MATEMÁTICA DISCRETA: APLICAÇÕES DO PRINCÍPIO DA INCLUSÃO E EXCLUSÃO

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✿

❆♣❧✐❝❛çõ❡s ❞♦ Pr✐♥❝í♣✐♦ ❞❡

■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳

†

♣♦r

❙❊❇❆❙❚■➹❖ ❆▲❱❊❙ ❇❊❩❊❘❘❆ ◆❊❚❖

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦ Pr♦❢✳ ❊❞✉❛r❞♦ ●♦♥ç❛❧✈❡s ❞♦s ❙❛♥t♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ❞❡ ♦❜t❡♥çã♦ ❞♦ ●r❛✉ ❞❡ ▼❡str❡ ♥♦ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✳

✵✽✴✷✵✶✻ ❏♦ã♦ P❡ss♦❛ ✲ P❇

† _{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

B574m Bezerra Neto, Sebastião Alves.

Matemática discreta: aplicações do Princípio de Inclusão e Exclusão / Sebastião Alves Bezerra Neto.- João Pessoa, 2016. 60f. : il.

Orientador: Eduardo Gonçalves dos Santos Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Princípio da Inclusão e Exclusão. 3. Crivo de Eratóstenes. 4. Função Fi de Euler. 5. Permutações

caóticas. 6. Número de Funções Sobrejetoras;

❆❣r❛❞❡❝✐♠❡♥t♦s

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛ ❉❡✉s q✉❡ ❡♠ ❝❛❞❛ ♠♦♠❡♥t♦ ❞❡ ❛❧❡❣r✐❛✱ ❞❡ ❛♣❡rt♦ ♦✉ ❞❡ ❢r✉st❛çã♦✱ s❡♠♣r❡ ❡st❡✈❡ ❝♦♠✐❣♦ ♠❡ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ♥♦✈❛ ❡s♣❡r❛♥ç❛✱ ♠♦t✐✈❛çã♦ ❡ ❜❛st❛♥t❡ ❢♦rç❛ ♣❛r❛ q✉❡ ❝❤❡❣❛ss❡ ❛té ♦ ✜♥❛❧ ❞❡st❡ ❝✉rs♦✳

❆ t♦❞♦s ❞❛ t✉r♠❛ P❘❖❋▼❆❚✲✷✵✶✹✱ ❛❧é♠ ❞❡ ♦✉tr♦s ❛❧✉♥♦s ❞♦ P❘❖❋▼❆❚ ❝♦♠ ♦s q✉❛✐s t✐✈❡ ♦ ♣r❛③❡r ❞❡ ❝♦♥✈✐✈❡r ♥♦s ❞♦✐s ú❧t✐♠♦s ❛♥♦s ❡ q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡st❛ ✐♠♣♦rt❛♥t❡ ❝♦♥q✉✐st❛ ♣❡ss♦❛❧✳

❆♦ ♠❡✉ ♣❛✐✱ ♠❡✉s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s q✉❡ s❡♠♣r❡ t✐✈❡r❛♠ ❝♦♠♣r❡❡♥sã♦ q✉❛♥❞♦ ♥ã♦ ♣✉❞❡ ❞❛r✲❧❤❡s ❛ ❛t❡♥çã♦ q✉❡ ♠❡r❡❝✐❛♠ ❞✉r❛♥t❡ ❛s ❞✉ríss✐♠❛s ❥♦r♥❛❞❛s ❞❡ ❡st✉✲ ❞♦s✳

❆♦ ♠❡✉ ✐r♠ã♦ ❈é❧✐♦ q✉❡ t❡♠ ❛♠♣❛r❛❞♦ ♥♦ss♦ ♣❛✐✱ ♠❡ ♣r♦♣♦r❝✐♦♥❛♥❞♦ tr❛♥q✉✐❧✐✲ ❞❛❞❡ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❞❡ t❡♠♣♦✱ ❞♦✐s ❢❛t♦r❡s ♣r✐♠♦r❞✐❛✐s ♣❛r❛ ❡st❡ ♥í✈❡❧ ❞❡ ❡st✉❞♦✳ ❆♦ ❛♠✐❣♦ ▲❡♦♥❛❧❞♦ ●♦♥③❛❣❛✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ q✉❛♥❞♦ ♥❡❝❡ss✐t❛✈❛ ♠❡ ❛❢❛st❛r ❞♦ tr❛❜❛❧❤♦ ❛❧é♠ ❞❛ ❛❥✉❞❛ ❝♦♠ ❛ t❡❝♥♦❧♦❣✐❛✳

❆ ❋❛❜✐❛♥♦ ❈❛str♦ q✉❡ ♠❡ ♦r✐❡♥t♦✉ ♠✉✐t♦ ❝♦♠ ♦ ❧❛t❡① s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐ ❞✉r❛♥t❡ ❛ ❡s❝r✐t❛ ❞❡st❡ tr❛❜❛❧❤♦✱ ♦ q✉❡ t♦r♥♦✉ ♣♦ssí✈❡❧ ❛ ♦r❣❛♥✐③❛çã♦ ❞♦ ♠❡s♠♦✳

❆♦s ❛♠✐❣♦s ❈❛r❧♦s ▼✉♥✐③ ❏ú♥✐♦r✱ ❋r❛♥❝✐s❝♦ ◆♦❣✉❡✐r❛✱ ❮t❛❧♦ ●✉s♠ã♦ ❡ ❘❛❢❛❡❧ ❚❛✈❛r❡s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ✈✐❛❣❡♥s à ❯❋P❇ ❡ ❞❡ ❛♣♦✐♦ ❡♠ ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡s ♥♦ ❝✉rs♦✳

❆♦s ❛♠✐❣♦s ❊❞s♦♥ ❆r❛ú❥♦✱ ❏♦sé ●♦♥③❛❧❡s ❡ ▼❛♥♦❡❧ ▼❛r❝♦s✱ ❝♦♠♣❛♥❤❡✐r♦ ❞❡ ❡st✉❞♦s ❛té ❛❧t❛s ❤♦r❛s ❞❛ ♥♦✐t❡✳

❆ t♦❞♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❛s ❡s❝♦❧❛s q✉❡ ❧❡❝✐♦♥♦ ♣♦r ♠❡ ✐♥❝❡♥t✐✈❛r❡♠ ❡ s❡ ❛❧❡✲ ❣r❛r❡♠ ❝♦♠ ❝❛❞❛ ♦❜❥❡t✐✈♦ ❛❧❝❛♥ç❛❞♦ ❞✉r❛♥t❡ ♦ ❝✉rs♦ ❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧✳ ❆❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ■✈❡❧t♦♥ ▲✉st♦s❛ q✉❡ ❝♦♠ s❡✉ ✈❛st♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦ ❡ ❤✉♠❛♥✐❞❛❞❡ ❛❥✉❞♦✉ ♠✉✐t♦ ❛ t✉r♠❛✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❊❞✉❛r❞♦ ●♦♥ç❛❧✈❡s ❞♦s ❙❛♥t♦s✱ q✉❡ ❝♦♠ s❡✉ ♣r♦✜ss✐♦✲ ♥❛❧✐s♠♦✱ s✉❛ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ t♦r♥♦✉ ♣♦ssí✈❡❧ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ❈♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦ ❇r✉♥♦ ❘✐❜❡✐r♦ ❡ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ ❆❧❡①❛♥❞r❡ ❙✐♠❛s✱ ❇r✉♥♦ ❘✐❜❡✐r♦✱ ❈❛r❧♦s ❇♦❝❦❡r✱ ●✐❧♠❛r ❈♦rr❡✐❛✱ ❊❞✉❛r❞♦ ●♦♥ç❛❧✲ ✈❡s✱ ▲❡♥✐♠❛r ❆♥❞r❛❞❡✱ ▲✐③❛♥❞r♦ ❙❛♥❝❤❡③✱ ◆❛♣♦❧❡♦♥ ❚✉❡st❛ ❡ ❙ér❣✐♦ ❞❡ ❆❧❜✉q✉❡rq✉❡✱ q✉❡ ❝♦♠ ♣❛❝✐ê♥❝✐❛✱ ❞❡❞✐❝❛çã♦ ❡ ❛♠✐③❛❞❡✱ ❛❧✐✈✐❛r❛♠ ❛ t❡♥sã♦ ♣r♦❞✉③✐❞❛ ♣❡❧♦ ❝✉rs♦✳

➚ Pr❡❢❡✐t✉r❛ ▼✉♥✐❝✐♣❛❧ ❞♦ ❏❛❜♦❛tã♦ ❞♦s ●✉❛r❛r❛♣❡s q✉❡ ❛tr❛✈és ❞❡ ❧✐❜❡r❛çõ❡s ❞❡ ▲✐❝❡♥ç❛s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❡ ❝✉rs♦s✱ ♣r♦♣♦r❝✐♦♥❛ ✐♥❝❡♥t✐✈♦ ❛ q✉❛❧✐✜❝❛çã♦ ❞♦ ♣r♦❢❡ss♦r✳

❉❡❞✐❝❛tór✐❛

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ✜❧❤♦s q✉❡ s❡♠♣r❡ ❡stã♦ ❝♦♠✐❣♦ ❡♠ t♦❞♦s ♦s ♠♦✲ ♠❡♥t♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ q✉❡ ❝♦♠ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ s❡♠♣r❡ ♠❡ ❝♦♥❢♦rt♦✉ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s q✉❡ ♣❛ss❡✐ ❞✉r❛♥t❡ ♦ ❝✉rs♦✳

❊♠ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s ✐r♠ã♦s ❏❡❢❡rs♦♥ ❡ ❲✐❧t♦♥ ✭ ✐♥✲♠❡♠♦r✐❛♥ ✮ q✉❡ ♠❡ ✐♥❝❡♥t✐✲ ✈❛r❛♠ ❛ r❡❛❧✐③❛r ❡st❡ ▼❡str❛❞♦✱ ❛♣❡s❛r ❞❡ t♦❞♦s ♦s s❡✉s ♣r♦❜❧❡♠❛s ❞❡ s❛ú❞❡✳

❆ t♦❞♦s q✉❡ s❡ ❛❧❡❣r❛♠ ❝♦♠ ♦ ♥♦ss♦ s✉❝❡ss♦✳

❘❡s✉♠♦

❖ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ▼❛t❡♠át✐❝❛ ❡stá ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s t❡ór✐❝♦s ❡ ♣rát✐❝♦s✱ ♦s q✉❛✐s ❣❡r❛❧♠❡♥t❡ ❡♥✈♦❧✈❡♠ s✐t✉❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦ ❞❡ ♥♦ss❛ s♦❝✐❡❞❛❞❡✳ ❊ss❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡✲ s❡♥t❛r ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ✈á✲ r✐♦s ♠♦❞❡❧♦s ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ❛ ❝♦♥t❛❣❡♠ ❞❡ ❡❧❡♠❡♥t♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛q✉❡❧❛s q✉❡ ❛♣❛r❡❝❡♠ ❝♦♥t❛❣❡♠ ❞✉♣❧❛s✱ tr✐♣❧❛s✱ ❞❡♥tr❡ ♦✉tr❛s✳ ❆❧é♠ ❞✐ss♦✱ ❜✉s❝❛ r❡❧❛❝✐♦♥á✲❧♦ ❝♦♠ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❞❡ ✉♠ ♥ú♠❡r♦ ❡ ❝♦♠ ♦ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s✱ ✉t✐❧✐③á✲❧♦ ♣❛r❛ s✐st❡♠❛t✐③❛r ❛ ❋ór♠✉❧❛ ❞❛ ❋✉♥çã♦ ❋✐ ✭Φ✮ ❞❡ ❊✉❧❡r✱

