦ ❝♦♥t✐❞♦s ❡♠ A✳
◆❡st❡ ❝❛s♦✱ s❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ A ❡stã♦ ❝♦♥t✐❞♦s ❡♠ B✱ ❞✐③❡♠♦s q✉❡ ❆ é
✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ❇✱ ♦✉ q✉❡ ❇ ❝♦♥té♠ ❆✳
❆ r❡❧❛çã♦ ❞❡ ✐♥❝❧✉sã♦ é ❞❡ ♠✉✐t❛ ✐♠♣♦rtâ♥❝✐❛✱ ♣♦✐s q✉❛s❡ t♦❞❛s ❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❆ ❡ ❇✱ ♣♦❞❡♠ s❡r s❡♣❛r❛❞❛s ❡♠ ❞✉❛s ♣❛rt❡s✿ ♣r✐♠❡✐r♦ ♠♦str❛♠♦s q✉❡ A⊆B ❡✱ ❛ s❡❣✉✐r✱ q✉❡B ⊆A✳
❖❜s❡r✈❛çã♦ ✶✳✷✳✶ ✿ ❈♦♥✈é♠ ❧❡♠❜r❛r q✉❡ ❛s r❡❧❛çõ❡s ❞❡ ♣❡rt✐♥ê♥❝✐❛∈❡ ❞❡ ✐♥❝❧✉sã♦ ⊆ sã♦ ❝♦♥❝❡✐t✉❛❧♠❡♥t❡ ❞✐❢❡r❡♥t❡s✱ ✉♠❛ ✈❡③ q✉❡ ❛s r❡❧❛çõ❡s ❞❡ ♣❡rt✐♥ê♥❝✐❛ r❡❢❡r❡ ✲ s❡ ❛ ❡❧❡♠❡♥t♦s ❡ ❝♦♥❥✉♥t♦s✱ ❡ ❛s ❞❡ ✐♥❝❧✉sã♦ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❡♥tr❡ ❝♦♥❥✉♥t♦s✳
❖❜s❡r✈❛çã♦ ✶✳✷✳✷ ❈♦♠ ❡st❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥❝❧✉sã♦✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♠♦✿ ❉❛❞♦s A ❡ B ❝♦♥❥✉♥t♦s✱ t❡♠♦s q✉❡ A = B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A⊆B ❡ B ⊆A✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ A=B ⇐⇒A⊆B ❡ B ⊆A✳
❆♦ ❛✜r♠❛r q✉❡ ❞♦✐s ❝♦♥❥✉♥t♦s sã♦ ✐❣✉❛✐s q✉❛♥❞♦ tê♠ ❡①❛t❛♠❡♥t❡ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s✱ ❛ t❡♦r✐❛ ❢♦r♥❡❝❡ ♥❛t✉r❛❧♠❡♥t❡ ❛ r❡❧❛çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❝♦♥❥✉♥t♦s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❡s❝r❡✈❡♠♦s q✉❡A=B ♣❛r❛ ✐♥❞✐❝❛r q✉❡ x∈As❡✱ ❡ s♦♠❡♥t❡ s❡✱x∈B✳
❆ ✐❣✉❛❧❞❛❞❡ é s✐♠étr✐❝❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ s❡ A=B✱ ❡♥tã♦ B =A✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
✷ ✲ ❆①✐♦♠❛ ❞♦ ❈♦♥❥✉♥t♦ ❱❛③✐♦✿
❖ ❛①✐♦♠❛ ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ X ♣❛r❛ ♦ q✉❛❧ ♥ã♦ ❤á ❡❧❡♠❡♥t♦s ♣❡r✲
t❡♥❝❡♥t❡s✳ ❚❛❧ ❝♦♥❥✉♥t♦ é ❞❡♥♦♠✐♥❛❞♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ é ❞❡s✐❣♥❛❞♦ ♣♦r ∅.
❊st❡ ❝♦♥❥✉♥t♦✱ q✉❡ ♥ã♦ t❡♠ ❡❧❡♠❡♥t♦s✱ t❡♠ ❝♦♠♦ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♥ã♦ ♣♦❞❡ s❡r s❛t✐s❢❡✐t❛✱ ♣♦r ❡①❡♠♣❧♦s✿
✲ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s t❛✐s q✉❡ x2 =−4✳
✲ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣♦s✐t✐✈♦s ♠ú❧t✐♣❧♦s ❞❡ ✼ ♠❡♥♦r❡s q✉❡ ✺✳
✸ ✲ ❆①✐♦♠❛ ❞♦ P❛r ✭♥ã♦✲♦r❞❡♥❛❞♦✮✿
P❛r❛ q✉❛❧q✉❡ra♦✉b❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦Z✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r{a, b}✱ q✉❡ ❝♦♥té♠ ❡①❛t❛♠❡♥t❡a ❡b✳
❖ ❝♦♥❥✉♥t♦ (x, y) = {{x},{x, y}} é ✉♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ♣❛r ❞❡♥♦♠✐♥❛❞♦ ♣❛r ♦r❞❡♥❛❞♦✱ ♠❛s q✉❡ é ♣❛ssí✈❡❧ ❞❡ ❝♦♥str✉çã♦ ❛ ♣❛rt✐r ❞♦ ❛①✐♦♠❛ ❛❝✐♠❛✳
✹ ✲ ❆①✐♦♠❛ ❞♦ ❈♦♥❥✉♥t♦ ❯♥✐ã♦✿
P❛r❛ ❝❛❞❛X❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦Y =S
X✱ ♦✉ s❡❥❛✱ ❛ ✉♥✐ã♦ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s
❝♦♥st✐t✉✐♥t❡s ❞❡ X✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡✱ ❞❛❞♦s X ❡ Y✱ ❡①✐st❡ Z ❞❡ ♠♦❞♦ q✉❡ Z =X∪Y✱ ✐st♦ é✱t é ❡❧❡♠❡♥t♦ ❞❡ Z s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t∈X ♦✉t ∈Y✳
✺ ✲ ❆①✐♦♠❛ ❞♦ ■♥✜♥✐t♦✿
❊①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ Y q✉❡ ❝♦♥té♠ ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ∅✱ ❡ ♣❛r❛ ❝❛❞❛ X ∈ Y✱ ♦
❝♦♥❥✉♥t♦ {X} t❛♠❜é♠ ♣❡rt❡♥❝❡ ❛Y✳
