❊st✉❞♦ ❡ ❆♣❧✐❝❛çõ❡s ❞❡
Pr♦❜❛❜✐❧✐❞❛❞❡ ●❡♦♠étr✐❝❛ ❡
P❛r❛❞♦①♦s
♣♦r
❋❡r♥❛♥❞♦ ❈❡s❛r ❞❡ ❆❜r❡✉ ❱✐❛♥❛
❊st✉❞♦ ❡ ❆♣❧✐❝❛çõ❡s ❞❡
Pr♦❜❛❜✐❧✐❞❛❞❡ ●❡♦♠étr✐❝❛ ❡
P❛r❛❞♦①♦s
†♣♦r
❋❡r♥❛♥❞♦ ❈❡s❛r ❞❡ ❆❜r❡✉ ❱✐❛♥❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲ t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
▼❛rç♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
❊st✉❞♦ ❡ ❆♣❧✐❝❛çõ❡s ❞❡
Pr♦❜❛❜✐❧✐❞❛❞❡ ●❡♦♠étr✐❝❛ ❡
P❛r❛❞♦①♦s
♣♦r❋❡r♥❛♥❞♦ ❈❡s❛r ❞❡ ❆❜r❡✉ ❱✐❛♥❛
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ Pr♦❜❛❜✐❧✐❞❛❞❡✳
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ●✐❧❜❡rt♦ ❋❡r♥❛♥❞❡s ❱✐❡✐r❛ ✲ ❯❋❈●
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r t♦❞❛s ❛s ❝♦✐s❛s ❜♦❛s q✉❡ t❡♠ ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ ❞✉r❛♥t❡ t♦❞❛ ❛ ♠✐♥❤❛ ✈✐❞❛✳
➚ ♠✐♥❤❛ ❡s♣♦s❛ ❘❛q✉❡❧ ❡ ♠✐♥❤❛ ✜❧❤❛ ◆❛tá❧✐❛ ♣♦r t♦❞♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣♦r t♦❞❛ ❝♦♠♣r❡❡♥sã♦ ♥♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ♥ã♦ ♣✉❞❡ ❡st❛r ♣r❡s❡♥t❡ ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✳
❆♦s ♠❡✉s ♣❛✐s ❏♦s✐♠❛r ❡ ❆♥❛ ❛ q✉❡♠ ❛❣r❛❞❡ç♦ ❞❡ ❝♦r❛çã♦ ♣♦r t❡r❡♠ ♠❡ ♣r♦✲ ♣♦r❝✐♦♥❛❞♦ ❝♦♥❞✐çõ❡s ♠♦r❛✐s ❡ ❡str✉t✉r❛✐s ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ♥❡ss❛ ❡t❛♣❛ ❛❝❛❞ê♠✐❝❛✳
❆♦s ♠❡✉s ✐r♠ã♦s ❆♥❛ P❛✉❧❛✱ ▼❛r❝♦ ❡ ❈❛r♦❧ ♣♦r t❡r❡♠ s✐❞♦s ♣❛r❝❡✐r♦s ❡ ❛♠✐❣♦s ❞❡ ✉♠❛ ✈✐❞❛ ✐♥t❡✐r❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦♦r❞❡♥❛❞♦r❡s ❞♦ ▼❡str❛❞♦ ♣❡❧❛s ót✐♠❛s ❛✉❧❛s ❡ ♣❡❧❛s ✈❛❧♦r♦s❛s ❝♦♥tr✐❜✉✐çõ❡s ♠❛t❡♠át✐❝❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ❇r✉♥♦ ❡ ◆❛♣♦❧❡ó♥✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ▼❡str❛❞♦✱ q✉❡ ❡st✐✈❡r❛♠ ❥✉♥t♦ ❝♦♠✐❣♦ ♥❡ss❛ ❝❛♠✐♥❤❛❞❛ ❞✉r❛♥t❡ ♦s ❞♦✐s ú❧t✐♠♦s ❛♥♦s✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♠✐③❛❞❡✱ ❡♠ ❡s♣❡❝✐❛❧ ❍❡r❜❡rt✱ ●❡r❛❧❞♦ ❡ ▼❛r❝❡❧♦✳
❆♦ ♠❡✉ ❛♠✐❣♦ ❙❤❡❧❞♦♥ ♣❡❧❛s ✐♠❡♥s✉rá✈❡✐s ❝♦♥tr✐❜✉✐çõ❡s ♥♦ ▲❛t❡① ❡ s❡♠♣r❡ ♣❡❧♦ t♦r❝✐❞❛ ❞♦ s✉❝❡ss♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❯❜❡r❧❛♥❞✐♦✱ ●✐❧❜❡rt♦ ❡ ❊❧✐s❛♥❞r❛✱ ♠❡♠❜r♦s ❞❛ ❇❛♥❝❛ ❊①❛♠✐♥❛✲ ❞♦r❛ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ▼❛♥❛ssés ❳❛✈✐❡r q✉❡✱ ❛❧é♠ ❞❡ t❡r s✐❞♦ ♦ ♣r♦❢❡ss♦r ❞❛ ❞✐s❝✐♣❧✐♥❛ q✉❡ ✐♥s♣✐r♦✉ ❡ss❡ tr❛❜❛❧❤♦✱ ❛❝♦♠♣❛♥❤♦✉✱ ❝♦rr✐❣✐✉ ❡ ♦ r❡❞✐r❡❝✐♦♥♦✉ ♣♦r ❞✐✈❡rs❛s ✈❡③❡s s❡♠♣r❡ ❝♦♠ ❞❡❞✐❝❛çã♦ ❡ s❛❜❡❞♦r✐❛✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦✱ ❛♣ós ✉♠❛ ❜r❡✈❡ r❡s✉♠♦ ❤✐stór✐❝♦ ❡ t❡ór✐❝♦ s♦❜r❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❛❜♦r❞❛ ♦ t❡♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛✳ ❊♥t❡♥❞❡♠♦s q✉❡ ❡ss❡ é ✉♠ r❛♠♦ ✐♠✲ ♣♦rt❛♥t❡ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ t✐✈❡♠♦s ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ❡st✉❞❛♠♦s ♦ ♠❛✐s ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦✲ ♠étr✐❝❛✱ q✉❡ é ♦ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥✳ ❆♣ós ❛❧❣✉♥s ❛♥♦s✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ ♣♦ss✐❜✐❧✐t♦✉ ❆❧❧❛♥ ▼❛❝▲❡♦❞ ❈♦r♠❛❝❦ ❡ ●♦❞❢r❡② ◆❡✇❜♦❧❞ ❍♦✉♥s✜❡❧❞✱ ❣❛✲ ♥❤❛❞♦r❡s ❞♦ Prê♠✐♦ ◆♦❜❡❧ ❞❛ ▼❡❞✐❝✐♥❛✱ ♦ ✐♥✈❡♥t♦ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t♦♠♦❣r❛✜❛ ❝♦♠♣✉t❛❞♦r✐③❛❞❛✳ ◆♦ tr❛❜❛❧❤♦ t❛♠❜é♠ é ❛♣r❡s❡♥t❛❞♦ ✉♠❛ ❢♦r♠❛ ✐♥t❡r❡ss❛♥t❡ ❞❡ ❝❛❧❝✉❧❛r ár❡❛s ❞❡ ✜❣✉r❛s ♥ã♦ ❡❧❡♠❡♥t❛r❡s ✉s❛♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛ ❛tr❛✈és ❞♦ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦✳ ❯♠ ♦✉tr♦ tó♣✐❝♦ ❛❜♦r❞❛❞♦ ❞✐③ r❡s♣❡✐t♦ ❛♦s ♣❛r❛❞♦①♦s ♣r♦❜❛❜✐❧íst✐❝♦s✳ ❖s ♣❛r❛❞♦①♦s ❛♣r❡s❡♥t❛❞♦s sã♦ ❛q✉❡❧❡s q✉❡ sã♦ ❝♦♥trár✐♦ ❛♦ s❡♥s♦ ❝♦♠✉♠✳
P❛❧❛✈r❛s✲❈❤❛✈❡✿ Pr♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛✱ ♣❛r❛❞♦①♦✱ ✐♥t✉✐çã♦✱ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥✳
❆❜str❛❝t