❜❡♠ ❝♦♠♦ ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s ❡ ❞♦ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✱ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s✱ ❋✉♥çã♦ ❋✐ ❞❡ ❊✉❧❡r✱ P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s ❡ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s✳

❆❜str❛❝t

❚❤❡ ♣r♦❝❡ss ♦❢ t❡❛❝❤✐♥❣ ❛♥❞ ❧❡❛r♥✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s ✐s ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ r❡s♦❧✉✲ t✐♦♥ ♦❢ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s✱ ✇❤✐❝❤ ♦❢t❡♥ ✐♥✈♦❧✈❡ s✐t✉❛t✐♦♥s ♦❢ ❡✈❡r②❞❛② ❧✐❢❡ ✐♥ ♦✉r s♦❝✐❡t②✳ ❚❤✐s ✇♦r❦ ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ ■♥❝❧✉s✐♦♥ ❛♥❞ ❊①❝❧✉s✐♦♥ Pr✐♥❝✐♣❧❡ ❛s ❛ t♦♦❧ ❢♦r s♦❧✈✐♥❣ ♠❛♥② ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ ❝♦✉♥t✐♥❣ ❡❧❡♠❡♥ts✱ ❡s♣❡❝✐❛❧❧② t❤♦s❡ t❤❛t ❛♣♣❡❛r ❞♦✉❜❧❡✱ tr✐♣❧❡ ❝♦✉♥t✐♥❣✱ ❛♠♦♥❣ ♦t❤❡rs✳ ■t ❛❧s♦ s❡❡❦s t♦ r❡❧❛t❡ ✐t ✇✐t❤ t❤❡ ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ♣r✐♠❡ ♥✉♠❜❡rs ♦❢ ❛ ♥✉♠❜❡r ❛♥❞ t❤❡ ❙✐❡✈❡ ♦❢ ❊r❛t♦st❤❡♥❡s✱ ✉s❡ ✐t t♦ s②st❡♠❛t✐③❡ t❤❡ ❋♦r♠✉❧❛ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❋✐ ✭ P hi✮ ❊✉❧❡r✱ ❛s ✇❡❧❧ ❛s ❢♦r ❞❡t❡r♠✐♥✐♥❣

t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ❈❤❛♦t✐❝ ❛♥❞ ♥✉♠❜❡r ♦❢ ❙♦❜r❡❥❡t♦r❛s ❢✉♥❝t✐♦♥s✳

❑❡②✇♦r❞s✿ ■♥❝❧✉s✐♦♥ ❛♥❞ ❊①❝❧✉s✐♦♥ Pr✐♥❝✐♣❧❡✱ ❙✐❡✈❡ ♦❢ ❊r❛t♦st❤❡♥❡s✱ ❢✉♥❝t✐♦♥ ❋✐ ❊✉❧❡r✱ P❡r♠✉t❛t✐♦♥s ❈❤❛♦t✐❝ ❛♥❞ ♥✉♠❜❡r ♦❢ ❙♦❜r❡❥❡t♦r❛s ❢✉♥❝t✐♦♥s✳

❙✉♠ár✐♦

✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ✻

✷ ❯♠❛ ✈❛r✐❛♥t❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ❡ ❛❧❣✉♠❛s ❛♣❧✐✲

❝❛çõ❡s ✶✼

✸ ▼❛✐s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ✷✽

✸✳✶ ❖ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✷ ❆ ❢✉♥çã♦ ❋✐ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹ ❉❡t❡r♠✐♥❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❢✉♥çõ❡s s♦❜r❡❥❡t♦r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✾

❆♣ê♥❞✐❝❡ ✺✵

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶ ❉❛♥✐❡❧ ❙✐❧✈❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶ ❯♥✐ã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✷ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❝❤ã♦ ❡ ❞♦ t❡t♦ ❞♦ ♥ú♠❡r♦x✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✸ ❯♥✐ã♦ ❡♥tr❡ três ❝♦♥❥✉♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✹ ❘❡tâ♥❣✉❧♦ ✷✵①✽✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶ ❯♥✐ã♦ ❡♥tr❡ q✉❛tr♦ ❝♦♥❥✉♥t♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✶ ❊r❛tóst❡♥❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❏♦❤♥ ◆❛♣✐❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶✳✶ Pr✐♠❡✐r♦ ❈❛s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❙❡❣✉♥❞♦ ❈❛s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❚❡r❝❡✐r♦ ❈❛s♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✸✳✶ ❆ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♣❡r♠✉t❛çã♦ ❝❛ót✐❝❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ P♦tê♥❝✐❛s ❞❡ ✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

■♥tr♦❞✉çã♦

Pr❡t❡♥❞❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ❝♦♠♦ ❢♦♥t❡ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ♣r♦❜❧❡♠❛s ♦♥❞❡ s✉r❣❡ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❝♦♥t❛❣❡♠ ❞✉♣❧❛✱ tr✐♣❧❛✱ ❡t❝✳✱ ♣r♦♣✐❝✐❛♥❞♦ ❛ ❛❧✉♥♦s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦ ✉♠❛ ♣❡r❝❡♣çã♦ ❞❛ ✉♥✐ã♦ ❡♥tr❡ ♠❛✐s ❞❡ três ❝♦♥❥✉♥t♦s✱ ✉♠❛ ✈❡③ q✉❡ ♥♦r♠❛❧♠❡♥t❡ ❝♦♥❤❡❝❡♠ ♦ ♠❡❝❛♥✐s♠♦ ✭♣♦ré♠✱ ♠✉✐t❛s ✈❡③❡s ♥ã♦ é ❛♣r❡s❡♥t❛❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✮ ♣❛r❛ ❞♦✐s ♦✉ ♥♦ ♠á①✐♠♦ três ❝♦♥❥✉♥t♦s✳ ❙ã♦ ♣r♦❜❧❡♠❛s tí♣✐❝♦s✿ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ♠❡♥♦r❡s ♦✉ ✐❣✉❛❧ ❛ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ❞❡ ❝❡rt❛s ❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s✳ ❍á ♠✉✐t♦s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ sã♦ r❡s♦❧✈✐❞♦s ❛tr❛✈és ❞❛ ❝♦♥t❛❣❡♠ ❞♦ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳ ❆ ❞❡t❡r♠✐♥❛çã♦ ❞❡st❡ ♥ú♠❡r♦ ❞❡ ❢✉♥çõ❡s ♣♦❞❡ s❡r ♦❜t✐❞♦ ❝♦♠ ❢❛❝✐❧✐❞❛❞❡ ❡ ❞❡ ♠❛♥❡✐r❛ ♣r❡❝✐s❛ ♣❡❧❛ ✉t✐❧✐③❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳ ❱❡r❡♠♦s ❛✐♥❞❛ q✉❡ ❡st❡ ♣r✐♥❝í♣✐♦ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ ❋✉♥çã♦ ❋✐ ❞❡ ❊✉❧❡r ❡ ❝♦♠ ❛s P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s✱ ♣♦rt❛♥t♦✱ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❝♦♥t❛❣❡♠✳

◆♦ss♦ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳

◆♦ ❈❛♣ít✉❧♦ 1✱ ❡st✉❞❛r❡♠♦s s✐t✉❛çõ❡s q✉❡ s❡r✈✐rã♦ ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦

❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❜r✐♥❝❛❞❡✐r❛ ❞♦ ✧❛♠✐❣♦ ♦❝✉❧t♦✧✳ ❉❡s❡♥✈♦❧✈❡r❡♠♦s ❛ ❋ór♠✉❧❛ ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣❛r❛ ❛ r❡✉♥✐ã♦ ❞❡ ❞♦✐s ❡ ❞❡ três ❝♦♥❥✉♥t♦s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ✈❡r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✳

◆♦ ❈❛♣ít✉❧♦ 2✱ ❛❜♦r❞❛r❡♠♦s ❛ r❡✉♥✐ã♦ ❡♥tr❡ q✉❛tr♦ ❝♦♥❥✉♥t♦s✱ ✉t✐❧✐③❛♥❞♦ ❛s

✐❞❡✐❛s ♦❜t✐❞❛s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❣❡♥❡r❛❧✐③❛r ❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣❛r❛ n ❝♦♥❥✉♥t♦s✳ ❱❡r❡♠♦s ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❛♣r❡s❡♥t❛r

❡st❡ ♣r✐♥❝í♣✐♦ ❡ ❢❛r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✳

◆♦ ❈❛♣ít✉❧♦ 3✱ ✐♥t✐t✉❧❛❞♦ ▼❛✐s ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s✱ ❡st✉❞❛r❡♠♦s ❖ ❈r✐✈♦ ❞❡

❊r❛tóst❡♥❡s✱ ❛ ❋✉♥çã♦ ❋✐ ❞❡ ❊✉❧❡r✱ ❛s P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s ❡ ❛ ❉❡t❡r♠✐♥❛çã♦ ❞♦ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s✳

♥ú♠❡r♦s ♣r✐♠♦s ❛té ❝❡rt♦ ✈❛❧♦r ❧✐♠✐t❡✳ ❋♦✐ ❝r✐❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❣r❡❣♦ ❊r❛tóst❡✲ ♥❡s✱ ♦ t❡r❝❡✐r♦ ❜✐❜❧✐♦t❡❝ár✐♦✲❝❤❡❢❡ ❞❛ ❇✐❜❧✐♦t❡❝❛ ❞❡ ❆❧❡①❛♥❞r✐❛✱ q✉❡ ✈✐✈❡✉ ❡♥tr❡ ♦s ❛♥♦s ✷✼✻ ❛✳❈✳ ❡ ✶✾✹ ❛✳❈✳ ❊❧❡ ❞❡s❡♥✈♦❧✈❡✉ ✉♠❛ t❛❜❡❧❛✱ q✉❡ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❈r✐✈♦ ❞❡ ❊r❛tóst❡♥❡s✱ ♦♥❞❡ ❡❧❡ ❝♦♥s❡❣✉✐✉✱ ♥ã♦ ❝♦♠ ✉♠❛ ❢ór♠✉❧❛ ✭♣♦✐s ❡st❡ é ✉♠ ❞♦s ❞❡s❛✜♦s ❞♦ ■♥st✐t✉t♦ ❈❧❛② ❞❡ ▼❛t❡♠át✐❝❛✱ ✈❡r ♣♦st❛❣❡♠ ❞♦ ❞✐❛ ✵✼✴✵✺✴✷✵✶✸ ♥♦ ✇✇✇✳❝❧❛②♠❛t❤✳♦r❣✮✱ ♠❛s ❝♦♠ ✉♠❛ t❛❜❡❧❛ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣r✐♠♦s✳ ❊r❛tós✲ t❡♥❡s ❛♣❧✐❝♦✉ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ❡s❝r❡✈❡♥❞♦ ♦s ♥ú♠❡r♦s ❞❡1❛1000.

