MÉTODOS DE CONTAGEM

▼ét♦❞♦s ❞❡ ❈♦♥t❛❣❡♠

♣♦r

▲✉✐s ❘♦❞r✐❣♦ ❉✬❆♥❞r❛❞❛ ❇❡③❡rr❛

♣♦r

▲✉✐s ❘♦❞r✐❣♦ ❉✬❆♥❞r❛❞❛ ❇❡③❡rr❛

s♦❜ ♦r✐❡♥t❛çã♦ ❞❛

Pr♦❢

_{✳ ❉r}

_{✳ ❊❧✐s❛♥❞r❛ ❞❡ ❋át✐♠❛ ●❧♦ss ❞❡ ▼♦r❛❡s}

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣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♦ ❞❡}

Pr✐♠❡✐r❛♠❡♥t❡✱ ♠❛♥✐❢❡st♦ t♦❞♦ ♦ ♠❡✉ ❝❛r✐♥❤♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✿ ✐r♠ã♦s✱ ❛✈ós✱ t✐♦s✱ ♣r✐♠♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♠✐♥❤❛ ♠ã❡ ❡ ♦ ♠❡✉ ♣❛✐✱ q✉❡ s❡♠♣r❡ ♠❡ ❝♦♥❝❡❞❡r❛♠ ❡str✉t✉r❛ ❡ tr❛♥q✉✐❧✐❞❛❞❡ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝♦♥❝❧✉✐r ♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ♦ s✉♣❡r✐♦r ❝♦♠ s✉❝❡ss♦✳

❉❡❞✐❝❛tór✐❛

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✱ ♥ã♦ ❛♣❡♥❛s ❛tr❛✈és ❞♦s ❝♦♥❝❡✐t♦s tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛❜♦r❞❛❞♦s ❡♠ ❝✉rs♦s ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛✱ t❛✐s ❝♦♠♦ ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s✱ ❛s ♣❡r♠✉t❛çõ❡s✱ ♦s ❛rr❛♥❥♦s✱ ❛s ❝♦♠❜✐♥❛çõ❡s✱ ❛s ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✉♥✐tár✐♦s ❡ ♦✉tr♦s✱ ♠❛s t❛♠❜é♠✱ ❢❡rr❛♠❡♥t❛s s♦✜st✐❝❛❞❛s ❞❡ ❝♦♥t❛❣❡♠✱ t❛❧ ❝♦♠♦ ♦ ✉s♦ ❞❡ ❣r❛❢♦s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠✱ ❈♦♠❜✐♥❛tór✐❛✱ ●r❛❢♦s✳

❆❜str❛❝t

❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ ❝♦✉♥t✐♥❣ ♣r♦❜❧❡♠s✱ ♥♦t ❥✉st t❤r♦✉❣❤ t❤❡ ❝♦♥❝❡♣ts tr❛❞✐t✐♦♥❛❧❧② ❝♦✈❡r❡❞ ✐♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❆♥❛❧②s✐s ❝♦✉rs❡s✱ s✉❝❤ ❛s t❤❡ ❜❛s✐❝ ♣r✐♥❝✐♣❧❡s✱ ♣❡r♠✉t❛t✐♦♥s✱ ❛rr❛♥❣❡♠❡♥ts✱ ❝♦♠❜✐♥❛t✐♦♥s✱ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ✇✐t❤ ✉♥✐t❛r② ❝♦❡✣❝✐❡♥ts✱ ❛♠♦♥❣ ♦t❤❡rs✱ ❜✉t ❛❧s♦ ✉s✐♥❣ s♦♣❤✐st✐❝❛t❡❞ t♦♦❧s✱ s✉❝❤ ❛s t❤❡ ✉s❡ ♦❢ ❣r❛♣❤s✳

❑❡②✇♦r❞s✿ ❈♦✉♥t✐♥❣ Pr♦❜❧❡♠s✱ ❈♦♠❜✐♥❛t♦r✐❛❧✱ ●r❛♣❤s✳

■♥tr♦❞✉çã♦ ✶

✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❡ ❈♦♠❜✐♥❛tór✐❛ ✸ ✶✳✶ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹ P❡r♠✉t❛çõ❡s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✺ P❡r♠✉t❛çõ❡s ❈✐r❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✻ ❈♦♠❜✐♥❛çã♦ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✼ P❡r♠✉t❛çõ❡s ❝♦♠ ❘❡♣❡t✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✶✳✽ ❆♣❧✐❝❛çõ❡s ❞♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✽✳✶ ❆rr❛♥❥♦ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✽✳✷ ❆rr❛♥❥♦ ❈♦♠♣❧❡t♦ ✭❝♦♠ r❡♣❡t✐çã♦✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✳✽✳✸ P❡r♠✉t❛çã♦ ❈❛ót✐❝❛s✭❞❡s❛rr❛♥❥♦s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✶✳✽✳✹ ❊q✉❛çõ❡s ▲✐♥❡❛r❡s ♥♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✶✳✽✳✺ ❊q✉❛çõ❡s ▲✐♥❡❛r❡s ♥♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✹✹ ✶✳✽✳✻ ❑❛♣❧❛♥s❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✷ ■♥tr♦❞✉çã♦ à t❡♦r✐❛ ❞♦s ●r❛❢♦s ✺✵

✷✳✶ ●r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✷ ❈♦❧♦r✐♥❞♦ ♠❛♣❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✷✳✸ Pr♦❜❧❡♠❛ ❞❛s ♣♦♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✷✳✹ ❖ ♣r♦❜❧❡♠❛ ❞❛s ❧✐❣❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✺ ❆t✐✈✐❞❛❞❡s s✉❣❡r✐❞❛s ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

❆ ❆♣ê♥❞✐❝❡ ✽✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✽✸

❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❈♦♠❜✐♥❛tór✐❛ ♦✉ ❛✐♥❞❛ ❞❡ ▼ét♦❞♦s ❞❡ ❈♦♥t❛❣❡♠✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ s❡❥❛ ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ q✉❡ ♦ ❝✐❞❛❞ã♦ ✐♥s❡r✐❞♦ ♥♦ ♠✉♥❞♦ ❞❛s ✐♥❢♦r♠❛çõ❡s✱ ❞❛s ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s ❡ ❞♦ ❞✐❛✲❛✲❞✐❛ ❞❛s tr❛♥s❛çõ❡s ✜♥❛♥❝❡✐r❛s ♥❡❝❡ss✐t❛ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s r❡❛✐s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ❡♥s✐♥♦ ❞❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ♥ã♦ ❡stá s❡♥❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❞❡ ❢♦r♠❛ ❛ ♣♦ss✐❜✐❧✐t❛r q✉❡ ♦s ❝✐❞❛❞ã♦s ❛ ✉t✐❧✐③❡♠ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s r❡❛✐s✳ ❖ ❡♥s✐♥♦ ❡s❝♦❧❛r ❧✐♠✐t❛✲s❡ q✉❛s❡ s❡♠♣r❡ ❛♦ tr❡✐♥❛♠❡♥t♦ ♥♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ❡ ❛❧❣♦r✐t♠♦s ♣❛r❛ ❡♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ ❞❡ ❛rr❛♥❥♦✱ ❝♦♠❜✐♥❛çõ❡s ♦✉ ♣❡r♠✉t❛çõ❡s✳ P❛r❛ ▼♦r❣❛❞♦ ❬✸❪✱ ❡♠❜♦r❛ ❝♦♠❜✐♥❛çõ❡s✱ ❛rr❛♥❥♦s ❡ ♣❡r♠✉t❛çõ❡s ❢❛ç❛♠ ♣❛rt❡ ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛✱ sã♦ ❝♦♥❝❡✐t♦s q✉❡ ♣❡r♠✐t❡♠ r❡s♦❧✈❡r ✉♠❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s✿ ♦s ❞❡ ❝♦♥t❛❣❡♠ ❞❡ ❝❡rt♦s t✐♣♦s ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ s❡♠ q✉❡ s❡❥❛ ♥❡❝❡ssár✐♦ ❡♥✉♠❡r❛r s❡✉s ❡❧❡♠❡♥t♦s✳ ◆♦ ❡♥t❛♥t♦ tr❛t❛r❡♠♦s ♥❡ss❡ ❚❈❈ ❞❡ ♦✉tr❛s té❝♥✐❝❛s ❞❡ ❝♦♥t❛❣❡♠✱ ❛❧❣✉♠❛s ❞❡ ❝❡rt❛ ❢♦r♠❛ ❜❡♠ s♦✜st✐❝❛❞❛s✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞♦ ✉s♦ ❞❡ ❣r❛❢♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✳

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ ❚❈❈ é ❡st✉❞❛r ❛❧❣✉♠❛s ❞❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ❞❡ ❝♦♥t❛❣❡♠✱ ♣r✐♦r✐③❛♥❞♦ ❛s ❛♣❧✐❝❛çõ❡s ♥❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s✳ ❊s♣❡r❛♠♦s ❛ss✐♠✱ q✉❡ ♦ ♠❡s♠♦ ♣♦ss❛ s❡r✈✐r ❝♦♠♦ ❢♦♥t❡ ❞❡ ❝♦♥s✉❧t❛ ♣❛r❛ ♣r♦❢❡ss♦r❡s✱ ❡st✉❞❛♥t❡s ❡ ❛❞♠✐r❛❞♦r❡s ❞❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛✳ ◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦

■♥tr♦❞✉çã♦

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

◆♦ ❈❛♣ít✉❧♦ ■✱ ❛♣ós ❛❧❣✉♠❛s ♥♦t❛çõ❡s ❡ ❞❡✜♥✐çõ❡s✱ ❛♣r❡s❡♥t❛♠♦s ❞♦✐s ♣r✐♥❝í♣✐♦s✿ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❡ ♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ✐♥tr♦❞✉③✐❞♦s ❛tr❛✈és ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣r♦❜❧❡♠❛s r❡s♦❧✈✐❞♦s✳ P♦st❡r✐♦r♠❡♥t❡ ❢♦r❛♠ ❝♦♥str✉í❞♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣❡r♠✉t❛çã♦ s✐♠♣❧❡s✱ ♣❡r♠✉t❛çã♦ ❝✐r❝✉❧❛r✱ ❝♦♠❜✐♥❛çã♦ s✐♠♣❧❡s ❡ ♣❡r♠✉t❛çã♦ ❝♦♠ r❡♣❡t✐çã♦✳ ❊♠ s❡❣✉✐❞❛✱ ❡st✉❞❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞♦s ❝♦♥❝❡✐t♦s ❛♥t❡r✐♦r♠❡♥t❡ ❧✐st❛❞♦s✱ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ r♦t✉❧❛❞❛s ❞❡ ♣r♦❜❧❡♠❛s ❞❡✿ ❆rr❛♥❥♦ ❙✐♠♣❧❡s✱ ❆rr❛♥❥♦ ❈♦♠♣❧❡t♦✱ P❡r♠✉t❛çã♦ ❈❛ót✐❝❛s✱ ♥ú♠❡r♦ ❞❡ s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❧✐♥❡❛r ♥♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♦✉ ♥♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s✱ ✶♦ _{▲❡♠❛ ❞❡ ❑❛♣❧❛♥s❦✱ ✷}♦ _{▲❡♠❛ ❞❡ ❑❛♣❧❛♥s❦ ❡ t❡r♠♦ ❣❡r❛❧ ❞♦}

❇✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✳

◆♦ ❈❛♣ít✉❧♦ ■■✱ tr❛t❛r❡♠♦s ❞❡ três ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ ❞✐s❝r❡t❛✱ ♦ ❞❡ ❝♦❧♦r❛çã♦ ❞❡ ♠❛♣❛s✱ ♦ ♣r♦❜❧❡♠❛ ❞❛s ♣♦♥t❡s ❡ ♦ ♣r♦❜❧❡♠❛ ❞❛s ❧✐❣❛çõ❡s ❞❡ á❣✉❛✱ ❧✉③ ❡ t❡❧❡❢♦♥❡✳ ❙❡rá ❞❛❞♦ ✉♠ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦ ❞❛ t❡♦r✐❛ ❞♦s ❣r❛❢♦s q✉❡ s❡r✈✐rá ❝♦♠♦ s✉❜sí❞✐♦ ♣❛r❛ ❛s r❡s♦❧✉çõ❡s ❞♦s três ♣r♦❜❧❡♠❛s ❛♣r❡s❡♥t❛❞♦s✳ ◆♦ ✜♥❛❧ ♣r♦♣♦♠♦s ❞✉❛s ❛t✐✈✐❞❛❞❡s ♣❛r❛ s❡r❡♠ ❛♣❧✐❝❛❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳

❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❞❡

❈♦♠❜✐♥❛tór✐❛

❊st✉❞❛r❡♠♦s ♥❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡ss❡ ❝❛♣ít✉❧♦ ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❞❡ ❝♦♠❜✐♥❛tór✐❛ t❛✐s ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦✱ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ♣❡r♠✉t❛çã♦ ✭s✐♠♣❧❡s✱ ❝✐r❝✉❧❛r ❡ ❝♦♠ r❡♣❡t✐çã♦✮ ❡ ❝♦♠❜✐♥❛çã♦ s✐♠♣❧❡s✱ q✉❡ s❡r✈✐rã♦ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s ❞❡ ❝♦♥str✉çã♦ ❞♦s ❞❡♠❛✐s tó♣✐❝♦s ❞❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ q✉❡ ❡♥❝♦♥tr❛r❡♠♦s ❡♠ s❡q✉ê♥❝✐❛ ♥❡ss❡ ❝❛♣ít✉❧♦✳ P❛r❛ q✉❡ ❝❛❞❛ ❝♦♥❝❡✐t♦ s❡❥❛ ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r❡❡♥❞✐❞♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ r❡s♦❧✈❡r❡♠♦s ♣r♦❜❧❡♠❛s ❡s♣❡❝í✜❝♦s q✉❡ ❢✉♥❝✐♦♥❛rã♦ ❝♦♠♦ ❢❛❝✐❧✐t❛❞♦r❡s ❞♦s ❝❛s♦s ❣❡r❛✐s✱ ❡①♣❧♦r❛♥❞♦✲♦s ♣♦r ♠❡✐♦ ❞❡ ❝♦♥t❡①t✉❛❧✐③❛çã♦✱ ✈❛❧♦r✐③❛♥❞♦ ❛ss✐♠ ♦ r❛❝✐♦❝í♥✐♦ ❡ ❛s ✐❞❡✐❛s ❣❡r❛✐s ❛♦ ✐♥✈és ❞♦ ✉s♦ ❡①❝❡ss✐✈♦ ❞❡ ❢ór♠✉❧❛s ❡ ❞❡ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✳ ◆ã♦ ❡st❛♠♦s ❛q✉✐ ♣r❡♦❝✉♣❛❞♦s ❡♠ ❝r✐❛r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❢ór♠✉❧❛s✱ ✉♠❛ ♣❛r❛ ❝❛❞❛ ♠♦❞❡❧♦ ❞❡ q✉❡stã♦✱ ❛♣❡♥❛s ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s t❡rã♦ s✉❛s ❡①♣r❡ssõ❡s ❣❡♥❡r❛❧✐③❛❞❛s✱ ♦s ❞❡♠❛✐s ❝♦♥❝❡✐t♦s s❡rã♦ tr❛t❛❞♦s ❝♦♠♦ ❛♣❧✐❝❛çõ❡s ❞✐r❡t❛s✳ ❉❡✈❡♠♦s ❡♥tã♦ ♥❡ss❡ ♠♦♠❡♥t♦ r❡s✐st✐r ❛ t❡♥t❛çã♦ ❞❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦s ❢❛t♦s✱ tã♦ ♣r❛③❡r♦s♦ ♣❛r❛ ♥ós ♠❛t❡♠át✐❝♦s ♠❛s ❢❛t❛❧♠❡♥t❡ s❡♠ ♦ ♠❡s♠♦ ❡❢❡✐t♦ ♣❛r❛ ❜♦❛ ♣❛rt❡ ❞♦s ❛❧✉♥♦s ❞❡ ✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ❞♦ ❡♥s✐♥♦

Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦ ❈❛♣ít✉❧♦ ✶

♠é❞✐♦ ❡①tr❡♠❛♠❡♥t❡ ❤❡t❡r♦❣ê♥❡❛✳ ❆✜♥❛❧✱ ❛✐♥❞❛ ✐♥s✐st♦ ❡♠ ❞✐③❡r q✉❡ ♥ã♦ q✉❡r❡♠♦s ❛q✉✐ tr♦❝❛r ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❞❛ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ♣♦r ❢ór♠✉❧❛s ❛ss♦❝✐❛❞❛s ❛♦s ♠♦❞❡❧♦s ❞❡ q✉❡stõ❡s✳ ❘❡s♦❧✈❡r❡♠♦s ❛ s❡❣✉✐r ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥t❛❣❡♠✱ ❢❛③❡♥❞♦ ✉s♦ ❞❡ ✉♠ ❞♦s ♣r✐♥❝í♣✐♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❈♦♠❜✐♥❛tór✐❛✳ P❛r❛ ❡ss❡ tr❛❜❛❧❤♦✱ ✉s❛r❡♠♦s ❛ ♥♦t❛çã♦ |A| ♣❛r❛ ✐♥❞✐❝❛r ❛

q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦ A ♣♦ss✉✐✳

✶✳✶ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦

❖❜s❡r✈❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

❊①❡♠♣❧♦ ✶ ◆✉♠ s♦rt❡✐♦ ❞❡ ✉♠❛ ♣r♦♠♦çã♦ ❞❛ rá❞✐♦ ◆♦✈❛ ❘❡❝✐❢❡✱ ❇✐r❛ ❢♦✐ ❝♦♥t❡♠♣❧❛❞♦ ❝♦♠ ✉♠ ♣❛r ❞❡ ❝♦♥✈✐t❡s ♣❛r❛ ♦ ❋♦rró ❞❛ ❈❛♣✐tá✱ ♣❛r❛ ❧❤❡ ❛❝♦♠♣❛♥❤❛r ♥❡ss❡ ❡✈❡♥t♦ ❡❧❡ ❞✐s♣õ❡ ❞❡ s❡✉s ✸ ✐r♠ã♦s ❡ ❞❡ ✺ ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦✳ ❉❡ q✉❛♥t❛s ❢♦r♠❛s ❞✐❢❡r❡♥t❡s ❡❧❡ ♣♦❞❡ ✐r ❛❝♦♠♣❛♥❤❛❞♦❄

❙♦❧✉çã♦✿ ❙❡ ❇✐r❛ ✐rá ❛❝♦♠♣❛♥❤❛❞♦ ❞❡ ✉♠❛ ú♥✐❝❛ ♣❡ss♦❛✱ ❡❧❡ ♣♦❞❡ ✐r ❛❝♦♠♣❛♥❤❛❞♦ ❞♦ s❡✉ ✶♦ _{✐r♠ã♦ ♦✉ ❞♦ s❡✉ ✷}♦ _{✐r♠ã♦ ♦✉ ❞♦ s❡✉ ✸}♦ _{✐r♠ã♦ ♦✉}

❛✐♥❞❛✱ ❞♦ s❡✉ ✶♦ _{❝♦❧❡❣❛ ♦✉ ❞♦ s❡✉ ✷}♦ _{❝♦❧❡❣❛ ♦✉ ❞♦ s❡✉ ✸}♦ _{❝♦❧❡❣❛ ♦✉ ❞♦ s❡✉}

✹♦ _{❝♦❧❡❣❛ ♦✉ ❞♦ s❡✉ ✺}♦ _{❝♦❧❡❣❛✳ ❚❡♠♦s ❡♥tã♦ ✽ ❝♦♠♣❛♥❤✐❛s ❞✐st✐♥t❛s ♣❛r❛ ✐r}

❛♦ ❋♦rró ❞❛ ❈❛♣✐tá ❝♦♠ ❇✐r❛✳

❖ ❡①❡♠♣❧♦ ❛❝✐♠❛ ✐❧✉str❛ ❜❡♠ ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦ ♣❛r❛ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✳ ◆♦t❡ q✉❡ ♥♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ♣❡r❣✉♥t❛ ✜♥❛❧ ♣♦r✿

✏❉❡ q✉❛♥t❛s ❢♦r♠❛s ❞✐❢❡r❡♥t❡s ❡❧❡ ♣♦❞❡ ✐r ❛❝♦♠♣❛♥❤❛❞♦ ❞❡ ✉♠ ❝♦❧❡❣❛ ❞❡ tr❛❜❛❧❤♦ ♦✉ ❞❡ ✉♠ ✐r♠ã♦❄✑

◆♦t❛✲s❡ q✉❡ ♦ ❝♦♥❡❝t✐✈♦ ✏♦✉✑ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞♦ ❛♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✱ s❡♥❞♦ ♣♦rt❛♥t♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♠♣♦rt❛♥t❡ ♥❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ss❡s ♣r♦❜❧❡♠❛s✳ P♦❞❡♠♦s ❛❣♦r❛ ❡♥tã♦ ❡♥✉♥❝✐❛r t❛❧ ♣r✐♥❝í♣✐♦✳

Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✿ ❙❡ ✉♠❛ ❞❡❝✐sã♦ ❆ ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡x♠❛♥❡✐r❛s✱ ✉♠❛

❞❡❝✐sã♦ ❇ ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ y ♠❛♥❡✐r❛s ❡ ❛s ❞❡❝✐sõ❡s sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✱

❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ s❡ t♦♠❛r❡♠ ❛s ❞❡❝✐sõ❡s ❆ ♦✉ ❇ éx+y✳

❯s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ ❝♦♥❥✉♥t♦s ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦ ❝♦♠♦ s❡❣✉❡✿

Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✿ ❙❡ ❆ ❡ ❇ sã♦ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✱ ❝♦♠ p ❡ q

❡❧❡♠❡♥t♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡♥tã♦ A∪B ♣♦ss✉✐p+q ❡❧❡♠❡♥t♦s✳

❆❣♦r❛ ✐r❡♠♦s ❣❡♥❡r❛❧✐③❛r ♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✱ ♣❛r❛ t❛♥t♦ ♦❜s❡r✈❡ ♦ ♣r♦❜❧❡♠❛ ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✷ ▼✐❣♥❛❝ ♦❢❡r❡❝❡✉ ❛ ▼❛r❝♦s ✸ ❧✐✈r♦s ❞❡ ➪❧❣❡❜r❛✱ ✼ ❧✐✈r♦s ❞❡ ❈♦♠❜✐♥❛tór✐❛ ❡ ✺ ❧✐✈r♦s ❞❡ ●❡♦♠❡tr✐❛ ❡ ♣❡❞✐✉✲❧❤❡ ♣❛r❛ ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ ❧✐✈r♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ▼❛r❝♦s ♣♦❞❡ r❡❛❧✐③❛r ❡ss❛ ❡s❝♦❧❤❛❄

❙♦❧✉çã♦✿ ❈♦♥s✐❞❡r❡ ♦s três ❝♦♥❥✉♥t♦s ❞❡ ❧✐✈r♦s✿

A={a₁, a₂, a₃}✱ ❝♦♥❥✉♥t♦ ❞♦s ❧✐✈r♦s ❞❡ ➪❧❣❡❜r❛✳

C ={c1, c2, c3, c4, c5, c6, c7}✱ ❝♦♥❥✉♥t♦ ❞♦s ❧✐✈r♦s ❞❡ ❈♦♠❜✐♥❛tór✐❛✳

G={g₁, g₂, g₃, g₄, g₅}✱ ❝♦♥❥✉♥t♦ ❞♦s ❧✐✈r♦s ❞❡ ●❡♦♠❡tr✐❛✳

▼❛r❝♦s ❞❡✈❡rá ❡s❝♦❧❤❡r ❛♣❡♥❛s ✉♠ ❧✐✈r♦ ❞❡♥tr❡ ♦s três ❝♦♥❥✉♥t♦s✱ ❝♦♠♦ t❡♠♦s ✸ ♦♣çõ❡s ❡♠ A✱ 7 ♦♣çõ❡s ❡♠ C ❡ ✺ ♦♣çõ❡s ❡♠ G✱ ♣♦rt❛♥t♦ ❛♦ t♦❞♦

♦❜t❡♠♦s ✶✺ ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛✳

P♦❞❡♠♦s ❛❣♦r❛ ❡♥✉♥❝✐❛r ♦ ❝❛s♦ ❣❡r❛❧ ❞♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✳

❊①t❡♥sã♦ ❞♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✿ ❙❡ A1, A2,· · ·, An sã♦ ❝♦♥❥✉♥t♦s

❞✐s❥✉♥t♦s ❞♦✐s ❛ ❞♦✐s ✭✐st♦ é Ai ∩Aj = ∅ ♣❛r❛ i 6= j✮✱ ❡ s❡ Ai ♣♦ss✉✐ ai

❡❧❡♠❡♥t♦s✱i= 1,2,3,· · · , n✱ ❡♥tã♦✿

|A₁∪A₂ ∪ · · · ∪An|=a1+a2+· · ·+an.

❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❈❛♣ít✉❧♦ ✶

✶✳✷ ❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦

◆❛ s❡çã♦ ❛♥t❡r✐♦r✱ ✜③❡♠♦s r❡❢❡rê♥❝✐❛ ❛ ✉♠ ♣r✐♥❝í♣✐♦ ❡❧❡♠❡♥t❛r ❞❡ ❝♦♥t❛❣❡♠ q✉❡ ❡st❛❜❡❧❡❝❡ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ✜♥✐t♦s é ❛ s♦♠❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝❛❞❛ ❝♦♥❥✉♥t♦✳ ◗✉❡r❡♠♦s ❛❣♦r❛ ❛♠♣❧✐❛r ♣❛r❛ ❝❛s♦s ♦♥❞❡ ♦s ❝♦♥❥✉♥t♦s ❡♥✈♦❧✈✐❞♦s ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐s❥✉♥t♦s✳ ❆❞♠✐t✐♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦ ♣♦❞❡♠♦s ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦✱ ❛♣❡s❛r ❞❡ ❜♦❛ ♣❛rt❡ ❞♦s ❧✐✈r♦s ❞❡♥♦♠✐♥á✲❧♦ Pr✐♥❝í♣✐♦ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦✳ ❙❡❣✉♥❞♦ ▼♦r❣❛❞♦ ❡t✳ ❛❧ ❬✸❪ ♦ Pr✐♥❝í♣✐♦ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ é ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ❝♦♥t❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ à ✉♥✐ã♦ ❞❡ ✈ár✐♦s ❝♦♥❥✉♥t♦s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐s❥✉♥t♦s✳ P❛r❛ ❡♥t❡♥❞❡r♠♦s ♠❡❧❤♦r ❞♦ q✉❡ s❡ tr❛t❛✱ ♦❜s❡r✈❡ ♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✸ ❊♠ ✉♠❛ ❝♦♦♣❡r❛t✐✈❛ ❞❡ ❛❣r✐❝✉❧t♦r❡s ❞♦ ♠✉♥✐❝í♣✐♦ ❞❡ ❱✐tór✐❛ ❞❡ ❙❛♥t♦ ❆♥tã♦✱ ✹✺ ❞♦s ❝♦♦♣❡r❛t✐✈❛❞♦s ❝✉❧t✐✈❛♠ ♦ s♦r❣♦✱ ✻✵ ❝✉❧t✐✈❛♠ ♦ ♠✐❧❤♦ ❡ ✸✵ ❝✉❧t✐✈❛♠ ❛♠❜♦s✳ ❙❛❜❡♥❞♦ q✉❡ t♦❞♦s ♦s ❝♦♦♣❡r❛t✐✈❛❞♦s ❝✉❧t✐✈❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ❞✉❛s ❝✉❧t✉r❛s✱ q✉❛❧ é ♦ ♥ú♠❡r♦ ❞❡ ❛❣r✐❝✉❧t♦r❡s ❞❛ ❝♦♦♣❡r❛t✐✈❛❄

❙♦❧✉çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ♣❡r❝❡❜❡♠♦s q✉❡ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦ ♥ã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ♥❡ss❡ ♣r♦❜❧❡♠❛✱ ♣♦✐s ♦s ❞♦✐s ❝♦♥❥✉♥t♦s ❡♥✈♦❧✈✐❞♦s ✭❞♦s ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ ♠✐❧❤♦ ❡ ❞♦s ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ s♦r❣♦✮ ♥ã♦ sã♦ ❞✐s❥✉♥t♦s✳ ❘❡s♦❧✈❡r❡♠♦s ♥♦ss♦ ♣r♦❜❧❡♠❛ ❞❡ ❞♦✐s ♠♦❞♦s ❞✐st✐♥t♦s✳

✭✐✮ ❈♦♥s✐❞❡r❡♠♦s ✉♠ ❞✐❛❣r❛♠❛✱ ♦♥❞❡ r❡♣r❡s❡♥t❛r❡♠♦s ♣❡❧❛ ❧❡tr❛ ❙ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ❝✉❧t✐✈❛♠ ♦ s♦r❣♦ ❡ ♣❡❧❛ ❧❡tr❛ ▼ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ❝✉❧t✐✈❛♠ ♠✐❧❤♦✱ s❛❜❡♠♦s q✉❡ ❤á ✸✵ ❛❣r✐❝✉❧t♦r❡s q✉❡ ❝✉❧t✐✈❛♠ ✏♠✐❧❤♦ ❡ s♦r❣♦✑✱ ❡♥tã♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❝♦❧♦❝❛r❡♠♦s ✸✵ ♥❛ ✐♥t❡rs❡çã♦ ❞♦s ❝♦♥❥✉♥t♦s✱

❋✐❣✉r❛ ✶✳✶✿

❖ s❡❣✉♥❞♦ ♣❛ss♦ é ♣r❡❡♥❝❤❡r♠♦s ❛s ♦✉tr❛s ❞✉❛s r❡❣✐õ❡s✳ ❈♦♠♦ t❡♠♦s ✻✵ ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ ♠✐❧❤♦✱ ❞❡✈❡♠♦s t❡r ❡ss❛ q✉❛♥t✐❞❛❞❡ ❞❡♥tr♦ ❞❛ r❡s♣❡❝t✐✈❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣♦ré♠ ❥á ❤á ✸✵ ♥❛ ✐♥t❡rs❡çã♦ ❞♦s ❝♦♥❥✉♥t♦s ✭❡ ♣♦rt❛♥t♦ ❞❡♥tr♦ ❞♦ ❝♦♥❥✉♥t♦ ▼✮ r❡st❛♥❞♦ ❛♣❡♥❛s ✸✵ ♣❛r❛ ❛ r❡❣✐ã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦s ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ ❛♣❡♥❛s ♠✐❧❤♦✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ❥á q✉❡ t❡♠♦s ✹✺ ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ s♦r❣♦✱ ❞♦s q✉❛✐s ✸✵ t❛♠❜é♠ ♣❧❛♥t❛♠ ♠✐❧❤♦ ❝♦♥❝❧✉í♠♦s q✉❡ ✶✺ ♣❧❛♥t❛♠ ❛♣❡♥❛s s♦r❣♦✳ ❖❜s❡r✈❡✿

❋✐❣✉r❛ ✶✳✷✿

❈♦♠♦ t♦❞♦s ♦s ❝♦♦♣❡r❛t✐✈❛❞♦s ❝✉❧t✐✈❛♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ss❛s ❞✉❛s ❝✉❧t✉r❛s✱ ♦ ♥ú♠❡r♦ ❞❡ ❛❣r✐❝✉❧t♦r❡s ❞❛ ❝♦♦♣❡r❛t✐✈❛ é

15 + 30 + 30 = 75

✭✐✐✮ ❈♦♥s✐❞❡r❡ S ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ❝✉❧t✐✈❛♠ ♦ s♦r❣♦ ❡ M ♦

❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ❝✉❧t✐✈❛♠ ♠✐❧❤♦✳ ◗✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝♦♥❥✉♥t♦s✳ P❛r❛ ❝♦♥t❛r ♦s ❡❧❡♠❡♥t♦s ❞❡S∪M ❝♦♥t❛♠♦s t♦❞♦s ♦s ✹✺ ❡❧❡♠❡♥t♦s ❞❡S (|S|)❡ t♦❞♦s ♦s ✻✵

❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❈❛♣ít✉❧♦ ✶

❡❧❡♠❡♥t♦s ❞❡ M (|M|)✳ ❆♦ ❢❛③❡r♠♦s ✐ss♦ ♦s ❛❣r✐❝✉❧t♦r❡s q✉❡ ♣❧❛♥t❛♠ s♦r❣♦

❡ ♠✐❧❤♦s ❢♦r❛♠ ❝♦♥t❛❞♦s ❞✉❛s ✈❡③❡s✱ ✉♠❛ ❡♠ |S| ❡ ♦✉tr❛ ❡♠ |M|✱ ♣♦rt❛♥t♦

❞❡✈❡♠♦s ❞❡s❝♦♥t❛r ❛ s❡❣✉♥❞❛ ❝♦♥t❛❣❡♠ ❞❡ss❡s ❡❧❡♠❡♥t♦s ❡ ♦❜t❡♠♦s

|S∪M|= 45 + 60−30 = 105−30 = 75.

❆ss✐♠ ❝♦♥❝❧✉í♠♦s q✉❡ ❤á ✼✺ ❛❣r✐❝✉❧t♦r❡s ♥❡ss❛ ❝♦♦♣❡r❛t✐✈❛✳

❖ q✉❡ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ❛❣♦r❛ é ✉♠ r❡s✉❧t❛❞♦✱ q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ♦ ❝❛s♦ ❣❡r❛❧ ❞♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✱ ♣❛r❛ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ♣♦ré♠ ❛❣♦r❛ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐s❥✉♥t♦s✳ ❚r❛t❛✲s❡ ❞❡ ✉♠ t❡♦r❡♠❛ ✭❛♣❡♥❛s q✉❛♥❞♦ ❛♥t❡r✐♦r♠❡♥t❡ é ❛ss✉♠✐❞♦ q✉❡ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦ é ✈á❧✐❞♦✮✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦✱ q✉❡ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ❞❡♠♦♥str❛❞♦ ✉t✐❧✐③❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦✳

❚❡♦r❡♠❛ ✶ ❙❡❥❛♠ A₁ ❡ A₂ ❞♦✐s ❝♦♥❥✉♥t♦s ✜♥✐t♦s q✉❛✐sq✉❡r✳ ❊♥tã♦✿

|A₁∪A₂|=|A₁|+|A₂| − |A₁∩A₂|.