❚❤✐s ✇♦r❦✱ ❛❢t❡r ❛ ❜r✐❡❢ ❤✐st♦r② ❛♥❞ t❤❡♦r② ♦❢ ♣r♦❜❛❜✐❧✐t②✱ ❛♣♣r♦❛❝❤❡s t❤❡ s✉❜❥❡❝t ❣❡♦♠❡tr✐❝ ♣r♦❜❛❜✐❧✐t②✳ ❲❡ ❜❡❧✐❡✈❡ ✐t ✐s ❛♥ ✐♠♣♦rt❛♥t ❜r❛♥❝❤ ♦❢ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ❛♥❞ ✇❡ ❤❛❞ t❤❡ ♦♣♣♦rt✉♥✐t② t♦ ♣r❡s❡♥t s♦♠❡ ❡①❛♠♣❧❡s✳ ■♥✐t✐❛❧❧②✱ ✇❡ st✉❞✐❡❞ t❤❡ ♠♦st ❢❛♠♦✉s ♣r♦❜❧❡♠ ✐♥ ❣❡♦♠❡tr✐❝ ♣r♦❜❛❜✐❧✐t②✱ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❜❧❡♠ ♦❢ ❇✉✛♦♥✬s ♥❡❡❞❧❡✳ ❆❢t❡r ❛ ❢❡✇ ②❡❛rs✱ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠ ❛❧❧♦✇❡❞ ❆❧❧❛♥ ▼❛❝▲❡♦❞ ❈♦r♠❛❝❦ ❛♥❞ ●♦❞❢r❡② ◆❡✇❜♦❧❞ ❍♦✉♥s✜❡❧❞✱ ◆♦❜❡❧ Pr✐③❡ ✇✐♥♥❡rs ✐♥ ▼❡❞✐❝✐♥❡✱ t❤❡ ✐♥✈❡♥t✐♦♥ ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❝♦♠♣✉t❡❞ t♦♠♦❣r❛♣❤②✳ ❚❤✐s ✇♦r❦ ❛❧s♦ ♣r❡s❡♥ts ❛♥ ✐♥t❡r❡st✐♥❣ ✇❛② t♦ ❝❛❧❝✉❧❛t❡ ❛r❡❛s ♦❢ ♥♦ ❡❧❡♠❡♥t❛r② ✜❣✉r❡s ❜② ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr✐❝ ♣r♦❜❛❜✐❧✐t② ✈✐❛ t❤❡ ▼♦♥t❡ ❈❛r❧♦ ▼❡t❤♦❞✳ ❆♥♦t❤❡r t♦♣✐❝ ❛❞❞r❡ss❡❞ ❝♦♥❝❡r♥s ♣r♦❜❛❜✐❧✐st✐❝ ♣❛r❛✲ ❞♦①❡s✳ ❚❤❡ ♣❛r❛❞♦①❡s ♣r❡s❡♥t❡❞ ❛r❡ t❤♦s❡ ✇❤✐❝❤ ❛r❡ ❝♦♥tr❛r② t♦ ❝♦♠♠♦♥ s❡♥s❡✳
❑❡②✇♦r❞s✿ ●❡♦♠❡tr✐❝ ♣r♦❜❛❜✐❧✐t②✱ ♣❛r❛❞♦① ❛♥❞ ✐♥t✉✐t✐♦♥✱ ❇✉✛♦♥✬s ♥❡❡❞❧❡✳
❙✉♠ár✐♦
✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ✶
✶✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❏♦❣♦s ❞❡ ❆③❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ●❡r♦❧❛♠♦ ❈❛r❞❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✹ P❛s❝❛❧ ❡ ❋❡r♠❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✷ ❇r❡✈❡ ❘❡✈✐sã♦ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s ✽
✷✳✶ ❊①♣❡r✐♠❡♥t♦ ❆❧❡❛tór✐♦ ❡ ❊①♣❡r✐♠❡♥t♦ ❉❡t❡r♠✐♥íst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❊s♣❛ç♦ ❆♠♦str❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✸ ❊✈❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡ ✭❉❡✜♥✐çã♦ ❈❧áss✐❝❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✺ ❊✈❡♥t♦s ❊q✉✐♣r♦✈á✈❡✐s ❡ ❊✈❡♥t♦s ♥ã♦ ❊q✉✐♣r♦✈á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✻ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✼ Pr♦❜❛❜✐❧✐❞❛❞❡ ❈♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✽ Pr♦♣r✐❡❞❛❞❡s ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✾ ❊✈❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸ Pr♦❜❛❜✐❧✐❞❛❞❡ ●❡♦♠étr✐❝❛ ✶✾
✸✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✷✳✶ ❱❛❧♦r❡s ❡①♣❡r✐♠❡♥t❛✐s ♣❛r❛π ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✷✳✷ ❆♣❧✐❝❛çã♦ ♥❛ ▼❡❞✐❝✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✸ ❯♠❛ ✈❛r✐❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✹ ❖ Pr♦❜❧❡♠❛ ❞❡ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ❊①t❡♥sã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✻ ▼ét♦❞♦ ❞❡ ▼♦♥t❡ ❈❛r❧♦ ♣❛r❛ ❝á❧❝✉❧♦ ❞❡ ár❡❛s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❡♠
s❛❧❛ ❞❡ ❛✉❧❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✻✳✶ ❊①♣❡r✐ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹ P❛r❛❞♦①♦s ✹✹
✹✳✶ P❛r❛❞♦①♦ ❞♦ ❆♥✐✈❡rsár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✷ P❛r❛❞♦①♦ ❞♦s ❚rês ❉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✸ ❖ ♣r♦❜❧❡♠❛ ❞❡ ▼♦♥t② ❍❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✹ P❛r❛❞♦①♦ ❞❛s ❈♦✐♥❝✐❞ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✹✳✶ ❖ ♣r♦❜❧❡♠❛ ❞❛ s❡❝r❡tár✐❛ ❞❡s❛t❡♥t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✺ ❖ P❛r❛❞♦①♦ ❞❡ ❇❡rtr❛♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✻ ❖ ♣r♦❜❧❡♠❛ ❞❛s ♠♦❡❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❆ ❆♣ê♥❞✐❝❡ ✻✾
❆✳✶ P❡r♠✉t❛çõ❡s ❝❛ót✐❝❛s ♦✉ ❞❡s❛rr❛♥❥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ❆✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ♦✉ Pr✐♥❝í♣✐♦ ❞❛s ●❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t ✳ ✼✷
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✹
■♥tr♦❞✉çã♦