❊ss❡ ❛①✐♦♠❛ ♣❡r♠✐t❡ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ♦♥❞❡ y ∪ {y} é ♦ s✉❝❡ss♦r ❞❡ y ❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ✐♥✐❝✐❛ ♣❡❧♦ ❝♦♥❥✉♥t♦ ∅✳
✻ ✲ ❆①✐♦♠❛ ❞❛ ❙✉❜st✐t✉✐çã♦✿
❙❡ ❛ r❡❧❛çã♦ ♦❜t✐❞❛ ❞❡P(x, y)é ✉♠❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ❡♠ x ❡y✱ ❡♥tã♦ ❞❛❞♦ ✉♠
❝♦♥❥✉♥t♦ B✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ A✱ ❝✉❥♦s ♦s ❡❧❡♠❡♥t♦s sã♦ ❛q✉❡❧❡s ❡❧❡♠❡♥t♦s ❞❡ B
q✉❡ s❛t✐s❢❛③❡♠ ❛ ❢ór♠✉❧❛ P(x, y)✱ ✐st♦ é✱A={z ∈B;P(x, z)}✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
✼ ✲ ❆①✐♦♠❛ ❞♦ ❈♦♥❥✉♥t♦ P♦tê♥❝✐❛✿
P❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝✉❥♦s ♠❡♠❜r♦s sã♦ ❡①❛t❛♠❡♥t❡ ♦s s✉❜✲ ❝♦♥❥✉♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❞❛❞♦✱ ♦✉ s❡❥❛✱ ❡st❡ ❛①✐♦♠❛ ♥♦s ❞✐③ q✉❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦y
♣❛r❛ ❝❛❞❛x❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡x✳ ❊♠❜♦r❛ ♦ ❝♦♥❥✉♥t♦ ❞❛ ♣♦tê♥✲
❝✐❛ y s❡❥❛ ❞❡✜♥✐❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ❡❧❡ ♥ã♦ ❡stá ✐♥❝❧✉í❞♦ ♥♦ ❆①✐♦♠❛ ❞❛
❙✉❜st✐t✉✐çã♦ ♣♦rq✉❡ y♥ã♦ é ❞❛❞♦ ❝♦♠♦ ❛♠♣❧✐t✉❞❡ ❞❡ q✉❛❧q✉❡r ❢✉♥çã♦✳ ❆❧é♠ ❞✐ss♦✱
❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ ♣♦tê♥❝✐❛ ❞❡ x s❡rá s❡♠♣r❡ s✉♣❡r✐♦r à ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦
❝♦♥❥✉♥t♦ x✳
✽ ✲ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✿
∀X,∃φ, ♦♥❞❡φ é ✉♠❛ ❢✉♥çã♦ t❛❧ q✉❡Dom(φ) =x− {∅} ❡
∀y(y∈Dom(φ))→φ(y)∈y, ♦♥❞❡ x− {∅} r❡♣r❡s❡♥t❛ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ♥ã♦✲✈❛③✐♦s ❞❡X✳
❊st❡ é ♦ ❢❛♠♦s♦ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ q✉❡ ✈❡r❡♠♦s ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡ ♣♦st❡r✐♦r✲ ♠❡♥t❡ ❡ q✉❡ é ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡♠♦♥str❛r♠♦s ♦ ▲❡♠❛ ❞❡ ❩♦r♥✱ ❡❧❡ ❛✜r♠❛ q✉❡ s❡♠♣r❡ ♣♦❞❡♠♦s ❡❢❡t✉❛r ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡ ❡s❝♦❧❤❛s ♠❡s♠♦ s❡♠ t❡r♠♦s q✉❛❧q✉❡r ♣r♦♣r✐❡❞❛❞❡ q✉❡ ❞❡✜♥❛ ❛ ❢✉♥çã♦ ❞❡ ❡s❝♦❧❤❛✳
❖ ❛①✐♦♠❛ ❞❛ ❡s❝♦❧❤❛ t❡♠ ♠✉✐t❛s ❢♦r♠❛s ❡q✉✐✈❛❧❡♥t❡s✱ ♥♦r♠❛❧♠❡♥t❡ tr❛t❛❞❛s ♥♦s t❡①t♦s ❞❡ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✱ ♣♦r ❡①❡♠♣❧♦✿
❖ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ♥ã♦ ✈❛③✐❛ ❞❡ ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s é✱ ❛✐♥❞❛✱ ♥ã♦ ✈❛③✐♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❞❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ í♥❞✐❝❡sI ❡ ✉♠❛ ❢✉♥çã♦ φ ❝♦♠
❞♦♠í♥✐♦ ❡♠I✱ s❡✱ ♣❛r❛ t♦❞♦i∈I, φ(i)6=∅✱ ❡♥tã♦ Q
i∈I φ(i)6=∅✳
✾ ✲ ❆①✐♦♠❛ ❞❛ ❘❡❣✉❧❛r✐❞❛❞❡✿
❊st❡ ú❧t✐♠♦ ❛①✐♦♠❛ ❣❛r❛♥t❡ q✉❡ ❝❛❞❛ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦x ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦
♠✐♥✐♠❛❧ ❝♦♠ r❡s♣❡✐t♦ à r❡❧❛çã♦∈✳ ❆ ✐❞❡✐❛ ♣♦r trás ❞♦ ❛①✐♦♠❛ ❝♦♥s✐st❡ ♥♦ ❞❡s❡❥♦ ❞❡ q✉❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s s❡❥❛♠ ❝♦♥str✉í❞♦s ❛ ♣❛rt✐r ❞♦ ❝♦♥❥✉♥t♦ ∅✱ ❡✈✐t❛♥❞♦ ❛ ♦❝♦r✲ rê♥❝✐❛ ❞❡ ❝❛❞❡✐❛s ❞❡s❝❡♥❞❡♥t❡s ❡ ✐♥✜♥✐t❛s ❝♦♠ r❡❧❛çã♦ à ♣❡rt✐♥ê♥❝✐❛ ∈✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ✉♠ ❝♦♥❥✉♥t♦ é ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♦❜❥❡t♦s ❡ ❡ss❡s ♦❜❥❡t♦s sã♦ ❞❡♥♦♠✐♥❛❞♦s ❞❡ ♠❡♠❜r♦s ♦✉ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✳