❉❡♠♦♥str❛çã♦✿ ❖❜s❡r✈❡ q✉❡A1∪A2 = (A1−A2)∪A2✱ ♦♥❞❡ ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❛ ✐❣✉❛❧❞❛❞❡ t❡♠♦s ✉♠❛ ✉♥✐ã♦ ❞❡ ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✳ P❡❧♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦✱ t❡♠♦s q✉❡

|A1∪A2|=|A1 −A2|+|A2|. ✭✶✳✶✮

❆♥❛❧♦❣❛♠❡♥t❡✱ ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❡st❡ ♣r✐♥❝í♣✐♦✱ t❡♠♦s q✉❡

|A1|=|A1−A2|+|A1∩A2|=⇒ |A1−A2|=|A1| − |A1∩A2|. ✭✶✳✷✮

❙✉❜st✐t✉✐♥❞♦ ✭✶✳✷✮ ❡♠ ✭✶✳✶✮ t❡♠♦s

|A1∪A2|=|A1| − |A1 ∩A2|+|A2|,

❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

■r❡♠♦s ❛❣♦r❛ tr❛t❛r ❞♦ t❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ♣❛r❛ três ❝♦♥❥✉♥t♦s✳ ■♥✐❝✐❛❧♠❡♥t❡ ♣❡♥s❡♠♦s ♥♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✳

❊①❡♠♣❧♦ ✹ ◆✉♠❛ ❡♥q✉❡t❡ s♦❜r❡ ❛s ❛t✐✈✐❞❛❞❡s ❛❡ró❜✐❝❛s✱ ❝♦♥st❛t♦✉✲s❡ q✉❡✿ ✹✵✵ ♣❡ss♦❛s ♣r❛t✐❝❛♠ ♥❛t❛çã♦❀ ✷✼✵ ♣r❛t✐❝❛♠ ❝✐❝❧✐s♠♦❀ ✷✾✵ ♣r❛t✐❝❛♠ ❛t❧❡t✐s♠♦❀ ✶✹✵ ♣r❛t✐❝❛♠ ♥❛t❛çã♦ ❡ ❝✐❝❧✐s♠♦❀ ✾✵ ♣r❛t✐❝❛♠ ♥❛t❛çã♦ ❡ ❛t❧❡t✐s♠♦❀ ✶✵✵ ♣r❛t✐❝❛♠ ❝✐❝❧✐s♠♦ ❡ ❛t❧❡t✐s♠♦❀ ✷✵ ♣r❛t✐❝❛♠ ♦s três ❡s♣♦rt❡s ♣❡sq✉✐s❛❞♦s✳ ◗✉❛♥t❛s ♣❡ss♦❛s ♣r❛t✐❝❛♠ ♥❛t❛çã♦ ♦✉ ❝✐❝❧✐s♠♦ ♦✉ ❛t❧❡t✐s♠♦❄

❙♦❧✉çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦✱ ♦♥❞❡ r❡♣r❡s❡♥t❛♠♦s ♣❡❧❛ ❧❡tr❛ ◆ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ♥❛t❛çã♦✱ ❛ ❧❡tr❛ ❈ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ❝✐❝❧✐s♠♦ ❡ ❛ ❧❡tr❛ ❆ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ❛t❧❡t✐s♠♦✳ ❈♦♥s✐❞❡r❡♠♦s t❛♠❜é♠ ❛ s❡❣✉✐♥t❡ ♥✉♠❡r❛çã♦ ❞♦s s✉❜❝♦♥❥✉♥t♦s ❞❡N ∪C∪A✿

❋✐❣✉r❛ ✶✳✸✿

❘❡s♦❧✈❡r❡♠♦s ♥♦ss♦ ♣r♦❜❧❡♠❛ ❞❡ ❞♦✐s ♠♦❞♦s ❞✐st✐♥t♦s✳

✭✐✮ ❯s❛♥❞♦ ❞✐❛❣r❛♠❛✱ s❛❜❡♠♦s q✉❡ ❤á ✷✵ ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ♥❛t❛çã♦✱ ❝✐❝❧✐s♠♦ ❡ ❛t❧❡t✐s♠♦✱ ❡♥tã♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❝♦❧♦❝❛r❡♠♦s ✷✵ ♥❛ ✐♥t❡rs❡çã♦ ❞♦s três ❝♦♥❥✉♥t♦s ✭s✉❜❝♦♥❥✉♥t♦ ■✮✳ ❖ s❡❣✉♥❞♦ ♣❛ss♦ é ♣r❡❡♥❝❤❡r♠♦s ♦s s✉❜❝♦♥❥✉♥t♦sII✱III ❡IV ❞❛ ❋✐❣✉r❛ ✶✳✸✳ ❙❛❜❡♠♦s ❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠

♥❛t❛çã♦ ❡ ❝✐❝❧✐s♠♦ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s s✉❜❝♦♥❥✉♥t♦s I ❡ II✱ ♣♦rt❛♥t♦ t❡♠♦s

q✉❡ I ❡ II s♦♠❛♠ ✶✹✵✱ ❝♦♠♦ ❥á ❤á ✷✵ ♥♦ s✉❜❝♦♥❥✉♥t♦ I r❡st❛♠ ✶✷✵ ♣❛r❛

♦ s✉❜❝♦♥❥✉♥t♦ II ✭♣r❛t✐❝❛♠ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♥❛t❛çã♦ ❡ ❝✐❝❧✐s♠♦✮✳ ❉❡ ♠♦❞♦

❛♥á❧♦❣♦ t❡♠♦s q✉❡ ♦ s✉❜❝♦♥❥✉♥t♦ III t❡rá 90− 20 = 70 ❡❧❡♠❡♥t♦s ❡ ♦

❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❈❛♣ít✉❧♦ ✶

s✉❜❝♦♥❥✉♥t♦ IV 100−20 = 80 ❡❧❡♠❡♥t♦s✳

❋✐❣✉r❛ ✶✳✹✿

P♦r ✜♠ ♣r❡❡♥❝❤❡r❡♠♦s ♦s s✉❜❝♦♥❥✉♥t♦sV✱V I ❡V II✳ ❈♦♠♦ ✹✵✵ ♣❡ss♦❛s

♣r❛t✐❝❛♠ ♥❛t❛çã♦✱ t❡♠♦s q✉❡ ♦s s✉❜❝♦♥❥✉♥t♦s I✱ II✱ III ❡ V s♦♠❛♠ ✹✵✵

❡❧❡♠❡♥t♦s✳ ❈♦♠♦ ❥á ❤á ✷✵ ❡❧❡♠❡♥t♦s ♥♦ s✉❜❝♦♥❥✉♥t♦ I✱ ✶✷✵ ♥♦ II ❡ ✼✵

♥♦ III✱ r❡st❛♠ 400−20−120−70 = 190 ❡❧❡♠❡♥t♦s ♣❛r❛ ♦ s✉❜❝♦♥❥✉♥t♦

V ✭♣r❛t✐❝❛♠ ❛♣❡♥❛s ♥❛t❛çã♦✮✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦ t❡♠♦s q✉❡ ♦ s✉❜❝♦♥❥✉♥t♦ V I t❡rá 270 −20−120 − 80 = 50 ❡❧❡♠❡♥t♦s ❡ ♦ s✉❜❝♦♥❥✉♥t♦ V II t❡rá

❡①❛t❛♠❡♥t❡290−20−70−80 = 120 ❡❧❡♠❡♥t♦s✳ ❚❡♠♦s ♣♦rt❛♥t♦ ♦ s❡❣✉✐♥t❡

❞✐❛❣r❛♠❛

❋✐❣✉r❛ ✶✳✺✿

t♦t❛❧✐③❛♥❞♦190 + 120 + 50 + 70 + 20 + 80 + 120 = 650♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠

♥❛t❛çã♦ ♦✉ ❝✐❝❧✐s♠♦ ♦✉ ❛t❧❡t✐s♠♦✳

✭✐✐✮ ❈♦♥s✐❞❡r❡♠♦s N ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ♥❛t❛çã♦✱ C

♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ❝✐❝❧✐s♠♦ ❡ ❛ ❧❡tr❛ A ♦ ❝♦♥❥✉♥t♦

❞❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ❛t❧❡t✐s♠♦✳ ◗✉❡r❡♠♦s ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝♦♥❥✉♥t♦s✳ P❛r❛ ❝♦♥t❛r ♦s ❡❧❡♠❡♥t♦s ❞❡ N ∪C ∪A ❝♦♥t❛♠♦s t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ N (|N|)✱ t♦❞♦s

♦s ❡❧❡♠❡♥t♦s ❞❡ C (|C|) ❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ A (|C|)✳ ❉❡ ✉♠❛ ❢♦r♠❛

❡q✉✐✈♦❝❛❞❛ t❡rí❛♠♦s✿

|N∪C∪A|=|N|+|C|+|A|

❉❡ ❢❛t♦✱ ❡ss❛ s♦♠❛ ♥ã♦ r❡♣r❡s❡♥t❛ ❛ ✉♥✐ã♦✱ ♦❜s❡r✈❡ ♥❛ ✜❣✉r❛ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛❣❡♥s ❞❡ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ q✉❛♥❞♦ ❛ ♠❡s♠❛ é r❡❛❧✐③❛❞❛✳

❋✐❣✉r❛ ✶✳✻✿

❆♦ ❢❛③❡r♠♦s ✐ss♦ ❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ❛♣❡♥❛s ♥❛t❛çã♦ ❡ ❝✐❝❧✐s♠♦✱ ❛s q✉❡ ♣r❛t✐❝❛♠ ❛♣❡♥❛s ♥❛t❛çã♦ ❡ ❛t❧❡t✐s♠♦ ❡ ❛s q✉❡ ♣r❛t✐❝❛♠ ❛♣❡♥❛s ❝✐❝❧✐s♠♦ ❡ ❛t❧❡t✐s♠♦ ❢♦r❛♠ ❝♦♥t❛❞❛s ❞✉❛s ✈❡③❡s✳ P❛r❛ ❝♦rr✐❣✐r ❡ss❡ ❡①❝❡ss♦ ❞❡✈❡♠♦s ❞❡s❝♦♥t❛r ✉♠❛ ✈❡③ ❝❛❞❛ ✉♠❛ ❞❛s ✐♥t❡rs❡❝çõ❡s ❞✉❛s ❛ ❞✉❛s ❞♦s ❝♦♥❥✉♥t♦s✱ ❛ss✐♠ s❡♥❞♦✱ ♥♦ss❛ ❡①♣r❡ssã♦ s❡ t♦r♥❛✿

|N ∪C∪A|=|N|+|C|+|A| − |N ∩C| − |N ∩A| − |C∩A|

❡ ♦ ♥♦✈♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛❣❡♥s ❞❡ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ s❡rá✿

❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❈❛♣ít✉❧♦ ✶

❋✐❣✉r❛ ✶✳✼✿

❊♥tr❡t❛♥t♦ ❣❡r❛♠♦s ✉♠ ♥♦✈♦ ♣r♦❜❧❡♠❛✱ ❛ ✐♥t❡rs❡çã♦ ❞♦s três ❝♦♥❥✉♥t♦s ♥ã♦ ❡stá s❡♥❞♦ ❝♦♥t❛❞❛ ❡ ♣❛r❛ ❝♦rr✐❣✐r ❡st❛ ❛✉sê♥❝✐❛✱ ❞❡✈❡♠♦s ❛❝r❡s❝❡♥tá✲❧❛ ✉♠❛ ✈❡③✳ ❖❜t❡♥❞♦✿

|N∪C∪A|=|N|+|C|+|A| − |N∩C| − |N∩A| − |C∩A|+|N∩C∩A|.

❈❤❡❣❛♠♦s ❡♥tã♦ ♥❛ ❡①♣r❡ssã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ✉♥✐ã♦ ❞❡ três ❝♦♥❥✉♥t♦s✱ s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s✱ t❡♠♦s✿

|N ∪C∪A|= 400 + 270 + 290−140−90−100 + 20 = 650.

❆ss✐♠ ❝♦♥❝❧✉í♠♦s q✉❡ ❤á ✻✺✵ ♣❡ss♦❛s ♣❡ss♦❛s q✉❡ ♣r❛t✐❝❛♠ ♥❛t❛çã♦ ♦✉ ❝✐❝❧✐s♠♦ ♦✉ ❝♦rr✐❞❛✳

❉❡♠♦♥str❛r❡♠♦s ❡♠ s❡❣✉✐❞❛ ♦ ❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ♣❛r❛ três ❝♦♥❥✉♥t♦s✱ ♣❛r❛ ✐ss♦ ✉s❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ✶✱ ❞❡♠♦♥str❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛❧é♠ ❞♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✳

❚❡♦r❡♠❛ ✷ ❙❡❥❛♠ A₁, A₂ ❡ A₃ três ❝♦♥❥✉♥t♦s ✜♥✐t♦s q✉❛✐sq✉❡r✳ ❊♥tã♦✱

|A1∪A2∪A3| = |A1|+|A2|+|A3|+|A1∩A2∩A3|

−(|A₁∩A₂|+|A₁∩A₃|+|A₂∩A₃|).

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ✶ t❡♠♦s q✉❡✱

|A1∪A2 ∪A3|=|A1 ∪(A2∪A3)|=|A1|+|A2∪A3| − |A1∩(A2∪A3)|.