❆ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s é ✉♠ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡①tr❡♠❛♠❡♥t❡ ❛♣❧✐❝á✈❡❧ ♥❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s ❡ ✐♥✐❝✐❛♠♦s ♦ ♥♦ss♦ tr❛❜❛❧❤♦ ❢❛③❡♥❞♦ ✉♠ ♣❡q✉❡♥♦ r❡tr♦s♣❡❝t♦ ❞❛ s✉❛ ❤✐stór✐❛✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ✜③❡♠♦s ✉♠❛ r❡❢❡rê♥❝✐❛ à ♠♦t✐✈❛çã♦ ✐♥✐❝✐❛❧ ❞♦ ❡st✉❞♦ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✱ q✉❡ ❢♦r❛♠ ♦s ❥♦❣♦s ❞❡ ❛③❛r✱ ♠✉✐t♦ ❞✐❢✉♥❞✐❞♦s ♥❛ ✐❞❛❞❡ ♠é❞✐❛ ❡ ♥❛ ✐❞❛❞❡ ♠♦❞❡r♥❛ ❛té ♦s ❞✐❛s ❛t✉❛✐s✳
❖ ✐♥í❝✐♦ ❞❛ ❢♦r♠❛❧✐③❛çã♦ ❞♦ ❡st✉❞♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠ ❈❛r❞❛♥♦✱ P❛s❝❛❧ ❡ ❋❡r♠❛t t❛♠❜é♠ ❢♦✐ ❛❜♦r❞❛❞♦✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ✜③❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❛ t❡♦r✐❛✱ r❡❧❡♠❜r❛♠♦s ❛ ❞❡✜♥✐çã♦ ❝❧áss✐❝❛✱ ❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✱ ♦ ❚❡♦r❡♠❛ ❞♦ ♣r♦❞✉t♦ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s✳ ❖s ♣♦♥t♦s ❡st✉❞❛❞♦s s❡r✈❡♠ ❞❡ ❛❧✐❝❡r❝❡s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳
❏á ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ tr❛t❛♠♦s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛✳ ❋✐③❡♠♦s ✉♠❛ ❛♥á✲ ❧✐s❡ ❞♦ ♣r✐♥❝✐♣❛❧ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛ q✉❡ é ♦ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥ q✉❡ ♣❡❞❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❛❣✉❧❤❛ ❧❛♥ç❛❞❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♠ ✉♠ ❛ss♦❛❧❤♦ ❝♦♠ ❧✐♥❤❛s ♣❛r❛❧❡❧❛s ❝❛✐r s❡♠ ❤❛✈❡r ✐♥t❡rs❡çã♦ ❝♦♠ ❡ss❛s ❧✐♥❤❛s✳
❋✐③❡♠♦s ✉♠❛ ❛♣❧✐❝❛çã♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ ❢♦✐ ❡st✐♠❛r ♦ ✈❛❧♦r ❞❡ π ✉s❛♥❞♦ ♦ ♣r♦✲
❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥✳ ❯s❛♠♦s t❛♠❜é♠ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛ ♣❛r❛ r❡❛✲ ❧✐③❛r ❝á❧❝✉❧♦s ❞❡ ár❡❛s ♥ã♦ ❡❧❡♠❡♥t❛r❡s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✳
◆♦ q✉❛rt♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ tr❛t❛♠♦s ❛❧❣✉♥s ♣❛r❛❞♦①♦s ♣r♦❜❛❜✐❧íst✐❝♦s✳ ❆ ♣❛❧❛✲ ✈r❛ ♣❛r❛❞♦①♦ ❢♦✐ ✉s❛❞❛ ♥♦ s❡♥t✐❞♦ ❞❡ s❡r ❝♦♥tr❛ ❛ ♦♣✐♥✐ã♦ ❝♦♠✉♠✳ ❋✐③❡♠♦s ✉♠ ❡st✉❞♦
❞❡t❛❧❤❛❞♦ ❞♦s ♣❛r❛❞♦①♦s ❞♦ ❛♥✐✈❡rsár✐♦ ❡ ❞♦s três ❞❛❞♦s✳ ❊♠ s❡❣✉✐❞❛ ❞✐s❝✉t✐♠♦s ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❡ ▼♦♥t② ❍❛❧❧ ❡ ✉s❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❇❛②❡s ♥❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❛❧❣✉♥s ❞❡ss❡s r❡s✉❧t❛❞♦s✳ P♦r ✜♠✱ ✜③❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞♦ ♣❛r❛❞♦①♦ ❞❛s ❝♦✐♥❝✐❞ê♥✲ ❝✐❛s ❡ ❞♦ ♣r♦❜❧❡♠❛ ❞❛s ♠♦❡❞❛s✳ ❚❛♠❜é♠ ❢♦✐ ❡s❝r✐t♦ ✉♠ ❛♣ê♥❞✐❝❡ ♥♦ q✉❛❧ ♦s ❧❡✐t♦r❡s ♠❛✐s ✐♥t❡r❡ss❛❞♦s ♣♦❞❡♠ ♦❜t❡r ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♣❡r♠✉t❛çõ❡s ❝❛ót✐❝❛s ❡ s♦❜r❡ ♦ ♣r✐♥❝í♣✐♦ ❞❛s ❣❛✈❡t❛s ❞❡ ❉✐r✐❝❤❧❡t✳
❈❛♣ít✉❧♦ ✶
❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛
✶✳✶ ■♥tr♦❞✉çã♦
❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡stá ♠✉✐t♦ ♣r❡s❡♥t❡ ❡♠ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✳ ◆♦ ✐♥í❝✐♦ ❞❡ t♦❞❛ ♣❛rt✐❞❛ ❞♦ ❝❛♠♣❡♦♥❛t♦ ❜r❛s✐❧❡✐r♦ ❞❡ ❢✉t❡❜♦❧✱ ♦ ❥✉✐③ ❧❛♥ç❛ ✉♠❛ ♠♦❡❞❛ ♣❛r❛ ❝✐♠❛ ❡ ♦s ❝❛♣✐tã❡s ❞♦s ❝❧✉❜❡s ❢❛③❡♠ ❛ ❡s❝♦❧❤❛ ❡♥tr❡ ❝❛r❛ ❡ ❝♦r♦❛✳ ❆♦ ❞❡s❝♦❜r✐r q✉❡ ❡stá ❣rá✈✐❞❛✱ ✉♠❛ ♠✉❧❤❡r ❡ s❡✉ ♠❛r✐❞♦✱ ❡ ♠✉✐t♦s ❢❛♠✐❧✐❛r❡s ❝♦st✉♠❛♠ ✏❛♣♦st❛r✑ q✉❛❧ s❡rá ♦ s❡①♦ ❞♦ ❜❡❜ê✳ ❆♦ ✈❡r✐✜❝❛r♠♦s ❛ ♣r❡✈✐sã♦ ❞♦ t❡♠♣♦ ♣❛r❛ ♦ ✜♥❛❧ ❞❡ s❡♠❛♥❛✱ ✜❝❛♠♦s s❛❜❡♥❞♦ q✉❛❧ ❛ ❝❤❛♥❝❡ ❞❡ ❝❤♦✈❡r✳ ◗✉❛♥❞♦ ❥♦❣❛♠♦s ♥❛ ❧♦t❡r✐❛✱ s❛❜❡♠♦s q✉❡ ♦ ♥♦ss♦ ❜✐❧❤❡t❡ t❡♠✱ ❡♠❜♦r❛ ♠✉✐t♦ ♣❡q✉❡♥❛✱ ✉♠❛ ❝❤❛♥❝❡ ❞❡ s❡r ♦ ♣r❡♠✐❛❞♦✳
❆ ❡ss❡s ❡ ♠✉✐t♦s ♦✉tr♦s ❛❝♦♥t❡❝✐♠❡♥t♦s ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦ t❡♠♦s ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠♦ ✜❣✉r❛ ♣r✐♥❝✐♣❛❧✳ ▼❡s♠♦ ♦s ❧❡✐❣♦s ❡♠ ▼❛t❡♠át✐❝❛ ♣❛r❡❝❡♠ ❡♥t❡♥❞❡r q✉❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ q✉❡ q✉❛♥t✐✜❝❛ ❛ ❝❤❛♥❝❡ ❞❡ ❛❧❣♦ ✈✐r ❛ ♦❝♦rr❡r✳