❆ ❋✉♥çã♦ ❋✐ ❞❡ ❊✉❧❡r r❡❝❡❜❡✉ ❡st❡ ♥♦♠❡ ♣♦r q✉❡ ❢♦✐ ♦ ♠❛t❡♠át✐❝♦ s✉íç♦ ▲❡♦✲ ♥❤❛r❞ ❊✉❧❡r q✉❡♠ ❛ ❞❡t❡r♠✐♥♦✉✳ ❊❧❛ t❛♠❜é♠ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❋✉♥çã♦ ■♥❞✐❝❛❞♦r

Φ ❞❡ ❊✉❧❡r ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♣♦r ❢✉♥çã♦ ❋✐✳ ❊st❛ ❢✉♥çã♦ ✐♥❞✐❝❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú✲

♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♠❡♥♦r❡s q✉❡ ♦✉ ✐❣✉❛❧ ❛ ❝❡rt♦ ♥ú♠❡r♦ m ♥❛t✉r❛❧✱ q✉❡ sã♦

r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ m✳

❊♠ sí♠❜♦❧♦s✱ ❡s❝r❡✈❡♠♦s

Φ(m) =|{x∈N_{/x≤m e mdc(x, m) = 1}|,}

♦♥❞❡ ⑤❆⑤ ✐♥❞✐❝❛ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ A✳

❆ ▼❛t❡♠át✐❝❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡✱ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ t❡♠ ❣r❛♥❞❡ ❛♣❧✐❝❛çã♦ ♥❛ ❈r✐♣t♦❣r❛✜❛✳ ❆ ❈r✐♣t♦❣r❛✜❛ é ❛ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r♦♣r✐❝✐❛r ❛ tr♦❝❛ s❡❣✉r❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ❡♥tr❡ ✉♠ tr❛♥s♠✐ss♦r ❡ ✉♠ r❡❝❡♣t♦r✳ ❊❧❛ ♣r♦❝✉r❛ t♦r♥❛r ❛ ❝♦♠✉♥✐❝❛çã♦ ✐♥❞❡❝✐❢rá✈❡❧ ♣❛r❛ t♦❞♦s✱ ❡①❝❡t♦ ♣❛r❛ ór❣ã♦s ❛✉t♦r✐③❛✲ ❞♦s✱ s❡♥❞♦ ✉t✐❧✐③❛❞❛ ❡♠ tr❛♥s❛çõ❡s ❜❛♥❝ár✐❛s ♥♦ ✐♥t❡r♥❡t ❜❛♥❦✐♥❣✱ ♥❛s ❝♦♠♣r❛s ❝♦♠ ❝❛rtõ❡s ✈✐❛ ✐♥t❡r♥❡t✱ ❡♥tr❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✳ ❆ ❢✉♥çã♦ Φ ❞❡ ❊✉❧❡r é ❛♣❧✐❝❛❞❛ ❡♠

✉♠ ♠ét♦❞♦ ❞❡ ❈r✐♣t♦❣r❛✜❛ ❞❡♥♦♠✐♥❛❞♦ ❘❙❆✱ ♠ét♦❞♦ ❡ss❡ ❝r✐❛❞♦ ❡♠ ✶✾✼✼ ♣♦r ❘✳ ❘✐✈❡st✱ ❆✳ ❙❤❛♠✐r❡ ❡ ▲✳ ❆❞❧❡♠❛♥✳

❉❡s❞❡ q✉❡ ❛s ♣r✐♠❡✐r❛s s♦❝✐❡❞❛❞❡s ❢♦r❛♠ ❢♦r♠❛❞❛s✱ ❛ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ ♦s s❡r❡s ❤✉♠❛♥♦s t♦r♥♦✉✲s❡ ♣r✐♠♦r❞✐❛❧✳ ❆ ❛rt❡ ❞❡ ❝✐❢r❛r✴❞❡❝✐❢r❛r ♠❡♥s❛❣❡♥s t❡♠ ❞❡s❡♠♣❡✲ ♥❤❛❞♦ ♣❛♣❡❧ ✐♥❞✐s♣❡♥sá✈❡❧ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ✈✐❞❛ ❝♦t✐❞✐❛♥❛✳ P❡r❝❡❜❡♠♦s ❞❡st❛ ♠❛♥❡✐r❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❢✉♥çã♦Φ❞❡ ❊✉❧❡r✳ ✭P❛r❛ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞❡st❡ ❛ss✉♥t♦✱

r❡❝♦♠❡♥❞❛♠♦s ❬✺❪✮✳

❆ ❜r✐♥❝❛❞❡✐r❛ ❞♦ ✧❛♠✐❣♦ ♦❝✉❧t♦✧✱ ❜❛st❛♥t❡ ❞✐❢✉♥❞✐❞❛ ❡♠ ♥♦ss❛ s♦❝✐❡❞❛❞❡✱ ♥♦s r❡♠❡t❡ ❛ ✉♠❛ ❛♥t✐❣❛ ❡ ✐♥tr✐❣❛♥t❡ q✉❡stã♦ ❞♦ sé❝✉❧♦ ❳❱■■■ q✉❡ ♠♦t✐✈♦✉ ♦ ❝é❧❡❜r❡ ♠❛t❡♠át✐❝♦ ▲❡♦♥❤❛r❞ ❊✉❧❡r ❛ ❡♠♣❡♥❤❛r✲s❡ ❡♠ ❞✐❢í❝✐❧ ❡ s✉r♣r❡❡♥❞❡♥t❡ tr❛❜❛❧❤♦ ❝♦♠ ❛ ✐♥t❡♥çã♦ ❞❡ s♦❧✉❝✐♦♥á✲❧❛✳ ❆ q✉❡stã♦ ✜❝♦✉ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♦ ✧♣r♦❜❧❡♠❛ ❞❛s ❝❛rt❛s ♠❛❧ ❡♥❞❡r❡ç❛❞❛s✧ q✉❡ ❝♦♥s✐st✐❛ ❡♠ ❞❡s❝♦❜r✐r ❞❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ♣♦❞❡✲s❡ ❝♦❧♦❝❛r n ❝❛rt❛s ❡♠ n ❡♥✈❡❧♦♣❡s✱ ❡♥❞❡r❡ç❛❞♦s ❛ n ❞❡st✐♥❛tár✐♦s ❞✐❢❡r❡♥t❡s✱ ❞❡ ♠♦❞♦

q✉❡ ♥❡♥❤✉♠❛ ❞❛s ❝❛rt❛s s❡❥❛ ❝♦❧♦❝❛❞❛ ♥♦ ❡♥✈❡❧♦♣❡ ❝♦rr❡t♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ ♣r♦❜❧❡♠❛ ❡q✉✐✈❛❧❡ ❛✿

❙❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡n✐t❡♥s é ♣❡r♠✉t❛❞♦ ❛❧❡❛t♦r✐❛♠❡♥t❡✱ q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡

♥❡♥❤✉♠ ❞❡❧❡s ✈♦❧t❡ à s✉❛ ♣♦s✐çã♦ ♦r✐❣✐♥❛❧✳

❚❛♥t♦ ❛ ❜r✐♥❝❛❞❡✐r❛ ❞♦ ✧❛♠✐❣♦ ♦❝✉❧t♦✧ q✉❛♥t♦ ♦ ✧♣r♦❜❧❡♠❛ ❞❛s ❝❛rt❛s ♠❛❧ ❡♥✲ ❞❡r❡ç❛❞❛s✧✱ sã♦ s✐t✉❛çõ❡s ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ P❡r♠✉t❛çõ❡s ❈❛ót✐❝❛s ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✧❞❡s❛rr❛♥❥♦s✧✳ ❊st❛s s✐t✉❛çõ❡s s❡rã♦ ❛❜♦r❞❛❞❛s ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳

◆♦ ❊♥s✐♥♦ ▼é❞✐♦✱ é ♠❛✐s ❝♦♠✉♠ ❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s q✉❡ ❡♥✈♦❧✈❡♠ ♦ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❇✐❥❡t♦r❛s ❡ ❞❡ ❋✉♥çõ❡s ■♥❥❡t♦r❛s✱ ♣♦ré♠ ❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ ◆ú♠❡r♦ ❞❡ ❋✉♥çõ❡s ❙♦❜r❡❥❡t♦r❛s✱ sã♦ ♠❡♥♦s ❡①♣❧♦r❛❞❛s✳ Pr❡t❡♥❞❡♠♦s✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ♠♦str❛r q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣♦❞❡ ❛❥✉❞❛r ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❡st❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ♣rát✐❝❛✳

❱❡r❡♠♦s ❛ s❡❣✉✐r ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ ♠❛t❡♠át✐❝♦ q✉❡ ❡♥✉♥❝✐♦✉ ♣❡❧❛ ♣r✐✲ ♠❡✐r❛ ✈❡③ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳

❉❛♥✐❡❧ ❆✉❣✉st♦ ❞❛ ❙✐❧✈❛ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♠❛t❡♠át✐❝♦s ♣♦rt✉❣✉❡s❡s ❞♦ sé❝✉❧♦ ❳■❳✱ r❡❝❡❜❡✉ ♦ ♣r✐♠❡✐r♦ ♥♦♠❡ ❞♦ s❡✉ ♣❛❞r✐♥❤♦✱ ❉❛♥✐❡❧ ◆✉♥❡s ❘✐❜❡✐r♦✳ ❊❧❡ t❛♠❜é♠ ❢♦✐ ♦✜❝✐❛❧ ❞❛ ♠❛r✐♥❤❛ ♣♦rt✉❣✉❡s❛✱ ♥❛t✉r❛❧ ❞❡ ▲✐s❜♦❛✱ ♥❛s❝❡✉ ♥♦ ❞✐❛ ✶✻ ❞❡ ♠❛✐♦ ❞❡ ✶✽✶✹ ❡ ❢❛❧❡❝❡✉ ❡♠ ✻ ❞❡ ♦✉t✉❜r♦ ❞❡ ✶✽✼✽✳