▲❡♠❜r❛♥❞♦ q✉❡

A1∩(A2∪A3) = (A1∩A2)∪(A1∩A3)

♣❡❧♦ ❚❡♦r❡♠❛ ✶✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡

|A1∪A2∪A3| = |A1|+|A2∪A3| − |(A1∩A2)∪(A1∩A3)| = |A₁|+|A₂|+|A₃| − |A₂∩A₃|

−|(A₁∩A₂)∪(A₁∩A₃)|

✭✶✳✸✮

❡ ❛✐♥❞❛ q✉❡

|(A₁ ∩A2)∪(A₁∩A3)| = |A₁∩A₂|+|A₁∩A₃|− |(A1∩A2)∩(A1∩A3)|.

✭✶✳✹✮

❙✉❜st✐t✉✐♥❞♦ ✭✶✳✸✮ ❡♠ ✭✶✳✹✮✱ t❡♠♦s q✉❡

|A₁∪A₂∪A₃| = |A₁|+|A₂|+|A₃| − |A₂∩A₃| −(|A₁∩A₂|

+|A1∩A3| − |(A1∩A2)∩(A1∩A3)|).

❏á q✉❡ (A1∩A2)∩(A1∩A3) = A1∩A2∩A3 ❝♦♥❝❧✉í♠♦s q✉❡

|A1∪A2∪A3| = |A1|+|A2|+|A3| − |A1∩A2| − |A1∩A3|

−|A₂∩A₃|+|A₁∩A₂∩A₃|

❝♦♠♦ ❞❡s❡❥❛❞♦✳

❖ ♣ró①✐♠♦ ♣❛ss♦ é ❡♥❝♦♥tr❛r ✉♠❛ r❡❣r❛ ❣❡r❛❧ ❞♦ ❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ♣❛r❛ n ❝♦♥❥✉♥t♦s✳ ❯s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡ s♦♠❛tór✐♦ P ♣♦❞❡♠♦s

❡♥✉♥❝✐❛r ♦ ❚❡♦r❡♠❛ ✷ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

|∪3

i=1Ai|=

3

X

i=1

|Ai| − X

1≤i1<i2≤3

|Ai1 ∩Ai2|+

X

1≤i1<i2<i3≤3

|Ai1 ∩Ai2 ∩Ai3|. ❉❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ ❞❛❞♦s ♦s ❝♦♥❥✉♥t♦s ✜♥✐t♦sA₁, A₂,· · · , An✱ ❛s ❡①♣r❡ssõ❡s

❛♥t❡r✐♦r❡s ♥♦s ❧❡✈❛♠ ❛ ❞❡✜♥✐r ♦s ♥ú♠❡r♦s✿

S₁n =

n X

i=1

|Ai|

S₂n = X 1≤i1<i2≤n

❚❡♦r❡♠❛ ❞❛ ✐♥❝❧✉sã♦ ❡ ❡①❝❧✉sã♦ ❈❛♣ít✉❧♦ ✶

✳✳✳

S_kn= X

1≤i1<i2<···<ik≤n

|Ai1 ∩Ai2 ∩ · · · ∩Aik|

✳✳✳

S_nn=|A₁∩A₂∩ · · · ∩An|.

❆ss✐♠✱ ❛ ✈❡rsã♦ ♠❛✐s ❣❡r❛❧ ❞♦ Pr✐♥❝í♣✐♦ ❆❞✐t✐✈♦✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ■♥❝❧✉sã♦ ❡ ❊①❝❧✉sã♦✱ é✿

❚❡♦r❡♠❛ ✸ ❙❡❥❛♠ A1, A2,· · · , An ❝♦♥❥✉♥t♦s ✜♥✐t♦s q✉❛✐sq✉❡r✳ ❊♥tã♦✱

| ∪n

i=1Ai|=S₁n−S₂n+S₃n−S₄n+· · ·+ (−1)n−1Snn.

❆ss✉♠✐♥❞♦✲s❡ ♦ ♣r✐♥❝í♣✐♦ ❛❞✐t✐✈♦✱ ♣r♦✈❛♠♦s ❡st❡ r❡s✉❧t❛❞♦ ♣♦r ✐♥❞✉çã♦✳ ❊st❛ ❞❡♠♦♥str❛çã♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❆♣ê♥❞✐❝❡✳

✶✳✸ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦

■♥✐❝✐❛r❡♠♦s ❡st❛ s❡çã♦ tr❛t❛♥❞♦ ❞❡ ✉♠❛ ❞❛s ♠❛✐s ♣♦❞❡r♦s❛s ❢❡rr❛♠❡♥t❛s ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✱ ♦ ✏Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✑ ♦✉ ✏Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✑ ♦✉ ❛✐♥❞❛ s✐♠♣❧❡s♠❡♥t❡ ♦ ✏P✳❋✳❈✑✱ ❝♦♠♦ s❡rá ❝❤❛♠❛❞♦ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥t♦s ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✳ P❛r❛ t❛♥t♦✱ ❝♦♠❡ç❛r❡♠♦s ❝♦♠ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠✳ ❚r❛t❛r❡♠♦s ❛ r❡s♦❧✉çã♦ ❞❡st❡s ❡①❡r❝í❝✐♦s ✉t✐❧✐③❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ár✈♦r❡ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ✭✉♠❛ ❡s♣é❝✐❡ ❞❡ ❧✐st❛❣❡♠ ❞❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✮ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ❛❧❣✉♠❛ té❝♥✐❝❛ ♣❛r❛ q✉❡ ❡✈✐t❡♠♦s ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❛♥t❡r✐♦r✱ ♣♦✐s ♦ ♠❡s♠♦ ♣♦❞❡ s❡r ♠✉✐♦ ❞❡s❣❛st❛♥t❡ ❛♦ ❞❡♣❡♥❞❡r ❞♦ ♥ú♠❡r♦ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ P♦❞❡rí❛♠♦s t❛♠❜é♠ ✉t✐❧✐③❛r t❛❜❡❧❛s ❛♦ ✐♥✈és ❞❡ ár✈♦r❡s✱ ♦ ❡❢❡✐t♦ ❞❡ ❧✐st❛❣❡♠ s❡r✐❛ ♦ ♠❡s♠♦✳

Pr♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠ sã♦✱ ♠✉✐t❛s ✈❡③❡s✱ ❝♦♥s✐❞❡r❛❞♦s ❞✐❢í❝❡✐s ❡♥tr❡ ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s✱ ❛♣❡s❛r ❞❡ ❛s té❝♥✐❝❛s ♠❛t❡♠át✐❝❛s ♥❡❝❡ssár✐❛s s❡r❡♠ ❜❛st❛♥t❡ ❡❧❡♠❡♥t❛r❡s✿ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❞❡ s♦♠❛✱ s✉❜tr❛çã♦✱ ♠✉❧t✐♣❧✐❝❛çã♦ ❡ ❞✐✈✐sã♦✳

❯♠ ❞♦s ♦❜❥❡t✐✈♦s ❞❡st❛ s❡çã♦ é ❤❛❜✐t✉❛r ♦ ❛❧✉♥♦ ❛ tr❛❜❛❧❤❛r ❝♦♠ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠ ❡ ❛ ✈❡r q✉❡✱ ❛✜♥❛❧ ❞❡ ❝♦♥t❛s✱ t❛✐s ♣r♦❜❧❡♠❛s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ❝♦♠ r❛❝✐♦❝í♥✐♦s s✐♠♣❧❡s ♥❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ s❡♠ ❡①✐❣✐r ♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ❝♦♠♣❧✐❝❛❞❛s✳ ➱ ✐st♦ ♦ q✉❡ ♣r♦❝✉r❛♠♦s ♠♦str❛r ♥♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✺ P❛r❛ ✐r à ❢❡st❛ ❞♦ ♣❛❞r♦❡✐r♦ ❞❡ ❙ã♦ ▲♦✉r❡♥ç♦ ❞❛ ♠❛t❛✱ ❈❤✐q✉✐♥❤❛ ❞✐s♣õ❡ ❞❡ ✷ ✈❡st✐❞♦s ❞❡ ❝❤✐t❛ ❡ ❞❡ ✸ ♣❛r❡s ❞❡ ❝❤✐♥❡❧❛ ❞❡ ❝♦✉r♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❈❤✐q✉✐♥❤❛ ♣♦❞❡ ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ ✈❡st✐❞♦ ❞❡ ❝❤✐t❛ ❡ ✉♠ ú♥✐❝♦ ♣❛r ❞❡ ❝❤✐♥❡❧❛ ❞❡ ❝♦✉r♦❄

❙♦❧✉çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❡♠♦s ❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛ ❞♦ ✈❡st✐❞♦✱ q✉❡ sã♦ ♦ ♣r✐♠❡✐r♦ ♦✉ ♦ s❡❣✉♥❞♦ ✈❡st✐❞♦✱ t❡♠♦s ♣♦rt❛♥t♦ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✳

Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❈❛♣ít✉❧♦ ✶

❊♠ s❡❣✉✐❞❛✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s ♦♣çõ❡s ❞❡ ✈❡st✐❞♦ t❡♠♦s ✸ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❡s❝♦❧❤❡r ❛ ❝❤✐♥❡❧❛✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ár✈♦r❡✱ ♦♥❞❡V₁❡ V2 r❡♣r❡s❡♥t❛♠ ♦s ❞♦✐s ✈❡st✐❞♦s ❡C1✱C2 ❡C3 ❛s três ❝❤✐♥❡❧❛s ❞❡ ❈❤✐q✉✐♥❤❛✳

■r❡♠♦s ♥❡ss❡ ♠♦♠❡♥t♦ ❢♦r♠❛r t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞♦ ❝♦♥❥✉♥t♦ ✭✈❡st✐❞♦ ❡ ❝❤✐♥❡❧❛✮✳

❋✐❣✉r❛ ✶✳✽✿

❈❛❞❛ ❡①tr❡♠✐❞❛❞❡ ❞❛ ár✈♦r❡ r❡♣r❡s❡♥t❛ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❈❤✐q✉✐♥❤❛ s❡ ✈❡st✐r✱ ♣♦rt❛♥t♦ t❡♠♦s ✻ ❢♦r♠❛s ❞✐st✐♥t❛s ❞❡ ✈❡st✐♠❡♥t❛✳

❯♠❛ ♦✉tr❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛ s❡r✐❛ ❛ s❡❣✉✐♥t❡✿ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❞❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❈❤✐q✉✐♥❤❛ ♣♦❞❡ ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ ✈❡st✐❞♦ ❞❡ ❝❤✐t❛ ❡ ✉♠❛ ú♥✐❝❛ ❝❤✐♥❡❧❛ ❞❡ ❝♦✉r♦ t❡♠♦s ❞✉❛s ❞❡❝✐sõ❡s ❛ t♦♠❛r✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❡s❝♦❧❤❡r ✉♠❛ ❡♥tr❡ ❛s ❞✉❛s ♦♣çõ❡s ❞❡ ✈❡st✐❞♦ ❡ ✉♠❛ ✈❡③ ❡s❝♦❧❤✐❞♦ ♦ ✈❡st✐❞♦ ❡s❝♦❧❤❡r ✉♠❛ ❝❤✐♥❡❧❛ ❞❡♥tr❡ ❛s três ♣♦ssí✈❡✐s✳ ❈♦♠♦✱ ✉♠❛ ✈❡③ ❡s❝♦❧❤✐❞♦ ♦ ✈❡st✐❞♦✱ ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡❧❡s ♣♦❞❡♠♦s ❝♦♠❜✐♥❛r ❝♦♠ ❝❛❞❛ ✉♠❛ ❞❛s três ❝❤✐♥❡❧❛s✱ t❡♠♦s ♣♦rt❛♥t♦ 2×3 = 6 ♠♦❞♦s ❞✐st✐♥t♦s ❞❡

✈❡st✐♠❡♥t❛✳

❖ ♠ét♦❞♦ ❞❛ ár✈♦r❡ ❣❡r❛❧♠❡♥t❡ é ✉t✐❧✐③❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❡ ✉♠ ♣r✐♠❡✐r♦