❆ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s é ♦ r❛♠♦ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ♣❡sq✉✐s❛ ❡ ❞❡s❡♥✈♦❧✈❡ ♠♦❞❡❧♦s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ♠❛✐s ❞✐✈❡rs♦s ❢❡♥ô♠❡♥♦s ❛❧❡❛tór✐♦s✳ ❆ ♣❛❧❛✈r❛ ♣r♦❜❛❜✐❧✐✲ ❞❛❞❡ é ❞❡r✐✈❛❞❛ ❞♦ ❧❛t✐♠ ♣r♦❜❛r❡✱ q✉❡ s✐❣♥✐✜❝❛ ♣r♦✈❛r ♦✉ t❡st❛r✳ ➱ ❝♦♠✉♠ ✉s❛r♠♦s ❛ ♣❛❧❛✈r❛ ♣r♦✈á✈❡❧ ♣❛r❛ ✐♥❞✐❝❛r ❛❧❣♦ q✉❡ ♥ã♦ s❡ t❡♠ ❝❡rt❡③❛ q✉❡ ✈❛✐ ❛❝♦♥t❡❝❡r✳ ❚❛♠✲ ❜é♠ é ❢r❡q✉❡♥t❡ ❡st❛ ♣❛❧❛✈r❛ ❡st❛r ❛ss♦❝✐❛❞❛ às ♣❛❧❛✈r❛s s♦rt❡✱ ❛③❛r✱ ✐♥❝❡rt♦✱ ❝❤❛♥❝❡
✶✳✷✳ ❏❖●❖❙ ❉❊ ❆❩❆❘
❡ ❞✉✈✐❞♦s♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❝♦♥t❡①t♦ ❞❛ ❢r❛s❡✳
❆s ♣r✐♠❡✐r❛s ♠❛♥✐❢❡st❛çõ❡s ❝❛t❛❧♦❣❛❞❛s ❡♥✈♦❧✈❡♥❞♦ ♣r♦❜❛❜✐❧✐❞❛❞❡ r❡❢❡r❡♠✲s❡ ❛ ♣rát✐❝❛s ❞❡ ✏❥♦❣♦s ❞❡ ❛③❛r✑✳
✶✳✷ ❏♦❣♦s ❞❡ ❆③❛r
❏♦❣♦s ❞❡ ❛③❛r sã♦ ❛q✉❡❧❡s q✉❡ ♥ã♦ ❞❡♣❡♥❞❡♠ t♦t❛❧♠❡♥t❡ ❞❛ ❤❛❜✐❧✐❞❛❞❡ ❞♦ ❥♦❣❛✲ ❞♦r✱ ♠❛s ❡①❝❧✉s✐✈❛ ♦✉ ♣r❡❞♦♠✐♥❛♥t❡♠❡♥t❡ ❞❡ s✉❛ s♦rt❡ ♦✉ ❞❡ s❡✉ ❛③❛r✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ ✈♦❝ê r❡♣❡t✐r ❛ r❡❛❧✐③❛çã♦ ❞♦ ❡✈❡♥t♦ ❡♠ ❝♦♥❞✐çõ❡s ✐❞ê♥t✐❝❛s ❛ ❛♥t❡r✐♦r✱ ❛✐♥❞❛ ❛ss✐♠ ♥ã♦ ♣♦❞❡rá ♣r❡✈❡r ❝♦♠ ❡①❛t✐❞ã♦ q✉❛❧ s❡rá ♦ r❡s✉❧t❛❞♦✳
❖s ❥♦❣♦s ❞❡ ❞❛❞♦s s✉r❣✐r❛♠ ❤á ♠✉✐t♦ t❡♠♣♦✱ ♠❛s ♥ã♦ ❡r❛♠ ❞❛ ❢♦r♠❛ ❝♦♠♦ ❝♦♥❤❡❝❡♠♦s ❤♦❥❡✳ ❖s ❞❛❞♦s ❝♦♠✉♥s ♣♦ss✉❡♠ s❡✐s ❢❛❝❡s ♥✉♠❡r❛❞❛s ❞❡ 1 ❛ 6 ❡ tê♠
❛ ❢♦r♠❛ ❞❡ ✉♠ ❤❡①❛❡❞r♦ r❡❣✉❧❛r✳ ❖s ❞❛❞♦s ❛♥t✐❣♦s✱ q✉❡ ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❛strá❧❛❣♦s✱ ❡r❛♠ ❢♦r♠❛❞♦s ♣♦r ♦ss♦s ❞❡ ♣❛t❛s ❞❡ ❛♥✐♠❛✐s✱ t❛✐s ❝♦♠♦ ✈❡❛❞♦s✱ ❜❡③❡rr♦s✱ ♦✈❡❧❤❛s ♦✉ ❝❛❜r❛s ❡✱ ❡♠❜♦r❛ ♣♦ss✉íss❡♠ s❡✐s ❢❛❝❡s ♥ã♦ r❡❣✉❧❛r❡s✱ ❛♣❡♥❛s q✉❛tr♦ ♣♦ss✉✐❛♠ ❡st❛❜✐❧✐❞❛❞❡ s✉✜❝✐❡♥t❡ ♣❛r❛ s❡r✈✐r❡♠ ❞❡ ❛♣♦✐♦ ♥♦ ❝❤ã♦✳ P❛r❛ ❛s ❢❛❝❡s ♠❛✐♦r❡s✱ ❡r❛♠ ❛tr✐❜✉í❞♦s ♦s ♥ú♠❡r♦s 3 ❡ 4✱ ❡♥q✉❛♥t♦ q✉❡ ♣❛r❛ ❛s ♠❡♥♦r❡s ❡r❛♠
❛tr✐❜✉í❞♦s ♦s ♥ú♠❡r♦s 1 ❡6✳ ❉❡ss❛ ❢♦r♠❛✱ ❛s ❢r❡q✉ê♥❝✐❛s ❞♦s r❡s✉❧t❛❞♦s ♣❛r❛ ♦ ❚❛❧✐
✭❝♦♠♦ ❡r❛♠ ❝♦♥❤❡❝✐❞♦s ♦s ❥♦❣♦s ❞❡ ♦ss♦s✮ ❡r❛♠ ❜❡♠ ❞✐❢❡r❡♥t❡s✱ ❝♦♠♦ ♥♦s ♠♦str❛ ❛ t❛❜❡❧❛ ❛ s❡❣✉✐r✳ ❊ss❡s ❞❛❞♦s t❛♠❜é♠ ❡r❛♠ ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡❛❧✐③❛r ♣r❡✈✐sõ❡s ❛❝❡r❝❛ ❞♦ ❢✉t✉r♦✳ P❛r❛ ✐ss♦✱ ❡r❛♠ ❥♦❣❛❞♦s ✈ár✐♦s ♦ss♦s ❞❡ ✉♠❛ ✈❡③ ❡ r❡❛❧✐③❛❞♦ ✉♠❛ ✏❧❡✐t✉r❛✑✳
❋❛❝❡s
1
3
4
6
❋r❡q✉ê♥❝✐❛ ✵✱✶✷ ✵✱✸✼ ✵✱✸✾ ✵✱✶✷
❖s ❣r❡❣♦s ❡ ♦s r♦♠❛♥♦s t❛♠❜é♠ ✉t✐❧✐③❛✈❛♠ ♦ ❛strá❧❛❣♦ ❡♠ s❡✉s ❥♦❣♦s✱ ♣♦ré♠ ❝♦♠
4 ✉♥✐❞❛❞❡s ❞❡ ✉♠❛ ✈❡③✳ ❆ ❥♦❣❛❞❛ ❞❡ ✈ê♥✉s✱ ❛ ♠❛✐s ✈❛❧✐♦s❛ ❞♦ ❥♦❣♦✱ ❡r❛ ❛q✉❡❧❛ q✉❡
✶✳✷✳ ❏❖●❖❙ ❉❊ ❆❩❆❘
❛s q✉❛tr♦ ❢❛❝❡s ✈♦❧t❛❞❛s ♣❛r❛ ❝✐♠❛ ❡r❛♠ t♦❞❛s ❞✐st✐♥t❛s✳ ❏á ♦ ❧❛♥❝❡ ♠❡♥♦s ✈❛❧✐♦s♦ ❞♦ ❥♦❣♦✱ ❡r❛ ❛ ♦❜t❡♥çã♦ ❞❡ q✉❛tr♦ ✉♥s✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✏♦s ❝ã❡s✑✳ ❈✉r✐♦s❛♠❡♥t❡✱ ♦ ❧❛♥❝❡ ♠❡♥♦s ✈❛❧✐♦s♦ ❞♦ ❥♦❣♦ ❡r❛ t❛♠❜é♠ ♦ ♠❛✐s r❛r♦ ❞❡ ❛❝♦♥t❡❝❡r✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛♣r♦①✐♠❛❞❛ ❞❡1❡♠ 5000✱ ♠❛s ♦s ❣r❡❣♦s ❡ ♦s r♦♠❛♥♦s ♥ã♦ ♣♦ss✉✐❛♠ t❛❧ ✐♥❢♦r♠❛çã♦✳
◆♦ ❇r❛s✐❧✱ ❛ ▲❡✐ 3688 ❞❡ 1941 ✭▲❡✐ ❞♦s ❥♦❣♦s ❞❡ ❛③❛r✮ ♣r♦í❜❡ ✏♦ ❥♦❣♦ ❡♠ q✉❡ ♦
❋✐❣✉r❛ ✶✳✶✿ ❆strá❧❛❣♦s✳
❣❛♥❤♦ ❡ ❛ ♣❡r❞❛ ❞❡♣❡♥❞❡♠ ❡①❝❧✉s✐✈❛ ♦✉ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❛ s♦rt❡✑✳ ❆ ❡①❝❡çã♦ é ❢❡✐t❛ ❛♦s ❥♦❣♦s ❛✉t♦r✐③❛❞♦s ♣❡❧❛ ❈❛✐①❛ ❊❝♦♥ô♠✐❝❛ ❋❡❞❡r❛❧ ♦✉ ❥♦❣♦s q✉❡ ♥ã♦ ❡♥✈♦❧✈❛♠ ✈❛♥t❛❣❡♥s ✜♥❛♥❝❡✐r❛s✳ ❖s ❥♦❣♦s ❞❡ ❛③❛r ♠♦t✐✈❛r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✳
❊♠ 960 ♦ ❜✐s♣♦ ❜❡❧❣❛ ❲✐❜♦❧❞✱ ❞❛ ❝✐❞❛❞❡ ❞❡ ❈❛♠❜r❛✐✱ ❡♥✉♠❡r♦✉ ❝♦rr❡t❛♠❡♥t❡
✭s❡♠ ❛s ♣❡r♠✉t❛çõ❡s✮ ♦s 56r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ♥♦ ❥♦❣♦ ❞❡ 3 ❞❛❞♦s✳ P❛r❛ ❝❛❞❛ ✉♠
❞♦s r❡s✉❧t❛❞♦s✱ ❡❧❡ ❛tr✐❜✉✐✉ ✉♠❛ ✈✐rt✉❞❡ ❡ ❝r✐♦✉ ❛ss✐♠ ✉♠ ❥♦❣♦ ♠♦r❛❧✳ ❈♦♠ ✐ss♦ ♦ ❇✐s♣♦ ❞❡✉ ✉♠ ♣❛ss♦ ♥♦ ❝❛♠✐♥❤♦ ❞❛ ❢♦r♠❛❧✐③❛çã♦ ❞❛ t❡♦r✐❛ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
▼❛✐s t❛r❞❡✱ ❡♥tr❡ 1220 ❡ 1250✱ ❘✐❝❤❛r❞ ❞❡ ❋♦r♥✐✈❛❧ ❡s❝r❡✈❡✉ ✉♠ ♣♦❡♠❛ ❧❛t✐♥♦✱
❞✐✈✐❞✐❞♦ ❡♠ três ❧✐✈r♦s ❡ ❡♥t✐t✉❧❛❞♦ ❉❡ ❱❡t✉❧❛✳ ❆❜❛✐①♦ s❡❣✉❡ ❛ tr❛❞✉çã♦ ❞❡ ✉♠❛ ♣❛ss❛❣❡♠ ❞♦ ♣♦❡♠❛ ✭✈✐❞❡ ❬✶❪✮ q✉❡ ❛♣r❡s❡♥t❛ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ❧❛♥ç❛♠❡♥t♦ ❞❡
3 ❞❛❞♦s✱ ❛❣♦r❛ ❝♦♠ ❛ ♣❡r♠✉t❛çã♦✿
✶✳✸✳ ●❊❘❖▲❆▼❖ ❈❆❘❉❆◆❖