■♥✐❝✐♦✉ ❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛ ♥❛ ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❞❛ ▼❛r✐♥❤❛ ❡♠ ✶✽✷✾✱ ❝♦♥❝❧✉✐♥❞♦ s❡✉ ❝✉rs♦ ❡♠ ✶✽✸✷✳ ❊♠ s❡❣✉✐❞❛✱ ✐♥❣r❡ss♦✉ ♥❛ ❆❝❛❞❡♠✐❛ ❞♦s ●✉❛r❞❛s✲▼❛r✐♥❤❛s ❝♦♥❝❧✉✐♥❞♦ s❡✉s ❡st✉❞♦s ♥❡st❛ ❛❝❛❞❡♠✐❛ ❡♠ ✶✽✸✺✳ ❆ ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❞❛ ▼❛r✐♥❤❛ ♣❛ss♦✉ ❛ ✐♥t❡❣r❛r ❡♠ ✶✽✸✼ ❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛✳ ❊♠ ✶✽✹✺✱ ❛ ❆❝❛❞❡♠✐❛ ❞♦s ●✉❛r❞❛s✲ ▼❛r✐♥❤❛s ❞❡✉ ♦r✐❣❡♠ à ❊s❝♦❧❛ ◆❛✈❛❧✳ ❉❛♥✐❡❧ ❢♦✐ ♣❛r❛ ❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❈♦✐♠❜r❛ ❝✉rs❛r ❇❛❝❤❛r❡❧❛t♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♥❝❧✉✐♥❞♦✲♦ ❡♠ ✶✽✸✾✳ ◆❛ ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❞❛ ▼❛r✐♥❤❛✱ ❢r❡q✉❡♥t♦✉ ♦ ❈✉rs♦ ▼❛t❤❡♠❛t✐❝♦✱ tr✐❡♥❛❧✱ q✉❡ ♣♦ss✐❜✐❧✐t❛✈❛ ♦ ❛❝❡ss♦ ❛ ♣r♦✜ssõ❡s ♥ã♦ só ❞❛ ▼❛r✐♥❤❛ ❝♦♠♦ t❛♠❜é♠ ❞♦ ❊①ér✲ ❝✐t♦✳ ❋♦✐ ♥♦♠❡❛❞♦ ❧❡♥t❡ s✉❜st✐t✉t♦ ❞❛ ❊s❝♦❧❛ ◆❛✈❛❧ s❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥❝✉rs♦✱ ♣♦r ✐♥t❡r♠é❞✐♦ ❞❡ ❞✐❧✐❣ê♥❝✐❛s ❢❡✐t❛s ♣♦r ❧❡♥t❡s ❞♦ ❈♦♥s❡❧❤♦ ❞❛ ❊s❝♦❧❛ P♦❧✐té❝♥✐❝❛ ❞❡ ▲✐s❜♦❛ ❡♠ ✶✽✹✽✱ ♣❛ss❛♥❞♦ três ❛♥♦s ❞❡♣♦✐s ❛ ❧❡♥t❡ ♣r♦♣r✐❡tár✐♦✳ ❊①❡r❝❡✉ ❡st❡ ❝❛r❣♦ ❛té s❡ ❥✉❜✐❧❛r ❡♠ ✶✽✻✺✳ ◆♦s ✜♥❛✐s ❞❡ ✶✽✻✽✱ r❡❢♦r♠♦✉✲s❡ ❞❡ ♦✜❝✐❛❧ ❞❡ ♠❛r✐♥❤❛✱ ♥♦ ♣♦st♦ ❞❡ ❈❛♣✐tã♦✲❞❡✲❋r❛❣❛t❛✳ P❛ss♦✉ ❛ s❡r só❝✐♦ ❧✐✈r❡ ❞❛ ❆❝❛❞❡♠✐❛ ❘❡❛❧ ❞❛s ❈✐ê♥❝✐❛s ❞❡ ▲✐s❜♦❛ ❡♠ ✶✾ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✶✽✺✶✱ só❝✐♦ ❡❢❡❝t✐✈♦ ❡♠ ✼ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✽✺✷ ❡ ❡♠ ✷✵ ❞❡ ❥❛♥❡✐r♦ ❞❡ ✶✽✺✾ t♦r♥♦✉✲s❡ só❝✐♦ ❞❡ ♠ér✐t♦✱ ♣❛ss❛♥❞♦ ❛ r❡❝❡❜❡r ✉♠❛ ♣❡♥sã♦ ❛♥✉❛❧ ❡ ✈✐t❛❧í❝✐❛ ♥♦ ✈❛❧♦r ❞❡ ✷✵✵✩✵✵✵ ré✐s✳ ❊♠ ❝♦♥✈❡rs❛ ❝♦♠ ♦ ❛♠✐❣♦ ❏♦sé ▼❛r✐❛ ▲❛t✐♥♦ ❈♦❡❧❤♦✱ s❡❝r❡tár✐♦✲❣❡r❛❧ ❞❛ ❆❝❛❞❡♠✐❛✱ ❛♥✉♥❝✐❛ ❛ ❞❡❝✐sã♦ ❞❡ ❝♦♥tr❛✐r ❝❛s❛♠❡♥t♦✳ ❈❛s♦✉✲s❡ ❝♦♠ ❩❡✛❡r✐♥❛ ❞✬❆❣✉✐❛r ✭✶✽✷✺✲✶✾✶✸✮ ❛♦s ✹✹ ❛♥♦s ♥❛ ✐❣r❡❥❛ ❞❡ ❙✳ ❏♦sé ❞❡ ▲✐s❜♦❛✱ ❡♠ ✶✻ ❞❡ ❛❜r✐❧ ❞❡ ✶✽✺✾✳ ❉❡st❛ ✉♥✐ã♦✱ ♥❛s❝❡✉ ✉♠ ú♥✐❝♦ ✜❧❤♦✱ ❏ú❧✐♦ ❉❛♥✐❡❧ ❞❛ ❙✐❧✈❛ ✭✶✽✻✻✲✶✽✾✶✮✳ ❊♠ ✉♠❛ ❞❡ s✉❛s ♦❜r❛s ❞❡ ▼❛t❡♠át✐❝❛ ❡❧❡ ❡♥✉♥❝✐♦✉ ❡ ❡s❝r❡✈❡✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ✉♠ ✐♠♣♦rt❛♥t❡ ♠ét♦❞♦ ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s ❡ ❞❛

❈♦♠❜✐♥❛tór✐❛✱ q✉❡ ❢♦✐ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦✲❊①❝❧✉sã♦✳ ❖s ❝♦♥❝❡✐t♦s s✉❜❥❛❝❡♥t❡s ❛ ❡st❡ ♣r✐♥❝í♣✐♦ sã♦ ❛tr✐❜✉í❞♦s✱ ❢r❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❆❜r❛❤❛♠ ❞❡ ▼♦✐✈r❡✱ ♣♦ré♠ ❛ ❢ór✲ ♠✉❧❛ ♠❛t❡♠át✐❝❛ q✉❡ ♦ r❡♣r❡s❡♥t❛ ❛♣❛r❡❝❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♥❛ ▼❡♠ór✐❛ ❞❡ ❉❛♥✐❡❧ ❞❛ ❙✐❧✈❛✱ ❛♣r❡s❡♥t❛❞❛ ❡♠ ✶✽✺✷ à ❆❝❛❞❡♠✐❛ ❞❡ ❈✐ê♥❝✐❛s ❞❡ ▲✐s❜♦❛ ❡ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✽✺✹✳ ❖✉tr❛ ✐♠♣♦rt❛♥t❡ ♣✉❜❧✐❝❛çã♦ ♠❛t❡♠át✐❝❛ é ❛ s✉❛ ▼❡♠ór✐❛ s♦❜r❡ ❛ r♦t❛çã♦ ❞❛s ❢♦rç❛s ❡♠ t♦r♥♦ ❞♦s ♣♦♥t♦s ❞❡ ❛♣❧✐❝❛çã♦✱ ❡♠ q✉❡ ❡❧❡ ❝♦rr✐❣❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❆✉❣✉st ▼ö❜✐✉s✱ ❛♣r❡s❡♥t❛❞❛ ❡♠ ✶✽✺✵ à ❆❝❛❞❡♠✐❛ ❞❡ ❈✐ê♥❝✐❛s ❞❡ ▲✐s❜♦❛ ♦♥❞❡ ❛♥✲ t❡❝✐♣❛ ❡♠ ✈ár✐♦s ❛♥♦s ♦ tr❛❜❛❧❤♦ ❝✐❡♥tí✜❝♦ ❞❡ ●❛st♦♥ ❉❛r❜♦✉① s♦❜r❡ r♦t❛çã♦ ❞❛s ❢♦rç❛s ❡♠ t♦r♥♦ ❞♦s s❡✉s ♣♦♥t♦s ❞❡ ❛♣❧✐❝❛çã♦✳ ▼✉✐t♦ ❡♠❜♦r❛ t❡♥❤❛ s✐❞♦ ♣r♦❢❡ss♦r ♣♦r ✈✐♥t❡ ❛♥♦s✱ ❞❡s❞❡ ♠❡❛❞♦s ❞❡ ✶✽✺✷ ❛té ❞❡ ✶✽✺✾✱ ♣♦r ♠♦t✐✈♦s ❞❡ ❞♦❡♥ç❛✱ ❉❛♥✐❡❧ ❞❛ ❙✐❧✈❛ ♣r❡❝✐s♦✉ ❛❢❛st❛r✲s❡ t❛♥t♦ ❞❛ ❞♦❝ê♥❝✐❛ ♥❛ ❊s❝♦❧❛ ◆❛✈❛❧ q✉❛♥t♦ ❞❛ ❆❝❛❞❡♠✐❛ ❞❛s ❈✐ê♥❝✐❛s ❞❡ ▲✐s❜♦❛✳ ❖s ❧♦♥❣♦s ♣❡rí♦❞♦s ❞❡ ❛✉sê♥❝✐❛ ♣♦r ♠♦t✐✈♦s ❞❡ s❛ú❞❡ r❡❞✉③❡♠ ❛ s✉❛ ❝❛rr❡✐r❛ ❞❡ ♠❛❣✐stér✐♦ ❛ ❝❡r❝❛ ❞❡ s❡t❡ ❛♥♦s ❞❡ s❡r✈✐ç♦ ❡❢❡❝t✐✈♦✳

❆ ♣❛rt✐r ❞❡ ♠❡❛❞♦s ❞❛ ❞é❝❛❞❛ ❞❡ ✶✽✻✵ ❡st✉❞♦✉ q✉❡stõ❡s r❡❧❛❝✐♦♥❛❞❛s ❛ ✈✐❛❜✐❧✐✲ ❞❛❞❡ ❞❡ ♣❧❛♥♦s ❞❡ ♣❡♥sõ❡s ❞❡ ♠♦♥t❡♣✐♦s ❞❡ s♦❜r❡✈✐✈ê♥❝✐❛ q✉❡ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣❛❣❛r ♣❡♥sõ❡s ❛♦s ❤❡r❞❡✐r♦s ❞♦s s❡✉s só❝✐♦s✱ ❛♣ós ♦ s❡✉ ❢❛❧❡❝✐♠❡♥t♦✳ ❆ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ❡s❝r✐t♦s q✉❡ ❝♦♠♣ôs ♥ã♦ ❡stá ♥♦ ♥í✈❡❧ ❞♦ ✈❛❧♦r ❝✐❡♥tí✜❝♦ ❡ s✐♠ ♥❛ ✐♥tr♦❞✉çã♦ ❞❡ ♠ét♦❞♦s ❛❝♦♥s❡❧❤❛❞♦s ♣❡❧❛ ❝✐ê♥❝✐❛ ♥❛ ♦r❣❛♥✐③❛çã♦ ❞❡ ❢✉♥❞♦s ❞❡ ♣❡♥sõ❡s✱ ♠❛r❝❛♥❞♦ ♦ s❡✉ ♣✐♦♥❡✐r✐s♠♦ ♥❛ ✐♥tr♦❞✉çã♦ ❞❛ ▼❛t❡♠át✐❝❛ ❆❝t✉❛r✐❛❧ ❡♠ P♦rt✉❣❛❧✳

❋✐❣✉r❛ ✶✿ ❉❛♥✐❡❧ ❙✐❧✈❛✳

❈❛♣ít✉❧♦ ✶

Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦

❊st✉❞❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ s✐t✉❛çõ❡s q✉❡ s❡r✈✐rã♦ ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳ ❈♦♠❡❝❡♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿

❙❡❥❛ ✉♠❛ ❜r✐♥❝❛❞❡✐r❛ ❞❡ ✧❛♠✐❣♦ ♦❝✉❧t♦✧✱ ♥❛ q✉❛❧ n ♣❡ss♦❛s ❡s❝r❡✈❡♠

s❡✉ ♥♦♠❡ ♥✉♠ ♣❡❞❛ç♦ ❞❡ ♣❛♣❡❧ ❡ ♦ ❞❡♣♦s✐t❛♠ ♥✉♠ r❡❝✐♣✐❡♥t❡✱ ❞❡ ♦♥❞❡ ❝❛❞❛ ✉♠ r❡t✐r❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ ✉♠ ❞♦s ♣❡❞❛ç♦s ❞❡ ♣❛♣❡❧✳ ❖ ♣r♦❜❧❡♠❛ é ❞❡s❝♦❜r✐r q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♥✐♥❣✉é♠ ♣❡❣❛r s❡✉ ♣ró♣r✐♦ ♥♦♠❡✳

❱❡❥❛♠♦s ✉♠❛ ✐❧✉str❛çã♦ ❞❡st❡ ♣r♦❜❧❡♠❛ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s três ❛♠✐❣♦s✳ P❡❧♦ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✱ t❡rí❛♠♦s ✉♠ t♦t❛❧ ❞❡ s❡✐s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ r❡t✐r❛❞❛s ❞♦s ♣❛♣é✐s✱ ✉♠❛ ✈❡③ q✉❡✿ ❛ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ❛ r❡t✐r❛r t❡r✐❛ três ♥♦♠❡s à s✉❛ ❞✐s♣♦s✐çã♦✱ ❛ s❡❣✉♥❞❛ ♣❡ss♦❛ t❡r✐❛ ❞♦✐s ♥♦♠❡s à s✉❛ ❞✐s♣♦s✐çã♦ ❡✱ ❧♦❣✐❝❛♠❡♥t❡✱ s♦❜r❛r✐❛ ❛♣❡♥❛s ✉♠ ♥♦♠❡ ♣❛r❛ ❛ t❡r❝❡✐r❛ ♣❡ss♦❛ r❡t✐r❛r ❞♦ r❡❝✐♣✐❡♥t❡✳ ❆ss✐♠✱ t❡rí❛♠♦s 3×2×1 = 6