❝✉rs♦ ❞❡ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛ ❡ ✈✐s❛ ❝♦♥✈❡♥❝❡r ♦ ❡st✉❞❛♥t❡ ❞❛ ✈❡r❛❝✐❞❛❞❡ ❞♦ ✈❛❧♦r q✉❡ s❡rá ❡♥❝♦♥tr❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ♣❡❧❛ té❝♥✐❝❛✳ ❈♦♥t✐♥✉❡♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

❊①❡♠♣❧♦ ✻ ❘✐❝❛r❞♦ ❢❛rá ✉♠❛ ✈✐❛❣❡♠ ❝✉❥♦ tr❡❝❤♦ s❡rá ❏♦ã♦ P❡ss♦❛✴❘❡❝✐❢❡ ✴❏♦ã♦ P❡ss♦❛ ❞❡ ô♥✐❜✉s✱ ❛✈✐ã♦✱ ♥❛✈✐♦ ♦✉ tr❡♠✳ P❛r❛ ❝✉rt✐r ❞✐❢❡r❡♥t❡s ♣❛✐s❛❣❡♥s ♥❡ss❛ ✈✐❛❣❡♠✱ ❡❧❡ ❞❡❝✐❞❡ q✉❡ ♦ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡ ❞❛ ✐❞❛ ♥ã♦ é ♦ ♠❡s♠♦ ❞❛ ✈♦❧t❛✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❘✐❝❛r❞♦ ♣♦❞❡ r❡❛❧✐③❛r ❛ ✈✐❛❣❡♠❄

❙♦❧✉çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❡♠♦s ❛s ♦♣çõ❡s ❞❡ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡s ❞❡ ✐❞❛✱ q✉❡ sã♦ ô♥✐❜✉s✱ ❛✈✐ã♦✱ ♥❛✈✐♦ ♦✉ tr❡♠✱ t❡♠♦s ♣♦rt❛♥t♦ ✹ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❊♠ s❡❣✉✐❞❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ✹ ♦♣çõ❡s ❞❡ ❘✐❝❛r❞♦ ✐r ❛ r❡❝✐❢❡✳ ❚❡♠♦s ✸ ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛ ❞♦ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡ ❞❛ ✈♦❧t❛✳ P♦❞❡♠♦s ❝♦♥str✉✐r ❛ s❡❣✉✐♥t❡ ár✈♦r❡ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ✐❧✉str❛r ♥♦ss❛ r❡s♦❧✉çã♦✿

❋✐❣✉r❛ ✶✳✾✿

Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❈❛♣ít✉❧♦ ✶

❈❛❞❛ ❡①tr❡♠✐❞❛❞❡ ❞❛ ár✈♦r❡ r❡♣r❡s❡♥t❛ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❘✐❝❛r❞♦ r❡❛❧✐③❛r ❛ ✈✐❛❣❡♠✱ ♣♦rt❛♥t♦ t❡♠♦s ✶✷ ❢♦r♠❛s ❞✐st✐♥t❛s ❞♦ ♠❡s♠♦ r❡❛❧✐③á✲❧❛✳ ❯♠❛ ♦✉tr❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❡ ♣r♦❜❧❡♠❛ s❡r✐❛ ❛ s❡❣✉✐♥t❡✿ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❞❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s ❘✐❝❛r❞♦ ♣♦❞❡ ❡s❝♦❧❤❡r ✉♠ ú♥✐❝♦ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ ✐❞❛ ❡ ✉♠ ú♥✐❝♦ ❞❡ ✈♦❧t❛✱ t❡♠♦s ❞✉❛s ❞❡❝✐sõ❡s ❛ t♦♠❛r✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❡s❝♦❧❤❡r ✉♠❛ ❡♥tr❡ ❛s q✉❛tr♦ ♦♣çõ❡s ❞❡ ✐❞❛ ❡ ✉♠❛ ✈❡③ ❡s❝♦❧❤✐❞♦ ♦ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡ ❞❡ ✐❞❛ ❡s❝♦❧❤❡r ♦ ❞❡ ✈♦❧t❛ ❞❡♥tr❡ ♦s três ♣♦ssí✈❡✐s ✭✉♠❛ ✈❡③ q✉❡ ❘✐❝❛r❞♦ ♥ã♦ ♣♦❞❡ t♦♠❛r ♥❛ ✈♦❧t❛ ♦ ♠❡s♠♦ ♠❡✐♦ ❞❡ tr❛♥s♣♦rt❡ ❞❛ ✐❞❛✮✳ ❚❡♠♦s ♣♦rt❛♥t♦4×3 = 12 ♠♦❞♦s ❞✐st✐♥t♦s ❞❡ ❘✐❝❛r❞♦

r❡❛❧✐③❛r ❡ss❛ ✈✐❛❣❡♠✳

P♦❞❡♠♦s ❛❣♦r❛ ❡♥✉♥❝✐❛r ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✱ ♦✉ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✱ s❡♥❞♦ ❡st❡ ❝♦♥❝❡✐t♦ ❛ ❢❡rr❛♠❡♥t❛ ♠❛✐s ♣♦❞❡r♦s❛ ♥❛ r❡s♦❧✉çã♦ ❞❛ ♠❛✐♦r✐❛ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠✱ ❛❧é♠ ❞❡ s❡r✈✐r ❝♦♠♦ ❢✉♥❞❛♠❡♥t❛çã♦ ❞❡ ❞✐✈❡rs♦s ♦✉tr♦s ❝♦♥❝❡✐t♦s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❞✐❛♥t❡✳ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❬✸❪✿ ❙❡ ✉♠❛ ❞❡❝✐sã♦ d1 ♣♦❞❡ s❡r t♦♠❛❞❛ ❞❡ n

♠❛♥❡✐r❛s ❡ s❡✱ ✉♠❛ ✈❡③ t♦♠❛❞❛ ❛ ❞❡❝✐sã♦ d₁✱ ❛ ❞❡❝✐sã♦d₂ ♣✉❞❡r s❡r t♦♠❛❞❛

❞❡m ♠❛♥❡✐r❛s ❡♥tã♦ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ s❡ t♦♠❛r❡♠ ❛s ❞❡❝✐sõ❡sd1 ❡

d₂ s✉❝❡ss✐✈❛♠❡♥t❡ é n×m✳

■r❡♠♦s ❛❣♦r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ♠❛✐s ❞❡ ❞✉❛s ❞❡❝✐sõ❡s✳

❊①❡♠♣❧♦ ✼ ✭❯❋❊❙ ✲ ❛❞❛♣t❛❞❛✮ ❯♠ ✏❙❤♦♣♣✐♥❣ ❈❡♥t❡r✑ ♣♦ss✉✐ ✷ ♣♦rt❛s ❞❡ ❡♥tr❛❞❛ ♣❛r❛ ♦ ❛♥❞❛r térr❡♦✱ ✹ ❡s❝❛❞❛s r♦❧❛♥t❡s ❧✐❣❛♥❞♦ ♦ térr❡♦ ❛♦ ♣r✐♠❡✐r♦ ♣❛✈✐♠❡♥t♦ ❡ ✸ ❡❧❡✈❛❞♦r❡s q✉❡ ❝♦♥❞✉③❡♠ ❞♦ ♣r✐♠❡✐r♦ ♣❛r❛ ♦ s❡❣✉♥❞♦ ♣❛✈✐♠❡♥t♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ✉♠❛ ♣❡ss♦❛✱ ♣❛rt✐♥❞♦ ❞❡ ❢♦r❛ ❞♦ ✏❙❤♦♣♣✐♥❣ ❈❡♥t❡r✑ ♣♦❞❡ ❛t✐♥❣✐r ♦ s❡❣✉♥❞♦ ♣❛✈✐♠❡♥t♦ ✉s❛♥❞♦ ♦s ❛❝❡ss♦s ♠❡♥❝✐♦♥❛❞♦s❄

❙♦❧✉çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❡♠♦s ❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛ ❞❡ ❡♥tr❛❞❛ ♣❛r❛ ♦ ❛♥❞❛r térr❡♦✱ q✉❡ sã♦ ❛ ♣r✐♠❡✐r❛ ♦✉ ❛ s❡❣✉♥❞❛ ♣♦rt❛✱ t❡♠♦s ♣♦rt❛♥t♦ ❞✉❛s

♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❊♠ s❡❣✉✐❞❛ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❡r ❛ ♣♦rt❛ ❞❡ ❡♥tr❛❞❛ t❡♠♦s ✹ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝♦❧❤❡r ❛ ❡s❝❛❞❛ r♦❧❛♥t❡ ❧✐❣❛♥❞♦ ♦ térr❡♦ ❛♦ ✶♦ _{♣❛✈✐♠❡♥t♦✳ ❉❡ss❛ ❢♦r♠❛ ✉♠❛ ✈❡③ ❡st❛♥❞♦ ♥♦ ✶}♦ _{♣❛✈✐♠❡♥t♦}

❞❡✈❡♠♦s ❡s❝♦❧❤❡r ✉♠ ❞♦s três ❡❧❡✈❛❞♦r❡s ❧✐❣❛♥❞♦ ♦ ✶♦ _{❝♦♠ ♦ ✷}♦ _{♣❛✈✐♠❡♥t♦✳}

❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ár✈♦r❡

❋✐❣✉r❛ ✶✳✶✵✿

❆q✉✐ P1 ❡ P2 r❡♣r❡s❡♥t❛♠ ♦s ❞♦✐s ♣♦rtõ❡s ❞❡ ❡♥tr❛❞❛✱ R1✱ R2 ✱ R3 ❡ R4

❛s q✉❛tr♦ ❡s❝❛❞❛s r♦❧❛♥t❡s ❡ E₁ ✱ E₂ ❡ E₃ ♦s três ❡❧❡✈❛❞♦r❡s✳ ■r❡♠♦s

❛❣♦r❛ ❢♦r♠❛r t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝♦♠❜✐♥❛çõ❡s✳ ❈❛❞❛ ❡①tr❡♠✐❞❛❞❡ ❞❛ ár✈♦r❡ r❡♣r❡s❡♥t❛ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡ss♦❛✱ ♣❛rt✐♥❞♦ ❞❡ ❢♦r❛ ❞♦ ❙❤♦♣♣✐♥❣ ❈❡♥t❡r✱ ❛t✐♥❣✐r ♦ s❡❣✉♥❞♦ ♣❛✈✐♠❡♥t♦ ✉s❛♥❞♦ ♦s ❛❝❡ss♦s ♠❡♥❝✐♦♥❛❞♦s✱ ♣♦rt❛♥t♦ t❡♠♦s ✉♠ t♦t❛❧ ❞❡ ✷✹ ❢♦r♠❛s ❞✐st✐♥t❛s✳

Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❈❛♣ít✉❧♦ ✶

✏❖ r❛❝✐♦❝í♥✐♦ ❛❝✐♠❛ ✐♥❞✐❝❛ q✉❡ ♦ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ♣♦❞❡✱ ♥❛ r❡❛❧✐❞❛❞❡✱ s❡r ❛♣❧✐❝❛❞♦ q✉❛♥❞♦ t❡♠♦s ❞✐✈❡rs❛s ❡t❛♣❛s ❞❡ ❞❡❝✐sã♦✿ ❞❡s❞❡ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ❡♠ ❝❛❞❛ ❡t❛♣❛ ♥ã♦ ❞❡♣❡♥❞❛ ❞❛s ❞❡❝✐sõ❡s ❛♥t❡r✐♦r❡s✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝á✲❧♦s ♣❛r❛ ❛❝❤❛r ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ P❛r❛ ♦ ♥♦ss♦ ❡①❡♠♣❧♦ t❡♠♦s 2×4×3 = 24 ♣♦ss✐❜✐❧✐❞❛❞❡s✑