❚❛❧✈❡③✱ ❞✐r❡♠♦s q✉❡ ❝❡rt♦s ♥ú♠❡r♦s sã♦ ♠❡❧❤♦r❡s ❉♦ q✉❡ ♦✉tr♦s ♣❛r❛ ✉s♦ ❡♠ ❥♦❣♦s✱ ♣❡❧❛ r❛③ã♦ q✉❡✱ ❉❡s❞❡ q✉❡ ✉♠ ❞❛❞♦ t❡♥❤❛ s❡✐s ❧❛❞♦s ❡ s❡✐s ♥ú♠❡r♦s ✉♥✐tár✐♦s✱
❊♠ três ❞❛❞♦s ❡①✐st❡♠ ❞❡③♦✐t♦✱
❉♦s q✉❛✐s ❛♣❡♥❛s três ♣♦❞❡♠ ❡st❛r ♥❛s ❢❛❝❡s s✉♣❡r✐♦r❡s ❞♦s ❞❛❞♦s✳ ❊❧❡s ✈❛r✐❛♠ ❡♠ ❞✐❢❡r❡♥t❡s ♠❛♥❡✐r❛s ❡ ❞❡❧❡s✱
❉❡③❡ss❡✐s ♥ú♠❡r♦s ❝♦♠♣♦st♦s sã♦ ♣r♦❞✉③✐❞♦s✳ ❊❧❡s ♥ã♦ sã♦✱ ♣♦ré♠✱ ❉❡ ✐❣✉❛❧ ✈❛❧♦r✱ ❞❡s❞❡ q✉❡ ♦ ♠❛✐♦r ❡ ♦ ♠❡♥♦r ❞❡❧❡s
❖❝♦rr❛ r❛r❛♠❡♥t❡ ❡ ♦s ❞♦ ♠❡✐♦ ♠❛✐s ❢r❡q✉❡♥t❡♠❡♥t❡✱ ❊ ♦ r❡st❛♥t❡✱ ♦ q✉❛♥t♦ ♠❛✐s ♣ró①✐♠♦ ❡stã♦ ❞❛q✉❡❧❡s ❞♦ ♠❡✐♦✱
▼❡❧❤♦r❡s sã♦ ❡ ♠❛✐s ❢r❡q✉❡♥t❡♠❡♥t❡ ♦❝♦rr❡♠✳
❊ss❡s✱ q✉❛♥❞♦ ♦❝♦rr❡♠✱ tê♠ ❛♣❡♥❛s ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❢❛❝❡s ♥♦s ❞❛❞♦s✱ ❆q✉❡❧❡s sã♦ s❡✐s✱ ❡ ♦s r❡st❛♥t❡s tê♠ ❝♦♥✜❣✉r❛çõ❡s ✐♥t❡r♠❡❞✐ár✐❛s ❡♥tr❡ ♦s ❞♦✐s✱
❚❛✐s q✉❡ ❡①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s ♠❛✐♦r❡s ❡ ❡①❛t❛♠❡♥t❡ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡♥♦r❡s✱
❊ ❡ss❡s tê♠ ✉♠❛ ❝♦♥✜❣✉r❛çã♦✳ ❖s ❞♦✐s s❡❣✉✐♥t❡s✱
❖ ♠❛✐♦r✱ ❡ ♦ ♦✉tr♦ ♠❡♥♦r✱ tê♠ ❞✉❛s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❢❛❝❡s ♥♦s ❞❛❞♦s ❝❛❞❛ ✉♠✳ ◆♦✈❛♠❡♥t❡✱ ❞❡♣♦✐s ❞❡❧❡s ❡①✐st❡♠ três ❝❛❞❛ ✉♠✱ ❡♥tã♦ q✉❛tr♦ ❝❛❞❛ ✉♠✳
❊ ❝✐♥❝♦ ❝❛❞❛ ✉♠✱ ❝♦♠♦ ❡❧❡s s❡❣✉❡♠ ❡♠ s✉❝❡ssã♦ ❞❡ ❛♣r♦①✐♠❛çã♦✳
❖s q✉❛tr♦ ♥ú♠❡r♦s ❞♦ ♠❡✐♦ tê♠ s❡✐s ❝♦♥✜❣✉r❛çõ❡s ❞❡ ❢❛❝❡s ♥♦s ❞❛❞♦s ❝❛❞❛ ✉♠✳
✶✳✸ ●❡r♦❧❛♠♦ ❈❛r❞❛♥♦
➱ ✐♠♣♦ssí✈❡❧ ❞❡s✈✐♥❝✉❧❛r ♦ ❡st✉❞♦ ❞♦s ❥♦❣♦s ❞❡ ❛③❛r ❞❡ ●❡r♦❧❛♠♦ ❈❛r❞❛♥♦ ✭1501
✲ 1576✮✳ ❋✐❧❤♦ ❞❡ ❋❛③✐♦ ❈❛r❞❛♥♦✱ q✉❡ ❡r❛ ✉♠ ❜❡♠ s✉❝❡❞✐❞♦ ❝♦♥s❡❧❤❡✐r♦ ❡ ❞❡ ❈❤✐❛r❛
✶✳✸✳ ●❊❘❖▲❆▼❖ ❈❆❘❉❆◆❖
❋✐❣✉r❛ ✶✳✷✿ ❚r❡❝❤♦ ❞♦ ♣♦❡♠❛ ❉❡ ❱❡t✉❧❛ ♥❛ ✈❡rsã♦ ✐♠♣r❡ss❛ ❡♠ ✶✺✸✹ ✭✈✐❞❡ ❬✶❪✮✳
▼✐❝❤❡r✐✱ ❈❛r❞❛♥♦ ♥❛s❝❡✉ ❡♠ P❛✈✐❛ ❞❡♣♦✐s ❞❡ ✉♠ ❞♦❧♦r♦s♦ tr❛❜❛❧❤♦ ❞❡ ♣❛rt♦ q✉❡ ❞✉✲ r♦✉ três ❞✐❛s✳ ❆ ♣❛rt❡✐r❛ ❢❡③ ✉♠❛ ♣r❡✈✐sã♦ ❞❡ ✉♠❛ ♠♦rt❡ ❜r❡✈❡✱ ❝❡r❝❛ ❞❡ ✉♠❛ ❤♦r❛ ❞❡ ✈✐❞❛ ♣❛r❛ ❈❛r❞❛♥♦✱ ♠❛s ♣❛r❛ ❛ s✉r♣r❡s❛ ❞❡ t♦❞♦s ❡ ♣❛r❛ ♦ ❜❡♠ ❞❛ ▼❛t❡♠át✐❝❛✱ ❡❧❡ s♦❜r❡✈✐✈❡✉ ♣♦r q✉❛s❡ 75 ❛♥♦s✳ ❆✐♥❞❛ ❜❡❜ê✱ ❈❛r❞❛♥♦ ❝♦♥tr❛✐✉ ❛ ♣❡st❡ ❜✉❜ô♥✐❝❛
❡ ❝♦♥tr❛r✐♦✉ ♠❛✐s ✉♠❛ ✈❡③ ❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ✉♠❛ ♠♦rt❡ ❜r❡✈❡✳ ❙❡✉s três ✐r♠ã♦s✱ q✉❡ t❛♠❜é♠ ❝♦♥tr❛ír❛♠ ❛ ♣❡st❡✱ ♥ã♦ t✐✈❡r❛♠ ❛ ♠❡s♠❛ s♦rt❡ ❡ ♠♦rr❡r❛♠✳ ❊♠ 1516
❈❛r❞❛♥♦ ❞❡❝✐❞✐✉ ❞❡✐①❛r ❛ ❝❛s❛ ❞❛ ❢❛♠í❧✐❛ ❡ ✈✐❛❥❛r ♣❛r❛ ❡st✉❞❛r ▼❡❞✐❝✐♥❛✱ ❝♦♥tr❛r✐✲ ❛♥❞♦ s❡✉ ♣❛✐ q✉❡ ❣♦st❛r✐❛ q✉❡ ❈❛r❞❛♥♦ ❡st✉❞❛ss❡ ❉✐r❡✐t♦ ❡ t✐✈❡ss❡ ❛ss✐♠ ♣❡r♠✐ssã♦ ❞❡ r❡❝❡❜❡r ✉♠❛ ❛❥✉❞❛ ❞❡ ❝✉st♦ ❛♥✉❛❧ ❞❡ 100 ❝♦r♦❛s✳
P❛r❛ ♠❛♥t❡r✲s❡ ❧♦♥❣❡ ❞❛ ❢❛♠í❧✐❛ ❡ ❝✉st❡❛r ♦s ❡st✉❞♦s✱ ❈❛r❞❛♥♦ ❝♦♠❡ç♦✉ ❛ ♣❛rt✐❝✐✲ ♣❛r ❞♦s ❥♦❣♦s ❞❡ ❛③❛r✳ P♦r t❡r ❝♦♠♣r❡❡♥❞✐❞♦ ✉♠ ♣♦✉❝♦ ❞♦s r❡s✉❧t❛❞♦s✱ ♠❛✐s ❡ ♠❡♥♦s ♣r♦✈á✈❡✐s ♥❛s ❛♣♦st❛s✱ ❡♠ ♣♦✉❝♦ t❡♠♣♦ ❥á ❤❛✈✐❛ ❥✉♥t❛❞♦1.000 ❝♦r♦❛s✱ ♦ ❡q✉✐✈❛❧❡♥t❡
❛ ✉♠❛ ❞é❝❛❞❛ ❞❛ ❛❥✉❞❛ ❞♦ ❝✉st♦ ❛♥✉❛❧ ♣❛r❛ ♦ ❝✉rs♦ ❞❡ ❉✐r❡✐t♦✳
❈❛r❞❛♥♦ ❝r✐♦✉ ✉♠❛ r❡❣r❛✱ ❡♠ s❡✉ ❧✐✈r♦ ▲í❜❡r ❉❡ ▲✉❞♦ ❆❧❡❛❡ ✭❖ ❧✐✈r♦ ❞♦s ❥♦❣♦s ❞❡ ❛③❛r✮ q✉❡ ❢♦✐ ♣✉❜❧✐❝❛❞♦ ❡♠ 1665✱ q✉❛s❡ ✉♠ sé❝✉❧♦ ❛♣ós ❛ s✉❛ ♠♦rt❡✱ q✉❡ ❡♠
✶✳✹✳ P❆❙❈❆▲ ❊ ❋❊❘▼❆❚
❧✐♥❣✉❛❣❡♠ ❛t✉❛❧ s❡ tr❛❞✉③ ♣♦r✿ ✏❙✉♣♦♥❤❛ q✉❡ ✉♠ ♣r♦❝❡ss♦ ❛❧❡❛tór✐♦ t❡♥❤❛ ♠✉✐t♦s r❡s✉❧t❛❞♦s ✐❣✉❛❧♠❡♥t❡ ♣r♦✈á✈❡✐s✱ ❛❧❣✉♥s ❢❛✈♦rá✈❡✐s ✭♦✉ s❡❥❛✱ ❣❛♥❤❛r✮✱ ♦✉tr♦s ❞❡s❢❛✲ ✈♦rá✈❡✐s ✭♣❡r❞❡r✮✳ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❜t❡r♠♦s ✉♠ r❡s✉❧t❛❞♦ ❢❛✈♦rá✈❡❧ é ✐❣✉❛❧ ❛ ♣r♦♣♦rçã♦ ❡♥tr❡ r❡s✉❧t❛❞♦s ❢❛✈♦rá✈❡✐s ❡ ♦ t♦t❛❧ ❞❡ r❡s✉❧t❛❞♦s✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❛♠♦str❛❧✑ ✭✈✐❞❡ ❬✷❪✮ ✳
❊ss❛ r❡❣r❛ ❞❡ ❈❛r❞❛♥♦ é ❛ ❞❡✜♥✐çã♦ ❝❧áss✐❝❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡ s❡rá r❡t♦♠❛❞❛ ❡♠ ❜r❡✈❡✳ ❈♦♠♦ s❡✉ ❧✐✈r♦ ❞❡♠♦r♦✉ ♠✉✐t♦ t❡♠♣♦ ♣❛r❛ s❡ t♦r♥❛r ♣ú❜❧✐❝♦✱ ❞♦✐s ♦✉tr♦s ♣❡rs♦♥❛❣❡♥s t✐✈❡r❛♠ ♣❛♣❡❧ ♣r❡❞♦♠✐♥❛♥t❡ ♣❛r❛ ✐♠♣✉❧s✐♦♥❛r ❛ ❢♦r♠❛❧✐③❛çã♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✿ ❇❧❛✐s❡ P❛s❝❛❧ ✭1623 ✲ 1662✮ ❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭1601 ✲ 1665✮✳
✶✳✹ P❛s❝❛❧ ❡ ❋❡r♠❛t
P❛s❝❛❧ ♥❛s❝❡✉ ❛♦ s✉❧ ❞❡ P❛r✐s ❡ ❛♦s 13 ❛♥♦s ❥á ♣❛rt✐❝✐♣❛✈❛ ❞❡ r❡✉♥✐õ❡s ❡♠ ✉♠