▼✉♥✐❞♦s ❞♦s ❆①✐♦♠❛s ❞♦ ❈♦♥❥✉♥t♦ ❱❛③✐♦ ❡ ❞❛ ❊①t❡♥sã♦✱ ♣♦❞❡♠♦s ❞❡♠♦♥str❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ q✉❡ ♥ã♦ t❡♠ ❡❧❡♠❡♥t♦s é ú♥✐❝♦✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ✿ ❊①✐st❡ ❛♣❡♥❛s ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ♥ã♦ t❡♠ ❡❧❡♠❡♥t♦s ❡ ❛ ❡st❡ ❝♦♥❥✉♥t♦ ❞á✲s❡ ♦ ♥♦♠❡ ❞❡ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ é ❞❡♥♦t❛❞♦ ♣♦r ∅✳
Pr♦✈❛✳ ❙❡❥❛♠A❡B❞♦✐s ❝♦♥❥✉♥t♦s s❡♠ ❡❧❡♠❡♥t♦s✳ ❙❡A❡Bsã♦ ❝♦♥❥✉♥t♦s ❞✐st✐♥t♦s✱
❡♥tã♦ ❡❧❡s ♥ã♦ ♣♦ss✉❡♠ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s✱ ♦✉ s❡❥❛✱ ❡①✐st❡x∈At❛❧ q✉❡x6∈B✱♦✉
❡①✐st❡ x∈B t❛❧ q✉❡x6∈A✳ ◆♦s ❞♦✐s ❝❛s♦s✱ t❡♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s A ❡B ♥ã♦
♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s✳ ❆ss✐♠✱A ❡ B ♥ã♦ ♣♦❞❡♠ s❡r ❝♦♥❥✉♥t♦s ❞✐st✐♥t♦s✳
✶✳✸ ❖♣❡r❛çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s
✶✳ ❯♥✐ã♦ ✭❘❡✉♥✐ã♦✮✿❉❛❞♦s ♦s ❝♦♥❥✉♥t♦sA❡B✱ ❡♥tã♦ ♣❡❧♦ ❛①✐♦♠❛ ❞❛ ✉♥✐ã♦✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦C t❛❧
q✉❡x∈Cs❡✱ ❡ s♦♠❡♥t❡ s❡✱x∈A♦✉x∈B✳ ❉❡♥♦t❛♠♦s t❛❧ ❝♦♥❥✉♥t♦ ♣♦rC =A∪B✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠A={1,2,3}❡B ={4,5,6}✱ ❡♥tã♦ t❡♠♦s q✉❡C ={1,2,3,4,5,6} é ♦ ❝♦♥❥✉♥t♦ ✉♥✐ã♦ ❞❡A ❡ B✱ ✐st♦ é C =A∪B✳
✷✳ ❉✐❢❡r❡♥ç❛ ❡ ■♥t❡rs❡çã♦✿
Pr♦♣♦s✐çã♦ ✶✳✸✳✶ ✿ ❙❡ ❆ ❡ ❇ sã♦ ❝♦♥❥✉♥t♦s✳
✭✐✮ ❊①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❈ t❛❧ q✉❡ x∈C s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x∈A ❡ x6∈B✳
✭✐✐✮ ❊①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❉ t❛❧ q✉❡ x∈D s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x∈A ❡ x∈B✳
Pr♦✈❛✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ R(x, B) ❞❡ x ❡ B ❝♦♠ ♦ s✐❣♥✐✜❝❛❞♦ ✬x 6∈ B✬✳
P❡❧♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣r❡❡♥sã♦✱ ♣❛r❛ t♦❞♦B ❡ ♣❛r❛ t♦❞♦ A✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ Dt❛❧
q✉❡ x ∈ D s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x ∈ A ❡ R(x, B)✱ ♦✉ s❡❥❛✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x ∈ A ❡ x6∈b ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ✭✐✮ ❡stá ♣r♦✈❛❞❛✳
✭✐✐✮ P❡❧♦ ❆①✐♦♠❛ ❞❛ ❯♥✐ã♦ ❡ ♣❡❧♦ ❆①✐♦♠❛ ❞❛ ❊s♣❡❝✐✜❝❛çã♦✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦
A∩B ={x ∈A∪Bx∈A ❡ x∈ B}✱ ❡ s❡rá ú♥✐❝♦ ♣❡❧♦ ❆①✐♦♠❛ ❞❛ ❊①t❡♥sã♦✳ ❙❡ t❡♠♦s ♦s ❝♦♥❥✉♥t♦sA✱B ❡C✱ ❜❛st❛ ♣r✐♠❡✐r♦ ❝♦♥s✐❞❡r❛r ♦ ❝♦♥❥✉♥t♦ A∩B q✉❡ t❡♠
s✉❛ ❡①✐stê♥❝✐❛ ❣❛r❛♥t✐❞❛ ♣❡❧❛ ♣❛rt❡ ❛♥t❡r✐♦r✱ ❡ ❝♦♠♦ ❡❧❡ é ✉♠ ❝♦♥❥✉♥t♦✱ ❜❛st❛ ❢❛③❡r ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A∩B ❡ C ♣❛r❛ ♦❜t❡r ♦ ❝♦♥❥✉♥t♦ A∩B ∩C✳ ❙❡♥❞♦
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❛ss✐♠✱ t❡♠♦s q✉❡ ♣❛r❛ ❞❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ A∩B ={x∈A∪Bx∈ A ❡ x∈B ={x∈Ax∈B}✳
❙❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r q✉❡ ♦s ❝♦♥❥✉♥t♦sC ❡Dsã♦ ú♥✐❝♦s ❡ sã♦ ❝❤❛♠❛❞♦s✱
r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ❞✐❢❡r❡♥ç❛ ❡ ✐♥t❡rs❡❝çã♦ ❞❡A❡B✱ ❡ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦rA−B
❡A∩B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❖❜s❡r✈❛çã♦ ✶✳✸✳✶ P♦❞❡ ♦❝♦rr❡r q✉❡ ♥ã♦ ❡①✐st❛ ❡❧❡♠❡♥t♦ ❛❧❣✉♠ x t❛❧ q✉❡ x∈ A ❡ x∈B✳ ◆❡st❡ ❝❛s♦✱ t❡♠✲s❡ q✉❡ A∩B =∅ ❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B ❞✐③❡♠✲s❡ ❞✐s❥✉♥t♦s✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠A ={x∈ N;x≤10} ❡ B ={x∈N;x > 5}✳ ❊♥tã♦ A∪B =N ❡A∩B ={6,7,8,9,10}✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ A = {x ∈ N;x > 2} ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♠❛✐♦r❡s ❞♦ q✉❡ ✷ ❡B ={x∈N;x <3} ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♠❡♥♦r❡s ❞♦ q✉❡ ✸✳ ❊♥tã♦A∩B ={∅}✱ ♣♦✐s ♥ã♦ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐sxt❛✐s q✉❡ 2< x <3✳ ❆ss✐♠ ♦s ❝♦♥❥✉♥t♦sA ❡ B sã♦ ❞✐s❥✉♥t♦s✳
❯♠❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛s ♦♣❡r❛çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s é q✉❡✿
n(A∪B) = n(A) +n(B)−n(A∩B)
❊①❡♠♣❧♦✿ ❙❡❥❛♠ A ={1,3,5,7,9} ❡ B ={2,3,4,5,6,8} t❡♠♦s q✉❡✿ (A∪B) = {1,2,3,4,5,6,7,8,9}✱ ♣♦ré♠ ♥♦t❡ q✉❡✱ n(A) = 5✱ n(B) = 6✱ n(A ∩ B) = 2 ❡
n(A∪B) = 9✱ ♦ q✉❡ s❛t✐s❢❛③ ❛ r❡❧❛çã♦ ❛♥t❡r✐♦r✳
✸✳ ❈♦♠♣❧❡♠❡♥t❛r ❞❡ B ❡♠ A✿
❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✱ t❛✐s q✉❡ B ⊂ A✱ ❝❤❛♠❛✲s❡ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ B
❡♠ r❡❧❛çã♦ ❛ A ♦ ❝♦♥❥✉♥t♦ A−B✱ ✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ A q✉❡ ♥ã♦
♣❡rt❡♥❝❡♠ ❛ B✳ ❯t✐❧✐③❛♠♦s ♦ sí♠❜♦❧♦ CB
A ♣❛r❛ ✐♥❞✐❝❛r ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ B ❡♠
r❡❧❛çã♦ ❛ A✳ ◆♦t❡♠♦s q✉❡ CB
A só é ❞❡✜♥✐❞♦ ♣❛r❛B ⊂A✱ ❡ ❛ss✐♠ t❡♠♦s✿
CB
A =A−B✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ A= {2,3,5,6,7,8,9} ❡ B ={3,5,6}✱ ♥♦t❡ q✉❡ B ⊂ A ❡ ❛ss✐♠ CB
A ❡stá ❞❡✜♥✐❞❛ ❡ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ♣♦r✿ CAB =A−B ={2,7,8,9}✳
P❡❧♦ ❛①✐♦♠❛ ❞♦ ❝♦♥❥✉♥t♦ ♣♦tê♥❝✐❛✱ t❡♠♦s q✉❡ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦y✱ ❡①✐st❡ ♦✉tr♦
❝♦♥❥✉♥t♦✱ ❝✉❥♦s ♠❡♠❜r♦s sã♦ ❡①❛t❛♠❡♥t❡ ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ y✳
❉❡✜♥✐çã♦ ✶✳✸✳✶ ❉❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❆✱ ❡♥tã♦ ❝❤❛♠❛♠♦s ❞❡ ❈♦♥❥✉♥t♦ ❞❛s P❛r✲ t❡s✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ ❆ ❡ ♦ ✐♥❞✐❝❛♠♦s ♣♦r ℘(A)✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❊①❡♠♣❧♦✿ ❉❛❞♦ ♦ ❝♦♥❥✉♥t♦A={2,3,4}✱ ❡♥tã♦ t❡♠♦s q✉❡✿
℘(A) ={∅,{2},{3},{4},{2,3},{2,4},{3,4},{2,3,4}}✳
P❡r❝❡❜❛ q✉❡ ♦ ❝♦♥❥✉♥t♦A ♣♦ss✉✐ ✸ ❡❧❡♠❡♥t♦s ❡ ♦ ❝♦♥❥✉♥t♦(A)❝♦♥té♠ ✽ ❡❧❡♠❡♥✲ t♦s✳ ❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ t❡♠♦s q✉❡✱ s❡ ✉♠ ❝♦♥❥✉♥t♦ A é ✜♥✐t♦✱ ❡♥tã♦ é ♣♦ssí✈❡❧
❞❡t❡r♠✐♥❛r ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ ℘(A) ❡ s❡✉ ✈❛❧♦r é ❞❛❞♦ ♣♦r2n✱ ♦♥❞❡ ♥ é ♦
♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦A✳
❈♦♠ ❡ss❡s ❡❧❡♠❡♥t♦s t❡ór✐❝♦s ♣♦❞❡♠♦s ❡✈✐t❛r ♦ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧❧✳
❖ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧❧✿
P❛r❛❞♦①♦ é ✉♠❛ ♣❛❧❛✈r❛ q✉❡ é ✉s❛❞❛ ♣❛r❛ s✐❣♥✐✜❝❛r ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❛♣❡♥❛s ❛♣❛r❡♥t❡✱ q✉❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛✳ ▼❛s✱ às ✈❡③❡s✱ t❡♠ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ❝♦♥tr❛❞✐çã♦ ✈❡r❞❛❞❡✐r❛ ❡ ✐♥s♦❧ú✈❡❧✳