➱ ✐♠♣♦rt❛♥t❡ ♥❡st❡ ♠♦♠❡♥t♦✱ ❡①✐❜✐r ❛❧❣✉♠❛s ❡str❛té❣✐❛s ✭té❝♥✐❝❛s✮✱ ❜❛s❡❛❞❛s ❡♠ ▼♦r❣❛❞♦✱ ❡t✳ ❛❧ ❬✸❪✱ ▲✐♠❛✱ ❡t✳ ❛❧ ❬✷❪✱ ❙❛♥t♦s✱ ❡t✳ ❛❧ ❬✼❪ ❡ ❖❧✐✈❡✐r❛✱ ❡t✳ ❛❧ ❬✹❪ q✉❡ ❢❛❝✐❧✐t❛♠ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠✳ ❙ã♦ ❡❧❛s✿

❊str❛té❣✐❛ ✶ P♦st✉r❛

➱ ♣r✐♠♦r❞✐❛❧ q✉❡ ✈♦❝ê t❡♥t❡ s❡ ❝♦❧♦❝❛r ♥♦ ❧✉❣❛r ❞❡ q✉❡♠ ❡stá ❡①❡❝✉t❛♥❞♦ ❛ ❛çã♦ ❞♦ ♣r♦❜❧❡♠❛✱ ♣❡♥s❡ s❡♠♣r❡ q✉❡ ❡stá ❞❡ ♣♦ss❡ ❞❛s ♦♣çõ❡s ❡ ❛♦ ♣r❡❡♥❝❤❡r ✉♠❛ ❡t❛♣❛ ✐♠❛❣✐♥❡ q✉❡ ✉♠❛ ❞❡ss❛s ♦♣çõ❡s ❥á ❢♦✐ ✉t✐❧✐③❛❞❛ ♥❛ ♠❡s♠❛✳

❊str❛té❣✐❛ ✷ ❉✐✈✐sã♦

❉❡✈❡♠♦s s❡♠♣r❡ ❞✐✈✐❞✐r ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡♠ ❡t❛♣❛s ✭♦✉ ❞❡❝✐sõ❡s✮ ♠❛✐s s✐♠♣❧❡s ❞❡ s❡r❡♠ ❞❡t❡r♠✐♥❛❞❛s ❛s s✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ♣❛r❛ q✉❡ ❞❡♣♦✐s ♣❡♥s❡♠♦s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❡t❛♣❛s ❝♦♠♦ ✉♠ t♦❞♦✱ ♣♦❞❡♥❞♦ t❡r ✉♠ ♦❧❤❛r s✐♠✉❧tâ♥❡♦ ♦✉ s✉❝❡ss✐✈♦ ❞❛s ❞❡❝✐sõ❡s✳

❊str❛té❣✐❛ ✸ ❖ q✉❡ ❞✐❢❡r❡♥❝✐❛ ✉♠❛ ❞❡❝✐sã♦ ❞❡ ♦✉tr❛

❊ss❡ é s❡♠ s♦♠❜r❛ ❞❡ ❞ú✈✐❞❛✱ ♦ ❣r❛♥❞❡ ❞✐❧❡♠❛ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠✳ ❈♦♠♦ ❥✉st✐✜❝❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❡❝✐sõ❡s ❡s❝♦❧❤✐❞♦ ♣♦r ✈♦❝ê t❡♠ ❞❡ ❢❛t♦ ♦ ♥ú♠❡r♦ ❝♦rr❡t♦ ❞❡ ❞❡❝✐sõ❡s ♦✉ ♣♦❞❡♠♦s ❛✐♥❞❛ s✉❜❞✐✈✐❞✐r ❛❧❣✉♠❛ ❞❡ss❛s ❞❡❝✐sõ❡s ❡♠ ♦✉tr❛s ♦✉ ❛té ❛❣r✉♣❛r ❛❧❣✉♠❛s ❝♦♠♣♦♥❞♦ ✉♠❛ ú♥✐❝❛ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ♦ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ❡t❛♣❛s ♥ã♦ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦ ♣r♦❜❧❡♠❛✳ ❋✉t✉r❛♠❡♥t❡ ✈❡r❡♠♦s ♦ ❝❧áss✐❝♦ ♣r♦❜❧❡♠❛ ❞❛ ♣✐♥t✉r❛ ❞❡ ✉♠ ♠❛♣❛✳ ❊ ❛í s❡ ♣❡r❣✉♥t❛✿ ♣♦rq✉❡ ♦ ❝♦rr❡t♦ é ❡s❝♦❧❤❡r ❛ ❝♦r ❞♦ ✶♦ _{♣❛ís ❡♠ s❡❣✉✐❞❛ ❛ ❝♦r ❞♦}

✷♦ _{♣❛ís ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❡ ♥ã♦ ❡s❝♦❧❤❡r♠♦s ♦ ♣❛ís q✉❡ t❡rá ❛ ✶}❛ _❝♦r

❡♠ s❡❣✉✐❞❛ ♦ ♦✉tr♦ ♣❛ís q✉❡ t❡rá ❛ ✷❛ _{❝♦r ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡❄ ❉❡ ❢❛t♦}

♥ã♦ ❤á ✉♠❛ r❡❝❡✐t❛ ❞❡ ❜♦❧♦ ♣❛r❛ ❛ ❞❡✜♥✐çã♦ ❞❛s ❡t❛♣❛s ♥❛ q✉❛❧ s❡rá ❞✐✈✐❞✐❞❛ ♦ s❡✉ ♣r♦❜❧❡♠❛✳ ❚❡r❡♠♦s q✉❡ ♦❜s❡r✈❛r ♦ q✉❡ ❢❛③ s❡♥t✐❞♦✳

❊str❛té❣✐❛ ✹ ◗✉❛♥❞♦ ✉♠❛ ❞❡❝✐sã♦ ❞❡✈❡ s❡r t♦♠❛❞❛ ❛♥t❡s ❞❡ ♦✉tr❛ ✭♥ã♦ ❛❞✐❛r ❞✐✜❝✉❧❞❛❞❡s✮

❉❡❝✐sõ❡s ❛❞✐❛❞❛s ♣♦r ❝♦♥t❡r ❛❧❣✉♠❛ ❞✐✜❝✉❧❞❛❞❡ t♦r♥❛♠✲s❡ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛s ♣♦st❡r✐♦r♠❡♥t❡✱ ♣♦r ✐ss♦ ❞❡✈❡♠ s❡r r❡s♦❧✈✐❞❛s ❞❡ ✐♠❡❞✐❛t♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛s ❡t❛♣❛s q✉❡ s♦❢r❡♠ ♠❛✐s ✐♥✢✉ê♥❝✐❛ ❡♠ r❡❧❛çã♦ ❛s ❞❡♠❛✐s ❞❡✈❡♠ s❡r r❡s♦❧✈✐❞❛s ♣r✐♦r✐t❛r✐❛♠❡♥t❡✳

❊str❛té❣✐❛ ✺ ▼ét♦❞♦s ❞❡ ❝♦♥t❛❣❡♠ ✭❞✐r❡t♦ ♦✉ ✐♥❞✐r❡t♦✮

❊♠ t♦❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥t❛❣❡♠✱ ♣♦❞❡♠♦s ♦♣t❛r ♣♦r ✉♠❛ ❞❛s ❞✉❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦✿

• ❞✐r❡t♦✿ ❊ss❡ ♠ét♦❞♦ ❝♦♥s✐st❡ ❡♠ ❝♦♥s✐❞❡r❛r ✐♥✐❝✐❛❧♠❡♥t❡ t♦❞❛s ❛s

♣❛rt✐❝✉❧❛r✐❞❛❞❡s ❞♦ ♣r♦❜❧❡♠❛ ❝❤❡❣❛♥❞♦ ❛ss✐♠ ❞✐r❡t❛♠❡♥t❡ à s♦❧✉çã♦ ♣r♦❝✉r❛❞❛✳

• ✐♥❞✐r❡t♦✿ ❊ss❡ ♠ét♦❞♦ ❝♦♥s✐st❡ ❡♠ ✐❣♥♦r❛r ✉♠❛ ✭♦✉ ♠❛✐s✮ ❞❛s

r❡str✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ ♦ q✉❡ ♥♦s ❢❛rá ❝♦♥t❛r ❡♠ ❞❡♠❛s✐❛✳ ❉❡♣♦✐s ❞❡s❝♦♥t❛r❡♠♦s ♦ q✉❡ ❤♦✉✈❡r s✐❞♦ ❝♦♥t❛❞♦ ✐♥❞❡✈✐❞❛♠❡♥t❡✳

❇❛s❡❛❞♦s ♥❡ss❛s té❝♥✐❝❛s✱ ✐r❡♠♦s ❛❣♦r❛ r❡s♦❧✈❡r ♦ ♣ró①✐♠♦ ♣r♦❜❧❡♠❛ s❡♠ ♦ ❛✉①í❧✐♦ ❞♦ ❞✐❛❣r❛♠❛ ❞❡ ár✈♦r❡✱ ♦❜s❡r✈❡✿

❊①❡♠♣❧♦ ✽ ❉❡s❞❡ ❞❡ ♦ ❞✐❛ ✸✶ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✷✱ ❝♦♠ ❛ ❛❝❡✐t❛çã♦ ❞❛ ❱❡♥❡③✉❡❧❛ ❝♦♠♦ ♠❡♠❜r♦ ♣❧❡♥♦ ❞♦ ▼❡r❝♦s✉❧✱ t♦r♥❛♥❞♦✲s❡ ♦ q✉✐♥t♦ ♣❛ís ❛ ❢❛③❡r ♣❛rt❡ ❞♦ ❜❧♦❝♦✱ ♦ ♥♦✈♦ ♠❛♣❛ ❞♦ ❝❤❛♠❛❞♦ ✏▼❡r❝❛❞♦ ❝♦♠✉♠ ❞♦ ❙✉❧✑ ❡stá r❡♣r❡s❡♥t❛❞♦ ❛❜❛✐①♦✿

P❡r♠✉t❛çõ❡s ❙✐♠♣❧❡s ❈❛♣ít✉❧♦ ✶

❋✐❣✉r❛ ✶✳✶✶✿

❯s❛♥❞♦ ❛♣❡♥❛s ✹ ❝♦r❡s✱ ❞❡ q✉❛♥t❛s ❢♦r♠❛s ❞✐st✐♥t❛s ♣♦❞❡♠♦s ❝♦❧♦r✐r ♦s ❝✐♥❝♦ ♣❛ís❡s q✉❡ ❝♦♠♣õ❡♠ ♦ ▼❡r❝♦s✉❧✱ ❝♦♠ ❛ ❝♦♥❞✐çã♦ q✉❡ ♣❛ís❡s ❝♦♠ ❢r♦♥t❡✐r❛ ❡♠ ❝♦♠✉♠ ♥ã♦ t❡♥❤❛♠ ❛ ♠❡s♠❛ ❝♦r❄