❣r✉♣♦ ❞❡ ❞✐s❝✉ssã♦ ❝♦♠ ✐♠♣♦rt❛♥t❡s ✐♥t❡❧❡❝t✉❛✐s ❝♦♠♦ ❘❡♥é ❉❡s❝❛rt❡s ✭1596✲1650✮✳
❏á ❡♠ 1651✱ ❝♦♠ ♦ ❢❛❧❡❝✐♠❡♥t♦ ❞❡ s❡✉ ♣❛✐✱ P❛s❝❛❧ ❤❡r❞♦✉ ✉♠❛ ❣r❛♥❞❡ ❤❡r❛♥ç❛ ❡
♣❛ss♦✉ ❛ ♣❛rt✐❝✐♣❛r ❞❡ ❞✐✈❡rs❛s ❢❡st❛s✳ ❯♠ ❞❡ s❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❢❡st❛✱ ❆♥t♦✐♥❡✲ ●♦♠❜❛✉❞ ✭❈❤❡✈❛❧✐❡r ❞❡ ▼ér❡✮✱ ✉♠ ❝♦♥❤❡❝✐❞♦ ❛♣♦st❛❞♦r✱ ❢❡③ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛ ❛ P❛s❝❛❧✿ s❡ ✈♦❝ê ❡ ♦✉tr♦ ❥♦❣❛❞♦r✱ ✐❣✉❛❧♠❡♥t❡ ❤á❜✐❧✱ ♣❛rt✐❝✐♣❛♠ ❞❡ ✉♠ ❥♦❣♦ ❡♠ q✉❡ ♦ ✈❡♥❝❡❞♦r é ❛q✉❡❧❡ q✉❡ ❛t✐♥❣✐r ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s✱ ❡ ❛♥t❡s ❞♦ ✜♥❛❧✱ ❝♦♠ ✉♠ ❥♦❣❛❞♦r ♥❛ ❧✐❞❡r❛♥ç❛✱ ♦ ❥♦❣♦ é ✐♥t❡rr♦♠♣✐❞♦✱ ❝♦♠♦ s❡r✐❛ ❛ ❞✐✈✐sã♦ ❥✉st❛ ❞♦ ❞✐♥❤❡✐r♦ ❛♣♦st❛❞♦❄ ❊ss❡ ♣r♦❜❧❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ♣r♦❜❧❡♠❛ ❞♦s ♣♦♥t♦s✳
P❛s❝❛❧ ♣❡r❝❡❜❡✉ q✉❡ ❛s ❢❡rr❛♠❡♥t❛s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ ❛✐♥❞❛ ♥ã♦ ❡①✐st✐❛♠ ❡ ❡s❝r❡✈❡✉ ♣❛r❛ s❡✉ ❛♠✐❣♦ P✐❡rr❡ ❞❡ ❋❡r♠❛t✱ ❝♦♠❡ç❛♥❞♦ ✉♠❛ ❞❛s tr♦❝❛s ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛ ❤✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ◆❡ss❛ é♣♦❝❛✱ ❋❡r✲ ♠❛t ♦❝✉♣❛✈❛ ✉♠ ❛❧t♦ ❝❛r❣♦ ♥❛ ❝♦rt❡ ❝r✐♠✐♥❛❧ ❞❡ ❚♦✉❧♦✉s❡ ❡ ♥ã♦ ❡r❛ ✉♠ ♣r♦✜ss✐♦♥❛❧ ❞❛ ▼❛t❡♠át✐❝❛✳ ❋❡r♠❛t✱ ♣♦r s✉❛ ❣❡♥✐❛❧✐❞❛❞❡✱ é ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❛✐♦r ♠❛t❡♠át✐❝♦ ❛♠❛❞♦r ❞❛ ❤✐stór✐❛✳ ❚❛♥t♦ P❛s❝❛❧ q✉❛♥t♦ ❋❡r♠❛t ❝❤❡❣❛r❛♠ à s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❝♦♥s✐❞❡r❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡♠❜♦r❛ ❝❛❞❛ ✉♠ t❡♥❤❛ ❡s❝♦❧❤✐❞♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞✐❢❡✲
✶✳✹✳ P❆❙❈❆▲ ❊ ❋❊❘▼❆❚
r❡♥t❡✳
❈♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ❝♦♠♦ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ ❛ ❡s♣❡r❛♥ç❛ ♠❛t❡♠át✐❝❛ ❢♦r❛♠ s❡♥❞♦ ❢♦r♠❛❧✐③❛❞♦s ❛ ♣❛rt✐r ❞❛s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❞❡ P❛s❝❛❧ ❡ ❋❡r♠❛t✳ ◆♦s ❛♥♦s q✉❡ s❡ s✉❝❡ss❡❞❡r❛♠✱ ♦✉tr♦s ✐♠♣♦rt❛♥t❡s ♠❛t❡♠át✐❝♦s ❝♦♠♦ ❏❛♠❡s ❇❡r♥♦✉❧❧✐✱ ▲❛✲ ♣❧❛❝❡✱ ●❛✉ss ❡ P♦✐ss♦♥ ❝♦♥tr✐❜✉ír❛♠ ❝♦♠ ♥♦✈♦s ❝♦♥❝❡✐t♦s✱ ❢♦r♠❛❧✐③❛çã♦ ❡ ❛♣❧✐❝❛çõ❡s ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
❈❛♣ít✉❧♦ ✷
❇r❡✈❡ ❘❡✈✐sã♦ ❞❡ Pr♦❜❛❜✐❧✐❞❛❞❡s
✷✳✶ ❊①♣❡r✐♠❡♥t♦ ❆❧❡❛tór✐♦ ❡ ❊①♣❡r✐♠❡♥t♦ ❉❡t❡r♠✐✲
♥íst✐❝♦
❙❡ ✜③❡r♠♦s r❡♣❡t✐❞♦s ❧❛♥ç❛♠❡♥t♦s ❞❡ ✉♠ ❞❛❞♦✱ ❝♦♠ s✉❛s ❢❛❝❡s ♥✉♠❡r❛❞❛s ❞❡1❛ 6✱ ❝♦♠ ❛ ♠❡s♠❛ ❛❧t✉r❛ ❞♦ s♦❧♦✱ ❥♦❣❛♥❞♦ s❡♠♣r❡ ❝♦♠ ❛ ♠❡s♠❛ ✐♥t❡♥s✐❞❛❞❡ ❞❡ ❢♦rç❛✱
❡❢❡t✉❛♥❞♦ ♦ ❧❛♥ç❛♠❡♥t♦ s❡♠♣r❡ ❞♦ ♠❡s♠♦ ❧♦❝❛❧✱ ♦✉ s❡❥❛✱ ❡♠ ❝♦♥❞✐çõ❡s ✐❞ê♥t✐❝❛s✱ ✈❡r❡♠♦s r❡s✉❧t❛❞♦s ❞✐❢❡r❡♥t❡s✳ ◆ã♦ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛♥t❡❝✐♣❛❞❛♠❡♥t❡ q✉❛❧ s❡rá ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦✱ ♣♦✐s ❡❧❡ ❞❡♣❡♥❞❡rá ❞♦ ❛❝❛s♦✳ ❊①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦ é ❛q✉❡❧❡ q✉❡ r❡♣❡t✐❞♦ ❞✐✈❡rs❛s ✈❡③❡s✱ s♦❜ ❝♦♥❞✐çõ❡s ✐❞ê♥t✐❝❛s✱ ♣r♦❞✉③❡♠ r❡s✉❧t❛❞♦s ❞✐❢❡r❡♥t❡s✳
❖ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠❛ ♠♦❡❞❛ é ✉♠ ❡①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦✱ ♣♦✐s ♥ã♦ ♣♦❞❡♠♦s ♣r❡✈❡r ❝♦♠ ❡①❛t✐❞ã♦ s❡ ♦ r❡s✉❧t❛❞♦ s❡rá ❝❛r❛ ♦✉ ❝♦r♦❛✳ ❘❡t✐r❛r ❛❧❡❛t♦r✐❛♠❡♥t❡ ✉♠❛ ❝❛rt❛ ❞❡ ✉♠ ❜❛r❛❧❤♦ q✉❡ ❝♦♥té♠ 52❝❛rt❛s✱ s♦rt❡❛r ✉♠ ♥ú♠❡r♦ ❡♥tr❡ 1❛10✱ s♦rt❡❛r
✉♠❛ ❜♦❧❛ ❞❡ ✉♠❛ ✉r♥❛ q✉❡ ❝♦♥té♠ 5 ❜♦❧❛s ❞❡ ❝♦r❡s ❞✐❢❡r❡♥t❡s sã♦ ♦✉tr♦s ❡①❡♠♣❧♦s
❞❡ ❡①♣❡r✐♠❡♥t♦s ❛❧❡❛tór✐♦s✳
❆❣♦r❛✱ s❡ ✈♦❝ê ❢❛③ ✉♠ ❡①♣❡r✐♠❡♥t♦ ♣❛r❛ s❛❜❡r q✉❛❧ s❡rá ❛ t❡♠♣❡r❛t✉r❛ ❡♠ q✉❡ ❛ á❣✉❛ ♣✉r❛✱ ❡♠ ❝♦♥❞✐çõ❡s ♥♦r♠❛✐s✱ ❢❡r✈❡✱ t❡rá ❝♦♠♦ r❡s✉❧t❛❞♦ 100♦C✳ ❙❡ ♦ ❡①✲
♣❡r✐♠❡♥t♦ ❢♦r ♣❛r❛ ❞❡s❝♦❜r✐r q✉❛❧ ♦ ♣♦♥t♦ ❞❡ ❢✉sã♦ ❞♦ ❣❡❧♦✱ ♦ r❡s✉❧t❛❞♦ s❡rá 0♦C✳
✷✳✷✳ ❊❙P❆➬❖ ❆▼❖❙❚❘❆▲
❊①♣❡r✐♠❡♥t♦ ❞❡t❡r♠✐♥íst✐❝♦ é ❛q✉❡❧❡ q✉❡ r❡♣❡t✐❞♦ ❡♠ ❝♦♥❞✐çõ❡s ✐❞ê♥t✐❝❛s ♣r♦❞✉③ r❡s✉❧t❛❞♦s ✐❞ê♥t✐❝♦s✳ ❖ ❡①♣❡r✐♠❡♥t♦ ❞❡t❡r♠✐♥íst✐❝♦ é ú♥✐❝♦ ❡ ♣r❡✈✐sí✈❡❧✳
✷✳✷ ❊s♣❛ç♦ ❆♠♦str❛❧
■♠❛❣✐♥❡ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ ✈♦❝ê ❞❡✈❡rá ❞❡s❝r❡✈❡r t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ❞❡ 6 ❢❛❝❡s✳ ❖s r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ❝♦♠ ❛ ❢❛❝❡ ✈♦❧t❛❞❛