❉❡♥tr❡ ♦s ♠✉✐t♦s ♣❛r❛❞♦①♦s q✉❡ ❢♦r❛♠ s❡♥❞♦ ❞❡s❝♦❜❡rt♦s✱ ♠❡r❡❝❡ ❡s♣❡❝✐❛❧ ❛t❡♥✲ çã♦ ♦ ❝❤❛♠❛❞♦ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧❧✱ q✉❡ ❡stá ❝♦♥t✐❞♦ ♥✉♠❛ ❝❛rt❛ q✉❡ ❇❡rtr❛♥❞ ❘✉ss❡❧❧ ✭✶✽✼✷✲✶✾✼✵✮ ❡s❝r❡✈❡✉ ❛ ●♦tt❧♦❜ ❋r❡❣❡ ✭✶✽✹✽✲✶✾✷✺✮ ❡♠ ✶✾✵✷✳ ❋r❡❣❡ r❡❝❡❜❡✉ ❛ ❝❛rt❛ ❞❡ ❘✉ss❡❧❧ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❡st❛✈❛ ♣❛r❛ ♣✉❜❧✐❝❛r ♦ s❡❣✉♥❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♦❜r❛ ❡♠ q✉❡ ❢✉♥❞❛♠❡♥t❛✈❛ t♦❞❛ ❛ ❛r✐t♠ét✐❝❛ ♥❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s✳
❊❧❡ r❡❛❣✐✉ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣❛❧❛✈r❛s✿ ✏◆❛❞❛ ♠❛✐s ✐♥❞❡s❡❥á✈❡❧ ♣❛r❛ ✉♠ ❝✐❡♥t✐st❛ ❞♦ q✉❡ ✈❡r r✉✐r ♦s ❢✉♥❞❛♠❡♥t♦s ❞♦ ❡❞✐❢í❝✐♦✱ ❥✉st❛♠❡♥t❡ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❡❧❡ ❡stá s❡♥❞♦ ❝♦♥❝❧✉í❞♦✳ ❋♦✐ ♥❡ss❛ ✐♥❝ô♠♦❞❛ s✐t✉❛çã♦ q✉❡ ♠❡ ❡♥❝♦♥tr❡✐ ❛♦ r❡❝❡❜❡r ✉♠❛ ❝❛rt❛ ❞♦ ❙r✳ ❇❡rtr❛♥❞ ❘✉ss❡❧❧ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ♠❡✉ tr❛❜❛❧❤♦ ❥á ❡st❛✈❛ ✐♥❞♦ ♣❛r❛ ♦ ♣r❡❧♦ ✑✳
❖s ❛①✐♦♠❛s ❞❡ ❈❛♥t♦r✱ ❡♥tr❡ ❡❧❡s ♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣r❡❡♥sã♦✱ ❡r❛♠ ♦s ❛①✐♦♠❛s ✉t✐❧✐③❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ❛♦s ❛①✐♦♠❛s ❞❡ ❩❡r♠❡❧♦✳ ❊ ❡st❡ s✐st❡♠❛ ❛①✐♦♠át✐❝♦ q✉❡ ❣❡r♦✉ ♦ P❛r❛❞♦①♦✳
✭P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧❧✮✿ ❊①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✳
P❛r❛ ❡①♣❧✐❝❛r ♦ ♣❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧❧✱ ❝♦♠❡ç❛♠♦s ♦❜s❡r✈❛♥❞♦ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r ❡❧❡♠❡♥t♦ ❞❡ ♦✉tr♦ ❝♦♥❥✉♥t♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛rt❡s ❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦❀ ✉♠❛ r❡t❛ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s❀ ❡ ♣♦❞❡♠♦s ❢♦r♠❛r ♦ ❝♦♥❥✉♥t♦ ❞❛s r❡t❛s ❞❡ ✉♠ ❞❛❞♦ ♣❧❛♥♦✱ ♣♦rt❛♥t♦✱ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❥✉♥t♦s✳ ❯♠ ❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r ❡❧❡♠❡♥t♦ ❞❡ s✐ ♠❡s♠♦✱ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✐❞❡✐❛s ❛❜str❛t❛s✱ ♣♦✐s t❛❧ ❝♦♥❥✉♥t♦ t❛♠❜é♠ é ✉♠❛ ✐❞❡✐❛ ❛❜str❛t❛❀ ♣♦rt❛♥t♦✱ ❡❧❡ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ s✐ ♠❡s♠♦✳
❖s ❆①✐♦♠❛s ❞❡ ❩❋❈ ❈❛♣ít✉❧♦ ✶
❖✉tr♦ ❡①❡♠♣❧♦✿ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝♦♥❥✉♥t♦s q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ s✐ ♠❡s♠♦✱ ♣♦✐s ❡❧❡✱ ❝♦♠ ❝❡rt❡③❛✱ ♣♦ss✉✐ ♠❛✐s ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s✳
❚❡♦r❡♠❛ ✶✳✸✳✶ ◆ã♦ ❡①✐st❡ ♦ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✳
Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ V✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✳ ❙❡❥❛ B = {x ∈ V /x 6∈ V}✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ B✱ t❡♠♦s q✉❡✿ B ∈ B ⇔ B 6∈ B✱ ♦ q✉❡ é ✉♠❛
❝♦♥tr❛❞✐çã♦✳ ❉❡st❛ ❢♦r♠❛✱ ♥ã♦ ❡①✐st❡ V✳
❈❛♣ít✉❧♦ ✷
❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ ▲❡♠❛ ❞❡ ❩♦r♥ ❡
❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦
✷✳✶ ◆♦çõ❡s ❇ás✐❝❛s
❆♥t❡s ❞❡ ❢❛❧❛r♠♦s ❡s♣❡❝✐✜❝❛♠❡♥t❡ s♦❜r❡ ❆①✐♦♠❛ ❞❛ ❊s❝♦❧❤❛✱ ▲❡♠❛ ❞❡ ❩♦r♥ ❡ ❚❡♦r❡♠❛ ❞❡ ❩❡r♠❡❧♦✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♥❡❝❡ssár✐♦s ♣❛r❛ t❛❧ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳
❊♠ ✶✾✷✶✱ ❑❛③✐♠✐❡r③ ❑✉r❛t♦✇s❦✐✱ ❡❧❛❜♦r♦✉ ❛ ♠❛✐s s✐♠♣❧❡s ❞❡✜♥✐çã♦ ❞❡ ♣❛r ♦r❞❡✲ ♥❛❞♦✿
❉❡✜♥✐çã♦ ✷✳✶✳✶ ❉✐③❡♠♦s q✉❡ (x, y) é ✉♠ ♣❛r ♦r❞❡♥❛❞♦✱ q✉❛♥❞♦✿
(x, y) ={{x} ,{x, y}}.