❙♦❧✉çã♦✿ P♦❞❡♠♦s r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛ ❢❛❝✐❧♠❡♥t❡ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❈♦♥t❛❣❡♠✳ P❛r❛ t❛♥t♦✱ s❡ ♣♦♥❤❛ ♥♦ ❧✉❣❛r ❞❛ ♣❡ss♦❛ q✉❡ ✐rá ♣✐♥t❛r ♦ ♠❛♣❛ ❞♦s ♣❛ís❡s ❞♦ ▼❡r❝♦s✉❧✳ ❱❛♠♦s ❞✐✈✐❞✐r ♥♦ss❛ r❡s♦❧✉çã♦ ❡♠ ✺ ❡t❛♣❛s✱ ❝❛❞❛ ✉♠❛ ❝♦rr❡s♣♦♥❞❡ ❛ ❡s❝♦❧❤❛ ❞❛ ❝♦r ❞❡ ❝❛❞❛ ♣❛ís✳ ❉❡✈❡♠♦s ❡♥tã♦ ✐♥✐❝✐❛r ♣❡❧❛ ❡t❛♣❛ q✉❡✱ s❡ ❞❡✐①❛❞❛ ♣❛r❛ ❞❡♣♦✐s ✐♠♣❧✐❝❛r✐❛ ❡♠ ✉♠❛ sér✐❡ ❞❡ r❡str✐çõ❡s✱ q✉❡ é ❛ ❡s❝♦❧❤❛ ❞❛ ❝♦r ❞♦ ❇r❛s✐❧ ✭❥á q✉❡ ❡st❡ ♣❛ís t❡♠ ❢r♦♥t❡✐r❛s ❡♠ ❝♦♠✉♠ ❝♦♠ t♦❞♦s ♦s ♦✉tr♦s✮✳ ■♥✐❝✐❛❧♠❡♥t❡ ♣❛r❛ ♣✐♥t❛r ♦ ❇r❛s✐❧ t❡♠♦s ✹ ♣♦ss✐❜✐❧✐❞❛❞❡s✱ ✉♠❛ ✈❡③ ♣✐♥t❛❞♦ ♦ ❇r❛s✐❧ t❡♠♦s ✸ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♣✐♥t❛r♠♦s ❛ ❆r❣❡♥t✐♥❛✳ ❯♠❛ ✈❡③ ♣✐♥t❛❞♦s ❇r❛s✐❧ ❡ ❆r❣❡♥t✐♥❛✱ t❡♠♦s ✷ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ♣✐♥t✉r❛ ❞♦ ❯r✉❣✉❛✐✳ ❯♠❛ ✈❡③ ♣✐♥t❛❞♦s ❇r❛s✐❧✱ ❆r❣❡♥t✐♥❛ ❡ ❯r✉❣✉❛✐✱ t❡♠♦s ✷ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ♣✐♥t✉r❛ ❞♦ P❛r❛❣✉❛✐ ✭✈✐st♦ q✉❡ ♣♦❞❡♠♦s ♣✐♥t❛r ❯r✉❣✉❛✐ ❡ P❛r❛❣✉❛✐ ❝♦♠ ❛ ♠❡s♠❛ ❝♦r✮ ❡ ✜♥❛❧♠❡♥t❡ ✉♠❛ ✈❡③ ♣✐♥t❛❞♦s ❇r❛s✐❧✱ ❆r❣❡♥t✐♥❛✱ ❯r✉❣✉❛✐ ❡ P❛r❛❣✉❛✐✱ t❡♠♦s ✸ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ♣✐♥t✉r❛ ❞❛ ❱❡♥❡③✉❡❧❛✳ P❡❧♦ P✳❋✳❈✳ t❡♠♦s4×3×2×2×3 = 144 ♣♦ss✐❜✐❧✐❞❛❞❡s✳

✶✳✹ P❡r♠✉t❛çõ❡s ❙✐♠♣❧❡s

◆❛ r❡s♦❧✉çã♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥t❛❣❡♠✱ é ❝♦♠✉♠ ♦ ❛♣❛r❡❝✐♠❡♥t♦ ❞❡ ❡①♣r❡ssõ❡s q✉❡ sã♦ ♣r♦❞✉t♦s ❞❡ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦s✐t✐✈♦s ❡ ❝♦♥s❡❝✉t✐✈♦s ♠❡♥♦r❡s ♦✉ ✐❣✉❛❧ ❛ ✉♠ ♥❛t✉r❛❧ n✳ P♦r

❡①❡♠♣❧♦✿ 4×3×2×1 ♦✉ 7×6×5×4×3×2×1✳ ❊ss❡s ♣r♦❞✉t♦s sã♦

❝❤❛♠❛❞♦s ❢❛t♦r✐❛✐s✳ ❆ ♥♦t❛çã♦ ✉s✉❛❧ ♣❛r❛ ❡❧❡s é✿

4! = 4×3×2×1; 7! = 7×6×5×4×3×2×1;

♦♥❞❡ s❡ ❧ê ♦ 4! ❝♦♠♦ ✏f atorial de 4✑ ❡ 7! ❝♦♠♦ ✏f atorial de 7✑✳ ❉❡ ✉♠❛

♠❛♥❡✐r❛ ❣❡r❛❧✱ s❡n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ n! é ❧✐❞♦ ❝♦♠ ✧f atorial de n”❡

n! =n×(n−1)×(n−2)×(n−3)· · · ×4×3×2×1

❖❜s❡r✈❡ q✉❡ 1! = 1 ❡✱ ♣♦r ❝♦♥✈❡♥çã♦✱ ❞❡♥♦t❛♠♦s0! = 1✳

◆♦s ♣r♦❜❧❡♠❛s ❞❡ ❈♦♥t❛❣❡♠ s✉r❣❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♣r♦❜❧❡♠❛s q✉❡ tr❛t❛♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❆♥❛❣r❛♠❛s ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣❛❧❛✈r❛ ❞❡ n ❧❡tr❛s✱

q✉❡ é ♥❛ ✈❡r❞❛❞❡ q✉❛❧q✉❡r ❛rr✉♠❛çã♦ ❞❛sn❧❡tr❛s ❝r✐❛♥❞♦ ✉♠❛ ♥♦✈❛ ♣❛❧❛✈r❛

✭q✉❡ ♣♦❞❡ t❡r s❡♥t✐❞♦ ♦✉ ♥ã♦✮✳ ❚❛♠❜é♠ ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❞❡ s❡ ♦r❣❛♥✐③❛r ✜❧❛s ✐♥❞✐❛♥❛s✱ ♦r❣❛♥✐③❛r ❧✐✈r♦s ❡♠ ♣r❛t❡❧❡✐r❛s✱ ❡♥✜♠✱ ✉♠❛ ✈❡③ t❡♥❞♦ n ❡❧❡♠❡♥t♦s ❞❡s❡❥❛♠♦s ❡♥❝♦♥tr❛r ♦ ♥ú♠❡r♦ ❞❡ ❢♦r♠❛s ❞❡ ♦r❞❡♥á✲❧♦s

❝♦♠♦ ❡♠ s❡q✉ê♥❝✐❛✳ P❛r❛ ✐♥tr♦❞✉③✐r♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ P❡r♠✉t❛çõ❡s s✐♠♣❧❡s✱ ♦❜s❡r✈❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

❊①❡♠♣❧♦ ✾ ◗✉❛♥t♦s sã♦ ♦s ❛♥❛❣r❛♠❛s ❞❛ ♣❛❧❛✈r❛ ▲■❉❊❘❄

❙♦❧✉çã♦✿ ❯s❛♥❞♦ ♦ P✳❋✳❈✱ ❞✐✈✐❞✐r❡♠♦s ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡♠ ❝✐♥❝♦ ❡t❛♣❛s✳ ❙❡ ❝♦❧♦q✉❡ ♥♦ ❧✉❣❛r ❞❛ ♣❡ss♦❛ q✉❡ ✐rá ❝♦♥str✉✐r ✉♠ ❛♥❛❣r❛♠❛✳ P❛r❛ ✐ss♦✱ ❜❛st❛ ❡s❝♦❧❤❡r♠♦s s✉❝❡ss✐✈❛♠❡♥t❡ ❛s ❧❡tr❛s q✉❡ s❡rã♦ ❝♦❧♦❝❛❞❛s ❡♠ ❝❛❞❛

P❡r♠✉t❛çõ❡s ❙✐♠♣❧❡s ❈❛♣ít✉❧♦ ✶

♣♦s✐çã♦ ❞♦ ❆♥❛❣r❛♠❛✳ P❛r❛ ❡s❝♦❧❤❡r♠♦s ❛ ❧❡tr❛ q✉❡ ♦❝✉♣❛rá ♦ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ t❡♠♦s ✺ ♣♦ss✐❜✐❧✐❞❛❞❡s❀ ♣❛r❛ ❛ s❡❣✉♥❞♦ ❧✉❣❛r ♣♦❞❡♠♦s ♣r❡❡♥❝❤ê✲❧♦ ❝♦♠ q✉❛❧q✉❡r ✉♠❛ ❞❛s q✉❛tr♦ ❧❡tr❛s r❡st❛♥t❡s❀ ♣❛r❛ ♦ t❡r❝❡✐r♦ ❧✉❣❛r t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s❀ ♣❛r❛ ♦ q✉❛rt♦✱ t❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡ ✜♥❛❧♠❡♥t❡ ♣❛r❛ ♦ q✉✐♥t♦ ❡ ú❧t✐♠♦ ❧✉❣❛r ❞♦ ❛♥❛❣r❛♠❛✱ t❡♠♦s ✉♠❛ ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡✳ ▲♦❣♦✱ ♣❡❧♦ P✳❋✳❈✱ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ❆♥❛❣r❛♠❛s é 5×4×3×2×1 = 120.

❆❣♦r❛✱ ✈❡❥❛ ♦✉tr❛ s✐t✉❛çã♦ q✉❡ ❡♥✈♦❧✈❡ ❛ ✏♦r❞❡♠✑ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥❥✉♥t♦✳

❊①❡♠♣❧♦ ✶✵ ❉❡ q✉❛♥t♦s ♠♦❞♦s ❞✐st✐♥t♦s ✻ ♣❡ss♦❛s ♣♦❞❡♠ ❢♦r♠❛r ✉♠❛ ✜❧❛ ✐♥❞✐❛♥❛❄

❙♦❧✉çã♦✿ ❊st❡ é ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❝♦♥t❛❣❡♠✱ q✉❡ é ❢❛❝✐❧♠❡♥t❡ r❡s♦❧✈✐❞♦ ♣❡❧♦ Pr✐♥❝í♣✐♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦✳ ❉❡ ❢❛t♦✱ ❜❛st❛ ❡s❝♦❧❤❡r♠♦s s✉❝❡ss✐✈❛♠❡♥t❡ ❛s ♣❡ss♦❛s ❝♦❧♦❝❛❞❛s ❡♠ ❝❛❞❛ ♣♦s✐çã♦ ❞❛ ✜❧❛✳ P❛r❛ ❡s❝♦❧❤❡r ♦ ♣r✐♠❡✐r♦ ❞❛ ✜❧❛✱ t❡♠♦s ✻ ♣♦ss✐❜✐❧✐❞❛❞❡s❀ ♦ s❡❣✉♥❞♦ ♣♦❞❡ s❡r q✉❛❧q✉❡r ✉♠❛ ❞❛s ✺ ♣❡ss♦❛s r❡st❛♥t❡s✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s é6×5×4×3×2×1 = 720

P♦❞❡♠♦s ❛❣♦r❛✱ ❛♣r❡s❡♥t❛r ✉♠❛ ✐♠♣♦rt❛♥t❡ ❞❡✜♥✐çã♦✳

❉❡✜♥✐çã♦ ✶ P❡r♠✉t❛çã♦ ❙✐♠♣❧❡s✿ ❯♠❛ ♣❡r♠✉t❛çã♦ ❞❡ ♥ ♦❜❥❡t♦s ❞✐st✐♥t♦s é q✉❛❧q✉❡r ❝♦❧❡çã♦ ♦r❞❡♥❛❞❛ ❞❡ss❡s ♥ ♦❜❥❡t♦s✳

❊①✐st❡♠ q✉❛♥t❛s ♣❡r♠✉t❛çõ❡s ♥✉♠ ❝♦♥❥✉♥t♦ ❞❡ n ♦❜❥❡t♦s❄ ❯s❛♥❞♦ ♦

P✳❋✳❈✱ ❞✐✈✐❞✐r❡♠♦s ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡♠ n ❡t❛♣❛s✳ ❈♦♠♦ ✉♠❛ ♣❡r♠✉t❛çã♦ é

❝♦❧❡çã♦ ♦r❞❡♥❛❞❛ ❞❡ss❡sn ♦❜❥❡t♦s✱ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ♣♦s✐çã♦ ❞❛ ♦r❞❡♠ ❞❡ss❡s

❡❧❡♠❡♥t♦s t❡♠♦s n ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❯♠❛ ✈❡③ ❡s❝♦❧❤✐❞♦ ♦ ♣r✐♠❡✐r♦ ❞❡ss❛

♦r❞❡♠✱ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞♦ s❡❣✉♥❞♦✱ t❡♠♦s n −1 ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❯♠❛ ✈❡③

❡s❝♦❧❤✐❞♦ ♦ s❡❣✉♥❞♦ ❡❧❡♠❡♥t♦ ❞❛ ♦r❞❡♠✱ ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞♦ t❡r❝❡✐r♦✱ t❡♠♦s