♠❛♥❡✐r❛s ❞♦s ♥♦♠❡s s❡r❡♠ r❡t✐r❛❞♦s ❞♦ r❡❝✐♣✐❡♥t❡ ❛❧❡❛t♦r✐❛♠❡♥t❡✳ P♦❞❡♠♦s s❡♣❛r❛r ❡ss❛s s❡✐s ♠❛♥❡✐r❛s ❡♠ três ❝❛s♦s✿

Pr✐♠❡✐r♦ ❝❛s♦ ✲ ❆ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ♣♦❞❡rá r❡t✐r❛r s❡✉ ♣ró♣r✐♦ ♥♦♠❡ ❡♠ ❞✉❛s s✐t✉❛çõ❡s✿

✶❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ ♣❡ss♦❛s t❛♠❜é♠ r❡t✐r❛♥❞♦ s❡✉s ♣ró♣r✐♦s}

♥♦♠❡s✳

✷❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ r❡t✐r❛♥❞♦ ♦ ♥♦♠❡ ❞❛ t❡r❝❡✐r❛ ❡ ❛ t❡r❝❡✐r❛ ♦ ♥♦♠❡ ❞❛}

s❡❣✉♥❞❛✳ ◆❡st❡ ❝❛s♦✱ ♥❡♥❤✉♠❛ ❞❛s s✐t✉❛çõ❡s s❛t✐s❢❛③❡♠ ♥♦ss♦ ♣r♦❜❧❡♠❛✳ ❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❝❛s♦ ♥❛ ❚❛❜❡❧❛ ✶✳✶✿

1

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

2

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

♥♦♠❡ r❡t✐r❛❞♦

n

p1

n

p2

n

p3

♥♦♠❡ r❡t✐r❛❞♦

n

p1

n

p3

n

p2 ❚❛❜❡❧❛ ✶✳✶✿ Pr✐♠❡✐r♦ ❈❛s♦✳

❙❡❣✉♥❞♦ ❝❛s♦ ✲ ❆ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ♣♦❞❡rá r❡t✐r❛r ♦ ♥♦♠❡ ❞❛ s❡❣✉♥❞❛ ♣❡ss♦❛ ❡♠ ❞✉❛s s✐t✉❛çõ❡s✿

✶❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ ♣❡ss♦❛ r❡t✐r❛♥❞♦ ♦ ♥♦♠❡ ❞❛ ♣r✐♠❡✐r❛ ❡ ❛ t❡r❝❡✐r❛}

♣❡ss♦❛ r❡t✐r❛♥❞♦ s❡✉ ♣ró♣r✐♦ ♥♦♠❡✳

✷❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ r❡t✐r❛♥❞♦ ♦ ♥♦♠❡ ❞❛ t❡r❝❡✐r❛ ❡ ❛ t❡r❝❡✐r❛ r❡t✐r❛♥❞♦ ♦}

♥♦♠❡ ❞❛ ♣r✐♠❡✐r❛✳ ◆❡st❡ ❝❛s♦ ❛♣❡♥❛s ❛ 2❛ s✐t✉❛çã♦ s❛t✐s❢❛③ ♥♦ss♦ ♣r♦❜❧❡♠❛✳

❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ❞♦ s❡❣✉♥❞♦ ❝❛s♦ ♥❛ ❚❛❜❡❧❛ ✶✳✷✿

1

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

2

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

♥♦♠❡ r❡t✐r❛❞♦

n

p2

n

p1

n

p3

♥♦♠❡ r❡t✐r❛❞♦

n

p2

n

p3

n

p1 ❚❛❜❡❧❛ ✶✳✷✿ ❙❡❣✉♥❞♦ ❈❛s♦✳

❚❡r❝❡✐r♦ ❝❛s♦ ✲ ❆ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ♣♦❞❡rá r❡t✐r❛r ♦ ♥♦♠❡ ❞❛ t❡r❝❡✐r❛ ♣❡ss♦❛ t❛♠❜é♠ ❡♠ ❞✉❛s s✐t✉❛çõ❡s✿

✶❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ r❡t✐r❛♥❞♦ ♦ ♥♦♠❡ ❞❛ ♣r✐♠❡✐r❛ ❡ ❛ t❡r❝❡✐r❛ ♣❡ss♦❛ ♦}

♥♦♠❡ ❞❛ s❡❣✉♥❞❛✳

✷❛ _{s✐t✉❛çã♦✿ ❈♦♠ ❛ s❡❣✉♥❞❛ r❡t✐r❛♥❞♦ s❡✉ ♣ró♣r✐♦ ♥♦♠❡ ❡ ❛ t❡r❝❡✐r❛ r❡t✐r❛♥❞♦ ♦}

♥♦♠❡ ❞❛ ♣r✐♠❡✐r❛✳ ❆♥❛❧✐s❛♥❞♦ ♦ t❡r❝❡✐r♦ ❝❛s♦✱ ❛♣❡♥❛s ❛ ✶❛ _{s✐t✉❛çã♦ r❡s♦❧✈❡}

♥♦ss♦ ♣r♦❜❧❡♠❛✳

❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ❞♦ t❡r❝❡✐r♦ ❝❛s♦ ♥❛ ❚❛❜❡❧❛ ✶✳✸✿

1

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

2

❛

♣❛rt✐❝✐♣❛♥t❡

p

1

p

2

p

3

♥♦♠❡ r❡t✐r❛❞♦

n

p3

n

p1

n

p2

♥♦♠❡ r❡t✐r❛❞♦

n

p3

n

p2

n

p1 ❚❛❜❡❧❛ ✶✳✸✿ ❚❡r❝❡✐r♦ ❈❛s♦✳

P♦rt❛♥t♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ❡♥tr❡ três ❛♠✐❣♦s ♥❡♥❤✉♠ ❞❡❧❡s ♣❡❣✉❡ s❡✉ ♣ró✲ ♣r✐♦ ♥♦♠❡ ❝♦♠ r❡t✐r❛❞❛s ❛❧❡❛tór✐❛s é 2

6 = 1 3✳

❱❡❥❛♠♦s ❛❣♦r❛ ♦ ✐♥í❝✐♦ ❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧ ❞❛ ❜r✐♥❝❛❞❡✐r❛ ❡♥tr❡ n ❛♠✐✲

❣♦s✳ ❆ s♦❧✉çã♦ ❝♦♠♣❧❡t❛✱ ✈❡r❡♠♦s ♥♦ ❈❛♣ít✉❧♦ 3✳ ❈❛❞❛ s♦rt❡✐♦ é ✉♠❛ ❜✐❥❡çã♦ ❞♦

❝♦♥❥✉♥t♦ ❞♦s ❛♠✐❣♦s s♦❜r❡ ❡❧❡ ♠❡s♠♦ ✲ ♦ q✉❡ é ❣❡r❛❧♠❡♥t❡ ❝❤❛♠❛❞♦ ❞❡ ♣❡r♠✉t❛çã♦✳ ❊♠ ♦✉tr♦s t❡r♠♦s✱ ❝❛❞❛ s♦rt❡✐♦ é ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛ ❞♦ ❝♦♥❥✉♥t♦A ={1,2,3, ..., n}

s♦❜r❡ ❡❧❡ ♠❡s♠♦✳

◆♦t❡♠♦s q✉❡ ♦ ♣r✐♠❡✐r♦ ❛ ❡s❝♦❧❤❡r t❡rán♥♦♠❡s à s✉❛ ❞✐s♣♦s✐çã♦✱ ♦ s❡❣✉♥❞♦n−1✱

♦ t❡r❝❡✐r♦ n−2 ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❛té q✉❡ ♦ ♣❡♥ú❧t✐♠♦ t❡rá ❞✉❛s ❡s❝♦❧❤❛s ❡ ♦

ú❧t✐♠♦ ❛♣❡♥❛s 1✳ P❡❧♦ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❞❛ ❈♦♥t❛❣❡♠ ♦❜t❡♠♦s ❛ q✉❛♥t✐❞❛❞❡ n×(n−1)×(n−2)×(n−3)×...×2×1 = n!

❢✉♥çõ❡s ❜✐❥❡t♦r❛s ❞❡ A={1,2,3, ..., n}s♦❜r❡ ❡❧❡ ♠❡s♠♦✳

❉❡st❛s n! ❢✉♥çõ❡s ♣♦ssí✈❡✐s ❞❡ r❡t✐r❛r ♦s ♥♦♠❡s ❞♦ r❡❝✐♣✐❡♥t❡✱ ❛❧❡❛t♦r✐❛♠❡♥t❡✱

♣r❡❝✐s❛r❡♠♦s ❞❡s❝♦♥t❛r ❛q✉❡❧❛s ❡♠ q✉❡ ✉♠❛ ♦✉ ♠❛✐s ♣❡ss♦❛s r❡t✐r❛♠ s❡✉s ♣ró♣r✐♦s ♥♦♠❡s✳ ❊①✐st❡♠ ❛❧❣✉♠❛s ♠❛♥❡✐r❛s ♣♦ssí✈❡✐s ♣❛r❛ r❡s♦❧✈❡r♠♦s ❡st❡ ♣r♦❜❧❡♠❛✳ ▼❛s✱ ❛♥t❡s ❞❡ ❝♦♥t✐♥✉❛r♠♦s ❛ s✉❛ r❡s♦❧✉çã♦✱ ✈❡r❡♠♦s ♦✉tr♦s ♣r♦❜❧❡♠❛s ♦♥❞❡ ❡ss❡ t✐♣♦ ❞❡ s✐t✉❛çã♦ ♦❝♦rr❡✳

❖s t✐♣♦s ❞❡ s✐t✉❛çõ❡s q✉❡ ❡①♣❧♦r❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦ s❡rã♦ ❛q✉❡❧❛s ❡♠ q✉❡ ❛♣❛✲ r❡❝❡rã♦ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠ ❞✉♣❧❛✱ tr✐♣❧❛✱ ❡t❝✳ P❛r❛ s♦❧✉❝✐♦♥á✲❧❛s✱ ✐r❡♠♦s ✉t✐❧✐③❛r ❛ té❝♥✐❝❛ ❞❡ s✉❜tr❛✐r ❛s r❡♣❡t✐çõ❡s ❡ ❡♠ s❡❣✉✐❞❛✱ r❡♣♦r ♦s ❞❡s❝♦♥t♦s ❛ ♠❛✐s ✭té❝♥✐❝❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✮✳

❱❡❥❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❯♥✐ã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✳ P♦r ❞❡✜♥✐çã♦✱ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s sã♦ ❛q✉❡❧❡s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ❝♦♠✉♥s✳ ▲♦❣♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❯♥✐ã♦ ❡♥tr❡ ❝♦♥❥✉♥t♦s ❞❡st❡ t✐♣♦ é ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝❛❞❛ ❝♦♥❥✉♥t♦✳

❊①❡♠♣❧♦ ✶✳✶ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A={1,3,5} ❡ B ={0,2,4,6,8}✳ ❱❛♠♦s ❛❣♦r❛✱

r❡♣r❡s❡♥t❛r ♦ ❝♦♥❥✉♥t♦ ✉♥✐ã♦ ❡♥tr❡ ❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ✈❡r✐✜❝❛r♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ♠❡s♠♦✳

❆ ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s é r❡♣r❡s❡♥t❛❞❛ ♣♦r A∪B ♣❛r❛ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ♣♦r ❡①❡♠✲

♣❧♦✱ s✐❣♥✐✜❝❛ ❛❣r✉♣❛r ❡♠ ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ♣❡rt❡♥ç❛♠ ❛♦s ❝♦♥❥✉♥t♦s ❡♥✈♦❧✈✐❞♦s✳ ❆ss✐♠✱ ♥♦ ♥♦ss♦ ❡①❡♠♣❧♦✱ t❡♠♦s

A∪B ={1,3,5} ∪ {0,2,4,6,8}={0,1,2,3,4,5,6,8}.