❆♦ ❝♦♥s✐❞❡r❛r♠♦s q✉❡ X ❡ Y sã♦ ❝♦♥❥✉♥t♦s✱ ♣❡❧♦ ❛①✐♦♠❛ ❞♦ ♣❛r✱ t❡♠♦s q✉❡
❡①✐st❡♠ ♦s ❝♦♥❥✉♥t♦s{x}={x, x} e {x, y}✳ ◆♦✈❛♠❡♥t❡ ❛♣❧✐❝❛♥❞♦ ♦ ❛①✐♦♠❛ ❞♦ ♣❛r✱ t❡♠♦s q✉❡(x, y) = {{x},{x, y}}é ✉♠ ❝♦♥❥✉♥t♦✳ ❈♦♠ ❡st❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛r ♦r❞❡♥❛❞♦ ❢❛③ ❞❡s♥❡❝❡ssár✐❛ ❛ ❝r✐❛çã♦ ❞❡ ✉♠ ♥♦✈♦ ❛①✐♦♠❛ q✉❡ ♣❛ss❡ ❛ ✐❞❡✐❛ ❞❡ ♣❛r ♦r❞❡♥❛❞♦✱ ❝♦♠ ✐st♦ ❝♦♥s❡❣✉✐♠♦s ♠♦str❛r ❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ♣❛r ♦r❞❡♥❛❞♦ ❛tr❛✈és ❞❛ ✐❣✉❛❧❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s q✉❡ ❥á ❢♦✐ ❛①✐♦♠❛❞♦✳ ❯♠ ❢❛t♦ ✐♠♣♦rt❛♥t❡ ❞❡ss❛ ❞❡✜♥✐çã♦ é q✉❡ ♣❛r❛ ✐♥❞✐❝❛r ✉♠❛ tr✐♣❧❛ ♦r❞❡♥❛❞❛ ❜❛st❛ r❡♣r❡s❡♥t❛r♠♦s ♣♦r✿
(x, y, z) = {{x},{x, y},{x, y, z}}.
❉❡✜♥✐çã♦ ✷✳✶✳✷ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ❆ ❡ ❇✱ ❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ✭①✱ ②✮ ❝♦♠ x∈A ❡ y∈B é ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡ ❆ ♣♦r ❇ ❡ é ❞❡♥♦t❛❞♦ ♣♦r✿
A×B ={(x, y) x∈Aey∈B}.
◆♦çõ❡s ❇ás✐❝❛s ❈❛♣ít✉❧♦ ✷
Pr♦♣♦s✐çã♦ ✷✳✶✳✶ ❖ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ A×B é ✉♠ ❝♦♥❥✉♥t♦✱ ♦✉ s❡❥❛✱ s❡ x ∈E
❡ y∈E✱ ❡♥tã♦ (x, y)∈℘(℘(E))✳
Pr♦✈❛✳ ❙❡ x∈ E ❡ y∈ E✱ ❡♥tã♦{x} ⊆ E {x, y} ⊆ E✳ ❉❛í✱ {x} ∈℘(E) ❡ t❛♠❜é♠ {x, y} ∈℘(E)✳ P♦rt❛♥t♦✱{{x},{x, y}} ⊆℘(E)❡✱ ✜♥❛❧♠❡♥t❡✱ t❡♠♦s {{x},{x, y}} ∈
℘(℘(E))✳
Pr♦♣♦s✐çã♦ ✷✳✶✳✷ P❛r❛ q✉❛✐sq✉❡r ❝♦♥❥✉♥t♦s A ❡ B✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝✉❥♦s ❡❧❡✲
♠❡♥t♦s sã♦ ❡①❛t❛♠❡♥t❡ ♦s ♣❛r❡s (x, y) ❝♦♠ x∈A ❡ y∈B✳
Pr♦✈❛✳ ❆ ♣❛rt✐r ❞♦ ❛①✐♦♠❛ ❞❛ ❝♦♠♣r❡❡♥sã♦✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ♦ ❝♦♥❥✉♥t♦ {z ∈
℘(℘A∪B))z = (x, y)}♣❛r❛ ❛❧❣✉♠x❡♠ A❡ ❛❧❣✉♠y❡♠ B}✳ ❈♦♠ ✐st♦✱ t❡♠♦s q✉❡ ❡ss❡ ❝♦♥❥✉♥t♦ ❝♦♥té♠ s♦♠❡♥t❡ ♣❛r❡s ❞♦ t✐♣♦ ❞❡s❡❥❛❞♦ ❡✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ♦ ❝♦♥❥✉♥t♦ ❝♦♥té♠ t♦❞♦s ❡❧❡s✳
❖❜s❡r✈❛çã♦ ✷✳✶✳✶ ❉❡✈✐❞♦ ❛s ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✱ t❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐✲ ❛♥♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s é ❛✐♥❞❛ ✉♠ ❝♦♥❥✉♥t♦✳
❉❡✜♥✐çã♦ ✷✳✶✳✸ ❯♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛ é q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ❞❡ A×B✳
❙❡❥❛R ✉♠❛ r❡❧❛çã♦ ❜✐♥ár✐❛✱ ❝♦♠R ⊆A×B✳ ●❡r❛❧♠❡♥t❡✱ ❡s❝r❡✈❡♠♦sxRy ♣❛r❛
r❡♣r❡s❡♥t❛r (x, y)∈A×B✳
❖❜s❡r✈❛çã♦ ✷✳✶✳✷ ❊♠ ❣❡r❛❧✱ q✉❛♥❞♦ tr❛t❛r♠♦s ❛ r❡s♣❡✐t♦ ❞❡ r❡❧❛çõ❡s ❜✐♥ár✐❛s✱ ❞✐r❡♠♦s ❛♣❡♥❛s r❡❧❛çõ❡s✳
❙❡❥❛ R ✉♠❛ r❡❧❛çã♦ ❞❡A ❡♠ B ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ r❡❧❛çã♦ ❡♠A×B✱ ❡♥tã♦
t❡♠♦s✿
❉❡✜♥✐çã♦ ✷✳✶✳✹ ❖ ❞♦♠í♥✐♦ ❞❡ ❘✱ ❞❡♥♦t❛❞♦ ♣♦r ❉♦♠✭❘✮✱ é ❞❡✜♥✐❞♦ ♣♦r✿
Dom(R) = {x∈A(x, y)∈R ♣❛r❛ ❛❧❣✉♠ y∈B}.