♣❛r❛ ❝✐♠❛ sã♦✿ 1,2,3,4,5✱ ♦✉ 6✳ ❈❤❛♠❛♠♦s ❞❡ ❡s♣❛ç♦ ❛♠♦str❛❧ ❡ r❡♣r❡s❡♥t❛r❡♠♦s
♣♦r Ω ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦✳
❊♥tã♦ t❡♠♦s q✉❡ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ss❡ ❞❛❞♦ ❞❡ s❡✐s ❢❛❝❡s é Ω =
{1,2,3,4,5,6}✳
◆♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ❞✉❛s ♠♦❡❞❛s✱ t❡r❡♠♦s ❝♦♠♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ♦ ❝♦♥❥✉♥t♦ Ω =
{(K, K),(K, C),(C, K),(C, C)}✱ ♦♥❞❡ ❛ ❧❡tr❛K r❡♣r❡s❡♥t❛ ♦ r❡s✉❧t❛❞♦ ❝❛r❛ ❡ ❛ ❧❡tr❛ C r❡♣r❡s❡♥t❛ ♦ r❡s✉❧t❛❞♦ ❝♦r♦❛✳
◆♦s ❞♦✐s ❡①❡♠♣❧♦s ❛❝✐♠❛✱ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ❡r❛ ❢♦r♠❛❞♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ ♣♦ré♠ s❡ ❛ s✐t✉❛çã♦ ❢♦ss❡ ❡s❝♦❧❤❡r ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥♦ ✐♥t❡r✈❛❧♦ [1,10]✱ t❡rí❛♠♦s ✉♠
❡s♣❛ç♦ ❛♠♦str❛❧ ❝♦♠ ✐♥✜♥✐t♦s ❡❧❡♠❡♥t♦s✳
✷✳✸ ❊✈❡♥t♦
■♠❛❣✐♥❡ ✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ ✈♦❝ê ♣❛rt✐❝✐♣❛ ❞❡ ✉♠ ❥♦❣♦ ❞❡ t❛❜✉❧❡✐r♦ ❝♦♠ ❛❧❣✉♥s ❝♦❧❡❣❛s ❡ ♣❛r❛ ❣❛♥❤❛r ♣r❡❝✐s❛ t✐r❛r ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ❝✐♥❝♦ ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ❝♦♠✉♠ ❞❡ s❡✐s ❢❛❝❡s✳ ❖ ❝♦♥❥✉♥t♦ q✉❡ r❡♣r❡s❡♥t❛ ♦s r❡s✉❧t❛❞♦s q✉❡ ❞❛rã♦ ❛ ✈✐tór✐❛ ❛ ✈♦❝ê éA={5,6}✳ ❖❜s❡r✈❡ q✉❡ ❡ss❡ ❝♦♥❥✉♥t♦ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❡s♣❛ç♦
❛♠♦str❛❧ Ω = {1,2,3,4,5,6}✳ ❊st❡ s✉❜❝♦♥❥✉♥t♦ é ♦ ❡✈❡♥t♦ ❞❡s❡❥❛❞♦ ♣♦r ✈♦❝ê ♣❛r❛
❣❛♥❤❛r ♦ ❥♦❣♦✳ ◗✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ❞♦ ❡s♣❛ç♦ ❛♠♦str❛❧ é ❝❤❛♠❛❞♦ ❞❡ ❡✈❡♥t♦ ❡ ❣❡r❛❧♠❡♥t❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ❧❡tr❛s ❞♦ ♥♦ss♦ ❛❧❢❛❜❡t♦✿ A, B, C, ...✳
✷✳✹✳ P❘❖❇❆❇■▲■❉❆❉❊ ✭❉❊❋■◆■➬➹❖ ❈▲➪❙❙■❈❆✮
✷✳✹ Pr♦❜❛❜✐❧✐❞❛❞❡ ✭❉❡✜♥✐çã♦ ❈❧áss✐❝❛✮
❆♦ ❧❛♥ç❛r♠♦s ✉♠❛ ♠♦❡❞❛✱ ✉s❛♥❞♦ ❛ ♥♦ss❛ ✐♥t✉✐çã♦✱ ❞✐③❡♠♦s q✉❡ ❛ ❝❤❛♥❝❡ ❞❡ ❝❛✐r q✉❛❧q✉❡r ✉♠❛ ❞❛s ❢❛❝❡s ✈♦❧t❛❞❛s ♣❛r❛ ❝✐♠❛ é ❛ ♠❡s♠❛✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❡ 50%
❞❡ ❝❤❛♥❝❡ ❞❡❧❛ ❝❛✐r ❝♦♠ ❛ ❢❛❝❡ ❝♦r♦❛ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ ❡ 50% ❞❡ ❝❤❛♥❝❡ ❞❡❧❛ ❝❛✐r
❝♦♠ ❛ ❢❛❝❡ ❝❛r❛ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛✳ ❆ ❝❤❛♥❝❡ ❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❞❡ss❡s ❡✈❡♥t♦s é ❝❤❛♠❛❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠ ❡✈❡♥t♦ A✱ ✐♥❞✐❝❛❞❛ ♣♦r P(A) é ❛ r❛③ã♦
❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❡✈❡♥t♦ n(A) ❡ ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞♦ ❡s♣❛ç♦
❛♠♦str❛❧ n(Ω)✱ ♦✉ s❡❥❛✱
P(A) = n(A)
n(Ω), ✭✷✳✶✮
♦✉✱ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱
P(A) = ♥ú♠❡r♦ ❞❡ ❝❛s♦s ❢❛✈♦rá✈❡✐s ❛♦ ❡✈❡♥t♦ ❆
♥ú♠❡r♦ ❞❡ r❡s✉❧t❛❞♦s ♣♦ssí✈❡✐s ❞♦ ❡①♣❡r✐♠❡♥t♦. ❊ss❛ ❢♦✐ ❛ ❞❡✜♥✐çã♦ ✉s❛❞❛ ♣♦r ❈❛r❞❛♥♦ ❡♠ s❡✉ ❧✐✈r♦ ▲í❜❡r ❉❡ ▲✉❞♦ ❆❧❡❛❡✳
❊①❡♠♣❧♦ ✶ ◆♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ❞❡ s❡✐s ❢❛❝❡s✱ q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❜✲ t❡r♠♦s ✉♠ ♥ú♠❡r♦ ♣❛r❄
❙♦❧✉çã♦✿ ❖ ❡✈❡♥t♦ A s❡rá ❞❛❞♦ ♣♦r A = {2,4,6}✳ ▲♦❣♦✱ n(A) = 3✳ ❖ ❡s♣❛ç♦
❛♠♦str❛❧ é Ω = {1,2,3,4,5,6}✳ ▲♦❣♦✱ n(Ω) = 6✳ P♦rt❛♥t♦✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡
♦❜t❡r♠♦s ✉♠ ♥ú♠❡r♦ ♣❛r ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ é ❝❛❧❝✉❧❛❞♦ ❢❛③❡♥❞♦ ✉s♦ ❞❛ ❢ór♠✉❧❛ ✭✷✳✶✮✱ ♦✉ s❡❥❛✱
P(A) = 3 6 =
1 2,
♦✉ s❡❥❛✱ 50%✳ ❖❜s❡r✈❡ q✉❡ ♦s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❞❛❞♦s ❡♠ ❢r❛çã♦✱ ❞❡❝✐♠❛❧ ♦✉
♣♦r❝❡♥t❛❣❡♠✱ s❡♥❞♦ ♦ ♣r✐♠❡✐r♦ ❣❡r❛❧♠❡♥t❡ ♦ ♠❛✐s ❢á❝✐❧ ❞❡ s❡r tr❛❜❛❧❤❛❞♦✳
✷✳✺✳ ❊❱❊◆❚❖❙ ❊◗❯■P❘❖❱➪❱❊■❙ ❊ ❊❱❊◆❚❖❙ ◆➹❖ ❊◗❯■P❘❖❱➪❱❊■❙
❉❛ ❞❡✜♥✐çã♦✱ ♣♦❞❡♠♦s ❡①tr❛✐r ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s✿
✭❛✮ P(A)≥0❀
✭❜✮ ❙❡ A ❡ B sã♦ ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ❞❡Ω✱ ✐st♦ é✱ A∩B =∅✱ t❡♠✲s❡ q✉❡
P(A∪B) =P(A) +P(B).