P♦rt❛♥t♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡B ❞♦ ❡①❡♠♣❧♦

é 8✳

❉❡♥♦t❛♥❞♦ ♣♦r|A| ❡|B|✱ ❛s ❝❛r❞✐♥❛❧✐❞❛❞❡s ↔quantidades de elementos ↔ ❞♦

❝♦♥❥✉♥t♦ A ❡ ❞♦ ❝♦♥❥✉♥t♦ B✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❛

✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B✱ ❞✐s❥✉♥t♦s ❛❝✐♠❛✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r |A∪B| ❞❛

s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

|A∪B|=|A|+|B|= 3 + 5 = 8.

❈♦♠♦ ♥ã♦ tr❛t❛♠♦s ❛♣❡♥❛s ❝♦♠ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✱ ✈❡r❡♠♦s ❛ s❡❣✉✐r q✉❡ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ❞♦✐s ♦✉ ♠❛✐s ❝♦♥❥✉♥t♦s q✉❡ ♥ã♦ s❡❥❛♠ ❞✐s❥✉♥t♦s ♥ã♦ é tã♦ s✐♠♣❧❡s ❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❛✉♠❡♥t❛ q✉❛♥❞♦ ❛ q✉❛♥✲ t✐❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s ❝r❡s❝❡✳ ◗✉❛♥❞♦ ❞❡s❡❥❛r♠♦s ❝♦♥t❛r ♦s ❡❧❡♠❡♥t♦s ❞❡ ❝♦♥❥✉♥t♦s ♥ã♦ ❞✐s❥✉♥t♦s✱ s❡rá ♥❡❝❡ssár✐♦✱ ❡✈✐❞❡♥t❡♠❡♥t❡✱ ❞❡s❝♦♥t❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s r❡♣❡t✐❞♦s ❡ ♦❜s❡r✈❛r s❡ ❤♦✉✈❡ ❞❡s❝♦♥t♦s ❛ ♠❛✐s q✉❡ ♥❡❝❡ss✐t❡♠ s❡r❡♠ r❡♣♦st♦s✳ ❊①❡♠♣❧♦ ✶✳✷ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A ={1,2,3,4,5} ❡ B ={0,2,4,6,8}✳ ❱❛♠♦s

❛❝♦♠♣❛♥❤❛r ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s✳ ❱❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❛❣r✉♣❛r ♦s ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s✱ ❝♦♠♦ s❡ ❡❧❡s ❢♦ss❡♠ ❞✐s❥✉♥t♦s✱ ✐st♦ é✱ r❡❛❧✐③❛r ❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B✳

A∪B ={1,2,3,4,5} ∪ {0,2,4,6,8}={0,1,2,2,3,4,4,5,6,8}.

❘❡t✐r❛♥❞♦ ❛s r❡♣❡t✐çõ❡s✱ ✐st♦ é✱ ♦s ❡❧❡♠❡♥t♦s 2❡ 4✱ ♦❜t❡♠♦s A∪B ={0,1,2,3,4,5,6,8}= 8 elementos.

➱ ❢❛t♦ q✉❡2❡4♣❡rt❡♥❝❡♠ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❛♦s ❞♦✐s ❝♦♥❥✉♥t♦s✱ ♦ q✉❡ ❞❡✜♥❡ ❛ ♦♣❡r❛✲

çã♦ ❞❡ ■♥t❡rs❡çã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s r❡♣r❡s❡♥t❛❞❛ ♣♦r A∩B✳ P❛r❛ ♥♦ss♦ ❡①❡♠♣❧♦ A∩B = {2,4}✱ ❛❝❛rr❡t❛♥❞♦ q✉❡ |A∩B| = 2 ❡❧❡♠❡♥t♦s✳ ❆ss✐♠ ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡

❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B✱ é ❞❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛✿

|A∪B|= |A|+|B| − |A∩B| ✭✶✳✶✮

= 5 + 5−2 = 8.

❆ ❢ór♠✉❧❛ ✭✶✳✶✮ é ❛ ❢ór♠✉❧❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣❛r❛ ❞♦✐s ❝♦♥✲ ❥✉♥t♦s ♥ã♦ ❞✐s❥✉♥t♦s✳ ❊st❛ ❢ór♠✉❧❛ s❡r✈❡ t❛♠❜é♠ ♣❛r❛ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✱ ✉♠❛ ✈❡③ q✉❡ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s é ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡✱ ❡♥tã♦✱ |A∩B|= 0✳

❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ♥❛ ✜❣✉r❛ 1.1✳

❋✐❣✉r❛ ✶✳✶✿ ❯♥✐ã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✳

❊①❡♠♣❧♦ ✶✳✸ ◆✉♠❛ ❝❧❛ss❡ ❞❡ 30❛❧✉♥♦s✱ 14❢❛❧❛♠ ✐♥❣❧ês✱ 5❢❛❧❛♠ ❛❧❡♠ã♦ ❡3❢❛❧❛♠

✐♥❣❧ês ❡ ❛❧❡♠ã♦✳ ❆ ♣❡r❣✉♥t❛ é✱ q✉❛♥t♦s ❛❧✉♥♦s ❢❛❧❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❧í♥❣✉❛ ❞❡♥tr❡ ❡❧❡s❄

P♦❞❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❞❡♥♦t❛r ♦s s❡❣✉✐♥t❡s ❝♦♥❥✉♥t♦s✿

■ ✲ ❈♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ✐♥❣❧ês✳ ❆ss✐♠✱|I| = 14❀

❆ ✲ ❈♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ❛❧❡♠ã♦✳ ▲♦❣♦✱|A| = 5✳

❆ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❡ss❡s ❝♦♥❥✉♥t♦s r❡♣r❡s❡♥t❛ ♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ❛s ❞✉❛s ❧í♥❣✉❛s ❡ q✉❡ ♣♦ss✉✐ 3 ❡❧❡♠❡♥t♦s✱ ✐st♦ é✱ |I ∩A| = 3✳ ❆♣❧✐❝❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡

❊①❝❧✉sã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ♦❜t❡♠♦s

|I∪A|= |I|+|A| − |I∩A|

= 14 + 5−3 = 16.

P♦rt❛♥t♦✱ 16❛❧✉♥♦s ❢❛❧❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❧í♥❣✉❛✳

❆♥t❡s ❞❡ s♦❧✉❝✐♦♥❛r♠♦s ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦✱ ✈❡r❡♠♦s ✉♠ ❝♦♥❝❡✐t♦ ❜❛st❛♥t❡ ✐♠✲ ♣♦rt❛♥t❡ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♥❛ r❡s♦❧✉çã♦ ❞♦ ♠❡s♠♦✳

❙❡❣✉♥❞♦ ❬✶❪✱ ❛ ♥♦t❛çã♦ ⌊x⌋ é ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ♠❛✐♦r ✐♥t❡✐r♦ ♠❡♥♦r

❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛♦ r❡❛❧ x ✭às ✈❡③❡s ❝❤❛♠❛❞♦ ❞❡ ❝❤ã♦ ❞❡ x✮✱ ❡♥q✉❛♥t♦ q✉❡ ⌈x⌉

✭❞❡♥♦♠✐♥❛❞♦ t❡t♦ ❞❡ x✮✱ ✐♥❞✐❝❛ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ x✳ ❖s

❝♦♥❝❡✐t♦s ❡stã♦ ✐❧✉str❛❞♦s ♥❛ ✜❣✉r❛1.2✱ ♦♥❞❡ ❛ r❡t❛ ❣r❛❞✉❛❞❛ r❡♣r❡s❡♥t❛ ♦ ❡✐①♦ ❞♦s

r❡❛✐s ❡ ❛s ♠❛r❝❛çõ❡s ♠❛✐♦r❡s r❡♣r❡s❡♥t❛♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s✳

❋✐❣✉r❛ ✶✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❝❤ã♦ ❡ ❞♦ t❡t♦ ❞♦ ♥ú♠❡r♦ x✳

P❛r❛ r❡s♦❧✈❡r♠♦s ❡st❡ ❡①❡♠♣❧♦✱ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠ú❧t✐♣❧♦s ❞❡

5✱ ❞❡7❡ ❞❡ 35q✉❡ é ♦ ♠❡♥♦r ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♥tr❡ 5❡7❡✱ ❡♠ s❡❣✉✐❞❛✱ ❛♣❧✐❝❛r♠♦s

♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳

❊①❡♠♣❧♦ ✶✳✹ ❉❡♥tr❡ ♦s ♥ú♠❡r♦s ❞❡ 1 ❛té 3600 ✐♥❝❧✉s✐✈❡✱ q✉❛♥t♦s sã♦ ❞✐✈✐sí✈❡✐s

♣♦r 5 ♦✉ ♣♦r 7❄

❙❡❥❛ Ai ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♠ú❧t✐♣❧♦s ❞❡ i ❡ ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s ❛

3600✳ ❖❜s❡r✈❡ q✉❡|Ai|=

₃₆₀₀

i

✳ ❖ ❝♦♥❥✉♥t♦Ai∩Aj ✐♥❞✐❝❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♠ú❧t✐♣❧♦s

❞❡i ❡ ❞❡j ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s ❛3600✱ ✐st♦ é✱ ✐♥❞✐❝❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♠ú❧t✐♣❧♦s ❝♦♠✉♥s

❛ i ❡ ❛ j✱ ♠❡♥♦r❡s ♦✉ ✐❣✉❛✐s ❛ 3600✳ ❊♥tã♦✱ t❡♠♦s q✉❡✿

|A5|=

3600 5

= 720;

|A7|=

3600 7

= 514;

|A5∩A7|=

3600 35

= 102.

❆ss✐♠✱ ❝♦♠♦ ❞❡s❡❥❛♠♦s ❝❛❧❝✉❧❛r |A5 ∪A7|✱ s❡❣✉❡ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①✲ ❝❧✉sã♦ q✉❡

|A5∪A7|=|A5|+|A7| − |A5∩A7|

= 720 + 514−102 = 1132.