❉❡✜♥✐çã♦ ✷✳✶✳✺ ❆ ✐♠❛❣❡♠ ❞❡ ❘✱ ❞❡♥♦t❛❞♦ ♣♦r ■♠✭❘✮✱ é ❞❡✜♥✐❞❛ ♣♦r✿
Im(R) = {y∈B(x, y)∈R ♣❛r❛ ❛❧❣✉♠ x∈A}.
❉❡✜♥✐çã♦ ✷✳✶✳✻ ❖ ❝❛♠♣♦ ❞❡ ❘✱ ❞❡♥♦t❛❞♦ ♣♦r Camp(R)✱ é ❞❛❞♦ ♣♦r✿
Camp(R) =Dom(R)∪Im(R).
❊①❡♠♣❧♦✿ ❙❡❥❛ R={(3,4),(5,6),(6,7)}✳ ❊♥tã♦✿
Dom(R) ={3,5,6}; Im(R) ={4,6,7}; Camp(R) = {3,4,5,6,7}.
◆♦çõ❡s ❇ás✐❝❛s ❈❛♣ít✉❧♦ ✷
Pr♦♣♦s✐çã♦ ✷✳✶✳✸ ❙❡ R é ✉♠❛ r❡❧❛çã♦✱ ❡♥tã♦ Camp(R)✱ Dom(R) ❡ Im(R) sã♦ ❝♦♥❥✉♥t♦s✳
Pr♦✈❛✳ ❖ ❛①✐♦♠❛ ❞❛ ❝♦♠♣r❡❡♥sã♦ ❣❛r❛♥t❡ q✉❡ Dom(R) ❡ Im(R) sã♦ ❝♦♥❥✉♥t♦s✱ ♣♦✐s sã♦ s✉❜❝♦♥❥✉♥t♦s ❞♦s ❝♦♥❥✉♥t♦s A ❡ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊ ❝♦♠♦ Camp(R) é ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s✱ ❡♥tã♦ t❛♠❜é♠ é ❝♦♥❥✉♥t♦✳
❙❡R é ✉♠❛ r❡❧❛çã♦ ❡♠A✱ ❡♥tã♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❞❡✜♥✐❞❛s ♣❛r❛ R✿
✭✐✮ ❘ é r❡✢❡①✐✈❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ x∈A ✱xRx❀
✭✐✐✮ ❘ é s✐♠étr✐❝❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦sx, y ∈A✱ s❡ xRy ✱ ❡♥tã♦ yRx❀
✭✐✐✐✮ ❘ é tr❛♥s✐t✐✈❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦sx, y, z ∈A✱ s❡ xRy ❡ yRz ✱ ❡♥tã♦ xRz❀
✭✐✈✮ ❘ é ❛♥t✐ss✐♠étr✐❝❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦s x, y ∈A✱ s❡ xRy ❡yRx✱ ❡♥tã♦ x=y❀
✭✈✮ ❘ é ✐rr❡✢❡①✐✈❛ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦x∈A✱(x, x)6∈R✱ ✐st♦ é✱x♥ã♦ s❡ r❡❧❛❝✐♦♥❛
❝♦♠ x✳
◗✉❛♥❞♦ ✉♠❛ r❡❧❛çã♦ ❘ s♦❜r❡ ❆ s❛t✐s❢❛③ ♦s ✐t❡♥s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✐✐✮✱ ❡♥tã♦ ❡ss❛ r❡❧❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ❘❡❧❛çã♦ ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛✳
❆ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠ ♠♦❞♦ ❞❡ ❣❡♥❡r❛❧✐③❛r ❛ r❡❧❛çã♦ ❞❡ ✐❣✉❛❧❞❛❞❡✱ ❡♠ s✐t✉❛çã♦ q✉❡ ✐♥❞✐✈í❞✉♦s ❡♠❜♦r❛ ❞✐st✐♥t♦s ♣♦ss❛♠ ❡①❡❝✉t❛r ✉♠ ♣❛♣❡❧ ❡q✉✐✈❛❧❡♥t❡✳
❊①❡♠♣❧♦✿ ❊♠ Z✱ ❛ r❡❧❛çã♦ xRy s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✏x−y é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 5✑ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♣♦✐s✿
✭✐✮ ✭r❡✢❡①✐✈❛✮✿ ♣❛r❛ t♦❞♦ x∈Z ✱ t❡♠♦s q✉❡ x−x= 0 = 0·5 ✱ ♦✉ s❡❥❛ xRx❀
✭✐✐✮ ✭s✐♠étr✐❝❛✮✿ s❡xRy✱ ❡♥tã♦ x−y = 5·n✱ ❝♦♠ n∈Z ❡✱ ❡♥tã♦y−x= 5·(−n) ✱ ♦✉ s❡❥❛✱ yRx❀
✭✐✐✐✮ ✭tr❛♥s✐t✐✈✐❞❛❞❡✮✿ s❡ xRy ❡ yRz ✱ ❡♥tã♦ x−y = 5·n ❡ y−z = 5·m ✱ ❝♦♠ n, m∈Z✳ ❊♥tã♦ x−z =x−y+y−z = 5·n+ 5·m= 5·(n+m)❡✱ ♣♦rt❛♥t♦✱
xRz✳
❉❡✜♥✐çã♦ ✷✳✶✳✼ ✿ ❙❡❥❛♠ X ❡ Y ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❢✉♥çã♦ é ✉♠❛ t❡r♥❛ (f, X, Y)✱ s❡♥❞♦ f ✉♠❛ r❡❧❛çã♦ ❡♠ X×Y✱ s❛t✐s❢❛③❡♥❞♦✿
(i) Dom(f) = X✱
✭✐✐✮ ❙❡ (x, y)∈f ❡ (x, z)∈f ❡♥tã♦ y=z✳ ❙❡(x, y)∈f✱ ❡s❝r❡✈❡♠♦s y=f(x)✳
❘❡❧❛çõ❡s ❞❡ ❖r❞❡♠ ❈❛♣ít✉❧♦ ✷
❉❡✜♥✐çã♦ ✷✳✶✳✽ ❈❤❛♠❛♠♦s ❞❡ ❢✉♥çã♦ ✐♥❥❡t♦r❛✱ q✉❛♥❞♦ ♣❛r❛ t♦❞♦ x ❡ z ❞✐❢❡✲
r❡♥t❡s✱ t❛✐s q✉❡ ❛♠❜♦s ♣❡rt❡♥ç❛♠ ❛ X✱ t❡♠♦s f(x) = y ❡ f(z) = y s❡✱ ❡ só s❡✱ x=z✳
❉❡✜♥✐çã♦ ✷✳✶✳✾ ❈❤❛♠❛♠♦s ❞❡ ❢✉♥çã♦ s♦❜r❡❥❡t♦r❛✱ q✉❛♥❞♦ ♣❛r❛ t♦❞♦ y ♣❡rt❡♥✲
❝❡♥t❡ ❛ Y ✱ ❡①✐st❡ ✉♠ x ♣❡rt❡♥❝❡♥t❡ ❛ X✱ t❛❧ q✉❡ f(x) = y✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ f
é ✉♠❛ s♦❜r❡❥❡çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Im(f) = Y✳
❉❡✜♥✐çã♦ ✷✳✶✳✶✵ ❈❤❛♠❛♠♦s ❞❡ ❢✉♥çã♦ ❜✐❥❡t♦r❛ ♦✉ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♠ ❛ ✉♠✱ q✉❛♥❞♦ ❡❧❛ ❢♦r ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✳
❉❡✜♥✐çã♦ ✷✳✶✳✶✶ ❯♠❛ r❡❧❛çã♦ IdA é ❞✐t❛ ✐❞❡♥t✐❞❛❞❡✱ q✉❛♥❞♦IdA={(x, x) :x∈
A}✳
❚❡♦r❡♠❛ ✷✳✶✳✶ xIdAx↔x∈A✳
Pr♦✈❛✳ ❉❡s❞❡
(x, x) = {{x},{x, x}={{x}},
✜❝❛ ❝❧❛r♦ q✉❡
IdA⊆℘(℘(A)).
❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡
x∈A → {{x}} ∈℘(℘(A)). (i)
❊♠ ✈✐rt✉❞❡ ❞♦ ❛①✐♦♠❛ ❞❛ s❡♣❛r❛çã♦✱ ♣♦❞❡♠♦s ✉s❛r (i) ♣❛r❛ ♦❜t❡r ♦ t❡♦r❡♠❛✳
✷✳✷ ❘❡❧❛çõ❡s ❞❡ ❖r❞❡♠
❉❡✜♥✐çã♦ ✷✳✷✳✶ ❯♠❛ r❡❧❛çã♦ é ❞✐t❛ ❞❡ ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞❛ q✉❛♥❞♦ ❛ r❡❧❛çã♦ é r❡✢❡①✐✈❛✱ ❛♥t✐ss✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛✳
❙❡♥❞♦ ❛ss✐♠✱ ❛ r❡❧❛çã♦ ❞❡ ✐♥❝❧✉sã♦ ❞❡ ❝♦♥❥✉♥t♦s (⊂) é ✉♠ ❡①❡♠♣❧♦ ❞❡ ♦r❞❡♠ ♣❛r❝✐❛❧✱ ♣♦✐s✿
❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A✱ B ❡ C✱ t❡♠♦s q✉❡✿
✭✐✮ A⊂A❀
✭✐✐✮ A⊂B ❡ B ⊂A✱ ❡♥tã♦ t❡♠♦s q✉❡A =B❀
✭✐✐✐✮ A⊂B ❡ B ⊂C✱ ❡♥tã♦ A⊂C✳
❖✉tr♦ ❡①❡♠♣❧♦ ❝❧áss✐❝♦ ❞❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ♣❛r❝✐❛❧ é ❛ r❡❧❛çã♦ ✏♠❡♥♦r ♦✉ ✐❣✉❛❧✑(≤)✿
❉❛❞♦s m, n ∈ N ✱ ❞✐③ ✲ s❡ m é ♠❡♥♦r ❞♦ q✉❡ n✱ ❡ ❡s❝r❡✈❡ ✲ s❡ m < n✱ ♣❛r❛
s✐❣♥✐✜❝❛r q✉❡ ❡①✐st❡ ❛❧❣✉♠ p∈N t❛❧ q✉❡ n =m+p✳