P♦rt❛♥t♦✱ ❞❡♥tr❡ ♦s ♥ú♠❡r♦s ❞❡ 1 ❛té 3600 ✐♥❝❧✉s✐✈❡✱ t❡♠♦s 1132 ♥ú♠❡r♦s q✉❡ sã♦

❞✐✈✐sí✈❡✐s ♣♦r 5♦✉ ♣♦r 7✳

P❛r❛ ❛♣❧✐❝❛r♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♥❛ ✉♥✐ã♦ ❞❡ três ❝♦♥❥✉♥t♦s✱ ❞❡✈❡♠♦s ✐❞❡♥t✐✜❝❛r ❛s ✐♥t❡rs❡çõ❡s ❞♦✐s ❛ ❞♦✐s ❡ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s três ❝♦♥❥✉♥t♦s✳ ❱❡❥❛♠♦s ❝♦♠♦ ❞❡✈❡♠♦s ♣r♦❝❡❞❡r ♣❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ❛ ❢ór♠✉❧❛ q✉❡ ♥♦s ❢♦r♥❡ç❛ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ três ❝♦♥❥✉♥t♦s ❛tr❛✈és ❞♦ ❡①❡♠♣❧♦ ❛❜❛✐①♦✳

❊①❡♠♣❧♦ ✶✳✺ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦s A={1,2,3,4,5}, B ={0,2,4,6,8} e C ={0,3,4,5,6,7,8,9}✳

❉❡t❡r♠✐♥❡♠♦s ❛s s❡❣✉✐♥t❡s ✐♥t❡rs❡çõ❡s✿ ✶✳ A∩B ={2,4}❀

✷✳ A∩C ={3,4,5}❀

✸✳ B∩C ={0,4,6,8}❀

✹✳ A∩B ∩C ={4}✳

❋❛ç❛♠♦s✱ ❛❣♦r❛✱ ❛ ✉♥✐ã♦ ❞♦s três ❝♦♥❥✉♥t♦s ❝♦♠♦ s❡ ❢♦ss❡♠ ❞✐s❥✉♥t♦s

A∪B∪C ={1,2,3,4,5} ∪ {0,2,4,6,7,8} ∪ {0,3,4,5,6,8,9}

={0,0,1,2,2,3,3,4,4,4,5,5,6,6,7,8,8,9}.

❘❡t✐r❛♥❞♦ ❛s r❡♣❡t✐çõ❡s ✭✐♥t❡rs❡çõ❡s✮ ❞♦✐s ❛ ❞♦✐s✱ ♦✉ s❡❥❛✱ ♦s ❡❧❡♠❡♥t♦s 0,2,3,4,4, 4,5,6e 8, ♦❜t❡♠♦s

A∪B∪C ={0,1,2,3,5,6,7,8,9}.

P♦❞❡♠♦s ♥♦t❛r q✉❡ ♦ ❡❧❡♠❡♥t♦4q✉❡ é ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s três ❝♦♥❥✉♥t♦s ❢♦✐ r❡t✐r❛❞♦

t♦t❛❧♠❡♥t❡ ❞❛ ✉♥✐ã♦✱ ❛ss✐♠ s❡♥❞♦✱ ❡❧❡ ♣r❡❝✐s❛ s❡r ✐♥❝❧✉í❞♦ ♥♦✈❛♠❡♥t❡✱ ❞❛í

A∪B ∪C ={0,1,2,3,4,5,6,7,8,9}possui 10elementos.

❊♥tã♦✱ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ três ❝♦♥❥✉♥t♦s é✿

|A∪B∪C|=|A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C|+|A∩B ∩C| ✭✶✳✷✮

= 5 + 6 + 7−2−3−4 + 1 = 10 elementos.

❆ ❢ór♠✉❧❛ ✭✶✳✷✮ é ❛ ❢ór♠✉❧❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣❛r❛ três ❝♦♥❥✉♥t♦s✳ ❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ♥❛ ✜❣✉r❛ 1.3.

❋✐❣✉r❛ ✶✳✸✿ ❯♥✐ã♦ ❡♥tr❡ três ❝♦♥❥✉♥t♦s✳

❊①❡♠♣❧♦ ✶✳✻ ◆✉♠❛ ❝❧❛ss❡ ❞❡ 30❛❧✉♥♦s✱ 14❢❛❧❛♠ ✐♥❣❧ês✱ 5❢❛❧❛♠ ❛❧❡♠ã♦ ❡7❢❛❧❛♠

❢r❛♥❝ês✳ ❙❛❜❡♥❞♦✲s❡ q✉❡ 3 ❢❛❧❛♠ ✐♥❣❧ês ❡ ❛❧❡♠ã♦✱ 2 ❢❛❧❛♠ ✐♥❣❧ês ❡ ❢r❛♥❝ês✱ 2 ❢❛❧❛♠

❢r❛♥❝ês ❡ ❛❧❡♠ã♦ ❡ q✉❡ 1 ❢❛❧❛ ❛s 3 ❧í♥❣✉❛s✱ ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦s

q✉❡ ❢❛❧❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡st❛s três ❧í♥❣✉❛s✳

P❛r❛ r❡s♦❧✈❡r♠♦s ❡st❡ ♣r♦❜❧❡♠❛ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①✲ ❝❧✉sã♦✳ ❉❡✈❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ✉♥✐ã♦ ❞♦s ❝♦♥❥✉♥t♦s ■✱ ❆✱ ❋✱ ❡♠ q✉❡ ■ r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ✐♥❣❧ês✱ ❆ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ❛❧❡♠ã♦ ❡ ❋ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛❧✉♥♦s q✉❡ ❢❛❧❛♠ ❢r❛♥❝ês✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s q✉❡✿

|I|= 14;

|A|= 5;

|F|= 7;

|I ∩A|= 3;

|I∩F|= 2;

|A∩F|= 2 e

|I∩A∩F|= 1.

❉❡st❛ ❢♦r♠❛✱

|I∪A∪F|=|I|+|A|+|F| − |I∩A| − |I ∩F| − |A∩F|+|I∩A∩F|

= 14 + 7 + 5−3−2−2 + 1 = 20.

P♦rt❛♥t♦ t❡♠♦s q✉❡ ✷✵ ❛❧✉♥♦s ❢❛❧❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ✐❞✐♦♠❛s✳

❊①❡♠♣❧♦ ✶✳✼ ◆❡st❡ ❡①❡♠♣❧♦✱ ❞❡✈❡♠♦s ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♠❡✲ ♥♦r❡s ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ 100 q✉❡ s❡❥❛♠ ♠ú❧t✐♣❧♦s ❞❡ 2✱ 3 ♦✉ 5✳

P❛r❛ ❝❤❡❣❛r♠♦s ❛ r❡s♣♦st❛✱ ❜❛st❛ ❝❛❧❝✉❧❛r♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ♠ú❧t✐♣❧♦s ❞❡ 2✱3 ♦✉5✳ P❛r❛ ✐st♦✱ ❛♣❧✐❝❛r❡♠♦s ♥♦✈❛♠❡♥t❡ ♦ Pr✐♥❝í♣✐♦

❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✳

|A2 ∪A3∪A5| = |A2|+|A3|+|A5| − |A2 ∩A3| − |A2∩A5| − |A3∩A5|+

+ |A2∩A3∩A5|

= 100 2 + 100 3 + 100 5 − 100 6 − 100 10 − 100 15 + 100 30

= 50 + 33 + 20−16−10−6 + 3 = 74.

▲♦❣♦✱ sã♦ 74♠ú❧t✐♣❧♦s ❞❡ 2✱ 3♦✉ 5♠❡♥♦r❡s ♦✉ ✐❣✉❛❧ ❛ 100✳

❊①❡♠♣❧♦ ✶✳✽ ❯♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❞✐♠❡♥sõ❡s 150× 324× 375 é ❢❡✐t♦ ❞❡ ❝✉❜♦s

✉♥✐tár✐♦s✳ P❡❧♦ ✐♥t❡r✐♦r ❞❡ q✉❛♥t♦s ❝✉❜♦s ✉♥✐tár✐♦s ❛ ❞✐❛❣♦♥❛❧ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♣❛ss❛❄

❊st❛ r❡s♦❧✉çã♦ é ✉♠❛ ❛❞❛♣t❛çã♦ ❞❡ ❬✶❪ ❡ ❞❡ ❬✻❪✱ ❛♥t❡s ❞❡ r❡s♦❧✈❡r♠♦s ❡st❡ ♣r♦✲ ❜❧❡♠❛✱ ❛♥❛❧✐s❛r❡♠♦s ✉♠❛ s✐t✉❛çã♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ q✉❡ ♥♦s ♣❡r♠✐t✐rá ❡♥t❡♥❞❡r ♠❡❧❤♦r ❛ s♦❧✉çã♦ ❞❡st❡ ❡①❡r❝í❝✐♦✳ ❈♦♥s✐❞❡r❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ s♦❧✉çã♦ ♣❛r❛ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡s20×8✳ ◆❛ ✜❣✉r❛1.4✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ♦ r❡tâ♥❣✉❧♦ ❡stá ❞✐✈✐❞✐❞♦

❡♠ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ♦ q✉❡ ♥♦s ❢♦r♥❡❝❡ 20 r❡t❛s ✈❡rt✐❝❛✐s ❡ 8 r❡t❛s ❤♦r✐③♦♥t❛✐s✳

❆ss✐♠✱ ♣♦❞❡rí❛♠♦s ✐♠❛❣✐♥❛r q✉❡ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s q✉❡ ❛ ❞✐❛❣♦♥❛❧ ♣❛ss❛r✐❛ s❡r✐❛ ❛ s♦♠❛ ❞❡ss❛s r❡t❛s✳

❆❝♦♥t❡❝❡ q✉❡ ♣❛rt✐♥❞♦ ❞♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❞♦ r❡tâ♥❣✉❧♦(0,0) ❛♦ ♣♦♥t♦ ✜♥❛❧ ✭✷✵✱✽✮✱

❛ ❞✐❛❣♦♥❛❧ ❛t✐♥❣❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❛s r❡t❛s ❤♦r✐③♦♥t❛❧ ❡ ✈❡rt✐❝❛❧ ❡♠ q✉❛tr♦ ♣♦♥t♦s ♦s q✉❛✐s sã♦✿ (5,2)✱ (10,4)✱(15,6)❡ (20,8)✳ ❈♦♠♦ ♥❡st❡s q✉❛❞r❛❞♦s ❛ ❞✐❛❣♦♥❛❧ ❛t✐♥❣❡

❛♦ ♠❡s♠♦ t❡♠♣♦ ❛s ❞✉❛s r❡t❛s✱ ❡❧❛s ♥ã♦ ♣♦❞❡r✐❛♠ t❡r s✐❞♦ ❝♦♥t❛❞❛s ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ❞❡✈❡♠♦s s✉❜tr❛✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉❛❞r❛❞♦s q✉❡ t❛❧ ❢❛t♦ ♦❝♦rr❡✳

❋✐❣✉r❛ ✶✳✹✿ ❘❡tâ♥❣✉❧♦ ✷✵①✽✳

❊st❛ q✉❛♥t✐❞❛❞❡✱ r❡❢❡r❡✲s❡ ❡①❛t❛♠❡♥t❡ ❛♦ ♠❞❝ ❡♥tr❡ ❛s ❞✐♠❡♥sõ❡s ❞♦ r❡tâ♥❣✉❧♦✳ P❛r❛ ♦ ❡①❡♠♣❧♦✱ t❡♠♦s mdc(8,20) = 4✳ ❖❜t❡♠♦s ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣r✐♠❡✐r♦ ♣♦♥t♦

❞✐✈✐❞✐♥❞♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ ✜♥❛❧ ♣❡❧♦ ♠❞❝✳ ❖s ❞❡♠❛✐s ♣♦♥t♦s s❡rã♦ ❡♥❝♦♥✲ tr❛❞♦s ♠✉❧t✐♣❧✐❝❛♥❞♦✲♦ ♣♦r k✱ ❝♦♠ k ∈ Z✱ t❛❧ q✉❡ _{k = 1,2, ...,(mdc−1)}✳ ❆ss✐♠✱

❞❡♥♦♠✐♥❛♥❞♦ ❞❡✿

❱ ♦ ❝♦♥❥✉♥t♦ ❞❛s r❡t❛s ✈❡rt✐❝❛✐s✱ t❡♠♦s |V| = 20❀

❍ ♦ ❝♦♥❥✉♥t♦ ❞❛s r❡t❛s ❤♦r✐③♦♥t❛✐s✱ t❡♠♦s |H| = 8❀

■ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧❛s ✐♥t❡rs❡çõ❡s ❡♥tr❡ ❛s r❡t❛s ❤♦r✐③♦♥t❛✐s ❡ ✈❡rt✐❝❛✐s✱ t❡✲ ♠♦s |I|= 4✳

▲♦❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ✉♥✐tár✐♦s q✉❡ ❛ ❞✐❛❣♦♥❛❧ ♣❛ss❛ é ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s ❱ ❡ ❍✱ ♦✉ s❡❥❛✱

|V ∪H|=|V|+|H| − |I|

= 20 + 8−4 = 24.

P♦rt❛♥t♦✱ t❡♠♦s ✉♠ t♦t❛❧ ❞❡ 24q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ❛t✐♥❣✐❞♦s ♣❡❧❛ ❞✐❛❣♦♥❛❧✳

●❡♥❡r❛❧✐③❛♥❞♦✱ ♣❛r❛ r❡tâ♥❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡sa ❡ b✱ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ✉♥✐✲

tár✐♦s q✉❡ ❛ ❞✐❛❣♦♥❛❧ ♣❛ss❛ é ❞❛❞♦ ♣♦r a + b − mdc✭a, b✮✳

❉❡ ♠♦❞♦ ❣❡r❛❧✱ ♣♦❞❡♠♦s t♦♠❛r ❝♦♠♦ ❜❛s❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞♦ t✐♣♦m×n×p✳

■♠❛❣✐♥❡♠♦s ❡st❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦rt❛❞♦ ❡♠ m ❢❛t✐❛s ❧♦♥❣✐t✉❞✐♥❛✐s ✱ n ❢❛t✐❛s tr❛♥s✲

✈❡rs❛✐s ❡p❢❛t✐❛s ✈❡rt✐❝❛✐s✳ ❆ ❞✐❛❣♦♥❛❧ q✉❡ ♣❛rt❡ ❞♦ ♣♦♥t♦(0,0,0)❛♦ ♣♦♥t♦(m, n, p)

t❡♦r✐❝❛♠❡♥t❡ ❝♦rt❛r✐❛ ❝❛❞❛ ✉♠❛ ❞❛s ❢❛t✐❛sm✱ n❡p✱ ❡♥tã♦ t❡rí❛♠♦sm+n+p❢❛t✐❛s✳

P♦ré♠ ❛ ❞✐❛❣♦♥❛❧ ♣♦❞❡rá ❝♦rt❛r ♦ ✐♥t❡r✐♦r ❞♦s ❝✉❜♦s ✉♥✐tár✐♦s ❞❛s s❡❣✉✐♥t❡s ♠❛♥❡✐r❛s✿ Pr✐♠❡✐r❛♠❡♥t❡ ♥❛ ❧♦♥❣✐t✉❞✐♥❛❧✱ ♥❛ tr❛♥s✈❡rs❛❧✱ ♦✉ ♥❛ ✈❡rt✐❝❛❧✱ ❡♠ ❞✉❛s ❞❡ss❛s ❛♦ ♠❡s♠♦ t❡♠♣♦ ♦✉ ❛✐♥❞❛ ♥❛s três s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❆♦ ❞❡s✐❣♥❛r♠♦sL❝♦♠♦ ❝♦♥❥✉♥t♦

❞♦s ❝✉❜♦s ❝♦rt❛❞♦s ♣r✐♠❡✐r❛♠❡♥t❡ ♥❛ ❧♦♥❣✐t✉❞✐♥❛❧✱T ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥✐❝✐❛❧♠❡♥t❡

❝♦rt❛❞♦s ♥❛ tr❛♥s✈❡rs❛❧ ❡ ♣♦rV ❛q✉❡❧❡s ❡♠ q✉❡ ♦ ❝♦rt❡ ♦❝♦rr❡ ♥❛ ✈❡rt✐❝❛❧ ♣♦r ♣r✐♠❡✐r♦

❡ ❝❛❧❝✉❧❛♥❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❡♥tr❡ ❡ss❡s ❝♦♥❥✉♥t♦s✱ ♦❜t❡♠♦s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝✉❜♦s ✉♥✐tár✐♦s ❝♦rt❛❞♦s ♣❡❧❛ ❞✐❛❣♦♥❛❧✳

P❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✱ t❡♠♦s q✉❡✿

|L∪T ∪V|=|L|+|T|+|V| − |L∩T| − |L∩V| − |T ∩V|+|L∩T ∩V|.

❆♣❧✐❝❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❞♦ ❝❛s♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❝❛❞❛ ✐♥t❡rs❡çã♦ r❡♣r❡s❡♥t❛ ✉♠ mdc ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s✳ ❊♥tã♦ ♣❛r❛ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦

❣❡♥ér✐❝♦ m×n×p✱ ♦ r❡s✉❧t❛❞♦ é ❞❛❞♦ ♣♦r

|L∪T ∪V|=m+n+p−mdc(m, n)−mdc(m, p)−mdc(n, p) +mdc(m, n, p).

❈♦♠♦ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ s♦❧✐❝✐t❛❞♦ t❡♠ ❞✐♠❡♥sõ❡s150×324×375✱ ♣♦❞❡♠♦s s✉♣♦r

q✉❡L♣♦ss✉✐150❡❧❡♠❡♥t♦s✱T ♣♦ss✉✐324❡❧❡♠❡♥t♦s ❡V ♣♦ss✉✐375❡❧❡♠❡♥t♦s✱ ❡♥tã♦✱

|L∪T ∪V|= 150 + 324 + 375−mdc(150,324)−mdc(150,375)−mdc(324,375) + +mdc(150,324,375)

= 150 + 324 + 375−6−75−3 + 3 = 768,

✐st♦ é✱ ❛ ❞✐❛❣♦♥❛❧ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♣❛ss❛ ♣❡❧♦ ✐♥t❡r✐♦r ❞❡ 768 ❝✉❜♦s ✉♥✐tár✐♦s✳

❈❛♣ít✉❧♦ ✷

❯♠❛ ✈❛r✐❛♥t❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❛

■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ❡ ❛❧❣✉♠❛s

❛♣❧✐❝❛çõ❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✉t✐❧✐③❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦ ♣❛r❛ ✉♥✐ã♦ ❡♥tr❡ q✉❛tr♦ ❝♦♥❥✉♥t♦s✳ ❊♠ s❡❣✉✐❞❛✱ ✐r❡♠♦s ❣❡♥❡r❛❧✐③❛r ❛ ❢ór♠✉❧❛ ♣❛r❛ ❡st❡ Pr✐♥❝í♣✐♦✳ ❱❡r❡♠♦s t❛♠❜é♠ q✉❡ ♦ ♠❡s♠♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s t✐♣♦s ❞❡ ❡q✉❛çõ❡s✳

❊①❡♠♣❧♦ ✷✳✶ ❙❡❥❛♠ ♦s ❝♦♥❥✉♥t♦sA={1,2,4,5,7,8,9,14}, B ={2,3,5,6,9,10,11,14}✱

C = {4,5,6,8,9,10,12,15} ❡ ❉ = {7,8,9,10,11,12,13,14}. ◗✉❛❧ ❛ q✉❛♥t✐❞❛❞❡ ❞❡

❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ss❡s q✉❛tr♦ ❝♦♥❥✉♥t♦s❄

❱❛♠♦s t❡♥t❛r ❞❡s❝♦❜r✐r✱ ✉t✐❧✐③❛♥❞♦ ✐❞❡✐❛s ❞♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ✉♠❛ ❢ór♠✉❧❛ q✉❡ ♥♦s ❢♦r♥❡ç❛ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ♣❛r❛ q✉❛tr♦ ❝♦♥❥✉♥t♦s✳ ❈♦♠ ♦s q✉❛tr♦ ❝♦♥❥✉♥t♦s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛s s❡❣✉✐♥t❡s ✐♥t❡rs❡çõ❡s✿

❊♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✿ ✶✳ A∩B ={2,5,9,14}❀

✷✳ A∩C ={4,5,8,9}❀

✸✳ A∩D={7,8,9,14}❀

✹✳ B∩C ={5,6,9,10}❀

✺✳ B∩D={9,10,11,14}❀

✻✳ C∩D={8,9,10,12}✳

❊♥tr❡ três ❝♦♥❥✉♥t♦s✿ ✶✳ A∩B ∩C ={5,9}❀

✷✳ A∩B ∩D❂{9,14}❀

✸✳ A∩C∩D={8,9}❀

✹✳ B∩C∩D={9,10}✳

❊♥tr❡ ♦s q✉❛tr♦ ❝♦♥❥✉♥t♦s✿ ✶✳ A∩B ∩C∩D={9}✳

❙❡❣✉✐♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞♦✐s ❡ três ❝♦♥❥✉♥t♦s✱ ❞❡✈❡♠♦s✿

• ■♥❝❧✉✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝❛❞❛ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s❀

• ❊①❝❧✉✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛s ✐♥t❡rs❡çõ❡s ❞❡ ❞♦✐s ❛ ❞♦✐s ❝♦♥❥✉♥t♦s❀ • ■♥❝❧✉✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛s ✐♥t❡rs❡çõ❡s ❞❡ três ❛ três ❝♦♥❥✉♥t♦s❀ • ❊①❝❧✉✐r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✐♥t❡rs❡çã♦ ❞♦s q✉❛tr♦ ❝♦♥❥✉♥t♦s✳

❱❡❥❛♠♦s ♣♦rq✉❡ ❡st❡ ú❧t✐♠♦ ♣❛ss♦✱ ❤❛❥❛ ✈✐st♦ q✉❡ ♦s três ♣r✐♠❡✐r♦s ♣❛ss♦s ❥á ❢♦r❛♠ ❡①♣❧✐❝❛❞♦s✳ ❆♦ s♦♠❛r♠♦s ♦s ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✱ ✐♥❝❧✉í✲ ♠♦s q✉❛tr♦ ✈❡③❡s ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s q✉❛tr♦ ❝♦♥❥✉♥t♦s✳ ❊♠ s❡❣✉✐❞❛✱ ❡ss❛ q✉❛♥t✐❞❛❞❡ é ❡①❝❧✉í❞❛ s❡✐s ✈❡③❡s ♥♦ s❡❣✉♥❞♦ ♣❛ss♦ ✭❛tr❛✈és ❞❛s ✐♥✲ t❡rs❡çõ❡s ❞♦✐s ❛ ❞♦✐s✮✳ ◆❛ t❡r❝❡✐r❛ ❡t❛♣❛ ✈♦❧t❛♠♦s ❛ ✐♥❝❧✉✐r q✉❛tr♦ ✈❡③❡s ❡st❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ✭❛tr❛✈és ❞❛s ✐♥t❡rs❡çõ❡s três ❛ três✮✳

❆té ❡st❡ ♠♦♠❡♥t♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❝♦♥t❛❜✐❧✐❞❛❞❡

4−6 + 4 = 2,

✐st♦ é✱ t❡♠♦s ✉♠❛ q✉❛♥t✐❞❛❞❡ ❛ ♠❛✐s ❞❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦s q✉❛tr♦ ❝♦♥❥✉♥t♦s ❛ q✉❛❧ é r❡t✐r❛❞❛ ♥♦ q✉❛rt♦ ♣❛ss♦✳ ❆ss✐♠✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ q✉❛tr♦ ❝♦♥❥✉♥t♦s é ❞❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛✿

|A∪B ∪C∪D|= |A|+|B|+|C|+|D| −

− |A∩B| − |A∩C| − |A∩D| − |B∩C| − |B∩D| − |C∩D| +

+ |A∩B∩C|+|A∩B∩D|+|A∩C∩D|+|B∩C∩D| −

− |A∩B∩C∩D|. ✭✷✳✶✮

❖❜s❡r✈❡ ❛ ✐❧✉str❛çã♦ ♥❛ ✜❣✉r❛ 2.1✳