✭❝✮ P(Ω) = 1✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❡✈❡♥t♦ é ❝❡rt♦✳
✭❞✮ P(∅) = 0✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❡✈❡♥t♦ é ✐♠♣♦ssí✈❡❧✳
✷✳✺ ❊✈❡♥t♦s ❊q✉✐♣r♦✈á✈❡✐s ❡ ❊✈❡♥t♦s ♥ã♦ ❊q✉✐♣r♦✲
✈á✈❡✐s
◗✉❛♥❞♦ ✈♦❝ê ❢❛③ ♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠❛ ♠♦❡❞❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s❛✐r ❛ ❢❛❝❡ ❝❛r❛ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ é ❞❡ 50% ❡ ❞❡ s❛✐r ❛ ❢❛❝❡ ❝♦r♦❛ é t❛♠❜é♠ ❞❡ 50%✱ ❝♦♠♦
✈✐♠♦s✳ ❏á ♥♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ ✉♠ ❞❛❞♦ ❞❡ s❡✐s ❢❛❝❡s✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ✐♥❞✐✈✐❞✉❛❧ ❞❡ s❛✐r q✉❛❧q✉❡r ✉♠❛ ❞❛s ❢❛❝❡s é1/6✳ ❆ ❡ss❡s ❡✈❡♥t♦s✱ q✉❡ t❡♠ ❛ ♠❡s♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡
❞❡ ♦❝♦rr❡r✱ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❡✈❡♥t♦s ❡q✉✐♣r♦✈á✈❡✐s✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❞❛❞♦ é ♣❡r❢❡✐t♦✱ ❤♦♥❡st♦ ♦✉ ♥ã♦ ✈✐❝✐❛❞♦✳
❆❣♦r❛✱ ✐♠❛❣✐♥❡ ✉♠ ❥♦❣❛❞♦r ❞❡s♦♥❡st♦ q✉❡ ❝♦♥❢❡❝❝✐♦♥❛ ✉♠ ❞❛❞♦ ❞❡ s❡✐s ❢❛❝❡s✱ ♣♦ré♠ s❡♠ ❛ ❢❛❝❡ ❝♦♠ ♦ ♥ú♠❡r♦1✱ ❝♦♠ ❞✉❛s ❢❛❝❡s ❝♦♠ ♦ ♥ú♠❡r♦6❡ ❛s ❞❡♠❛✐s ❢❛❝❡s
❝♦♠ ❛ ✐♥❝✐❞ê♥❝✐❛ ♥♦r♠❛❧✳ ◆♦ ❧❛♥ç❛♠❡♥t♦ ❞❡ss❡ ❞❛❞♦✱ q✉❛❧ s❡r✐❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ s❛✐r ❛ ❢❛❝❡ ❝♦♠ ♦ ♥ú♠❡r♦ 1 ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛❄ ❊ ❝♦♠ ❛ ❢❛❝❡ ♥ú♠❡r♦ 4❄ ❊ ❝♦♠ ❛
❢❛❝❡ ♥ú♠❡r♦ 6❄
➱ ❢á❝✐❧ ♦❜s❡r✈❛r q✉❡ ❛s r❡s♣♦st❛s sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ P(A) = 0 ✭❡✈❡♥t♦ ✐♠✲
♣♦ssí✈❡❧✮✱ P(B) = 1/6 ❡ P(C) = 2/6 = 1/3✳ ❊ss❡s ❡✈❡♥t♦s ♥ã♦ tê♠ ❛ ♠❡s♠❛
♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛❝♦♥t❡❝❡r ❡ sã♦ ❝❤❛♠❛♠♦s ❞❡ ❡✈❡♥t♦s ♥ã♦ ❡q✉✐♣r♦✈á✈❡✐s✳ ❖ ❞❛❞♦
✷✳✻✳ ❆▲●❯▼❆❙ P❘❖P❘■❊❉❆❉❊❙ ❉❊ P❘❖❇❆❇■▲■❉❆❉❊
❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r é ❝❤❛♠❛❞♦ ❞❡ ❞❛❞♦ ✐♠♣❡r❢❡✐t♦✱ ❞❡s♦♥❡st♦ ♦✉ ✈✐❝✐❛❞♦✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠♦s ❛ ♣❧❛♥✐✜❝❛çã♦ ❞❡ ❞♦✐s ❞❛❞♦s ✈✐❝✐❛❞♦s ❞❡ 6 ❢❛❝❡s✳
✷✳✻ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡
◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❞❡♠♦♥str❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❢❡r❡♥t❡s às ♣r♦❜❛❜✐❧✐❞❛✲ ❞❡s✳
Pr♦♣♦s✐çã♦ ✶ ❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✿
✭❛✮ P(Ac) = 1−P(A)✱ ♦♥❞❡ Ac é ♦ ❡✈❡♥t♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ A✳
✭❜✮ ❙❡ A⊂B✱ ❡♥tã♦ P(A)≤P(B)✳
✭❝✮ P(A∪B) =P(A) +P(B)−P(A∩B)✳
❉❡♠♦♥str❛çã♦✿ ✭❛✮ ❉❡ ❢❛t♦✱ ❝♦♠♦ ♦s ❝♦♥❥✉♥t♦s A ❡ Ac sã♦ ❞✐s❥✉♥t♦s✱ s❡❣✉❡ q✉❡ 1 =P(Ω) =P(A∪Ac) = P(A) +P(Ac)
❡✱ ♣♦rt❛♥t♦✱
P(Ac) = 1−P(A).
✷✳✼✳ P❘❖❇❆❇■▲■❉❆❉❊ ❈❖◆❉■❈■❖◆❆▲
✭❜✮ ❙❡ A⊂B✱ ❡♥tã♦ B =A∪(B−A)✱ ❞♦♥❞❡
P(B) =P[A∪(B−A)]
❡✱ ♣♦rt❛♥t♦✱
P(B) = P(A) +P(B−A).
❉❛í✱ P(A) =P(B)−P(B −A)≤P(B)✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✭❜✮✳
✭❝✮ ❱❡❥❛ q✉❡ A ∪ B = A ∪• (B − A)✳ ▲♦❣♦✱ P(A ∪ B) = P(A) + P(B − A)✳
❆❧é♠ ❞✐ss♦✱ ❝♦♠♦B = (A∩B)∪• (B−A)✱ t❡♠♦s q✉❡P(B) =P(A∩B) +P(B−A)✳
P♦rt❛♥t♦✱
P(A∪B)−P(B) = P(A) +P(B−A)−P(B)
= P(A) +P(B−A)−P(A∩B)−P(B−A) = P(A)−P(A∩B).
❉❛í✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦✳
✷✳✼ Pr♦❜❛❜✐❧✐❞❛❞❡ ❈♦♥❞✐❝✐♦♥❛❧
❈♦♥s✐❞❡r❡ ❛ s✐t✉❛çã♦ ❡♠ q✉❡ ✈♦❝ê ♣r❡❝✐s❛ ❛❝❡rt❛r ❛♥t❡❝✐♣❛❞❛♠❡♥t❡ ❡♠ ✉♠❛ ú♥✐❝❛ t❡♥t❛t✐✈❛✱ ♦ ♥ú♠❡r♦ ❞❛ ❢❛❝❡ q✉❡ ✜❝❛rá ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ ❞❡ ✉♠ ❞❛❞♦ ❤♦♥❡st♦ ❞❡ s❡✐s ❢❛❝❡s✳ ❖❜✈✐❛♠❡♥t❡ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✈♦❝ê ❛❝❡rt❛r ♦ r❡s✉❧t❛❞♦ é 1/6✱ ♣♦✐s Ω =
{1,2,3,4,5,6} ❡ ✈♦❝ê ♣♦ss✉✐ ❛♣❡♥❛s ✉♠❛ t❡♥t❛t✐✈❛✳ P♦ré♠✱ s❡ ❛♣ós ♦ ❧❛♥ç❛♠❡♥t♦✱
❛❧❣✉é♠ ♣❛ss❛ ❛ ✐♥❢♦r♠❛çã♦ ❞❡ q✉❡ ♦ r❡s✉❧t❛❞♦ ❢♦✐ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ✐♥t✉✐t✐✈❛♠❡♥t❡ ✈♦❝ê s❛❜❡ q✉❡ s✉❛ ❝❤❛♥❝❡ ❛✉♠❡♥t❛ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡✳ ❆ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛❝❡rt♦ s❛❧t❛ ♣❛r❛ 1/3✱ ♣♦✐s ♦ s❡✉ ♥♦✈♦ ❡s♣❛ç♦ ❛♠♦str❛❧ é Ω′ = {2,4,6}✳ ❉❡ss❛ ❢♦r♠❛✱
❞❡✜♥✐r❡♠♦s P(B|A) ❝♦♠♦ s❡♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r ♦ ❡✈❡♥t♦ B✱ s❛❜❡♥❞♦
✷✳✽✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆ P❘❖❇❆❇■▲■❉❆❉❊ ❈❖◆❉■❈■❖◆❆▲
q✉❡ ♦ ❡✈❡♥t♦ A ❥á ♦❝♦rr❡✉✳ ❉❛í✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡ P(B|A) = P(A∩B)
P(A) . ✭✷✳✷✮
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ✉♠ ❞❛❞♦ ❤♦♥❡st♦ ❞❡ 6 ❢❛❝❡s ❡ ♦s ❡✈❡♥t♦s A✿ s❛✐r ✉♠
♥ú♠❡r♦ ♣❛r ❡ B✿ s❛✐r ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳ ◗✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛❝♦♥t❡❝❡r ♦
❡✈❡♥t♦ B✱ s❛❜❡♥❞♦ q✉❡ ♦ ❡✈❡♥t♦ A❥á ♦❝♦rr❡✉❄ ❚❡♠♦s q✉❡ P(A) = 1/2✱P(B) = 1/2
❡ P(A∩B) = 1/6✳ P♦rt❛♥t♦✱ ♣♦r ✭✷✳✷✮✱ t❡♠♦s q✉❡
P(B|A) = 1/6 1/2 =
1 3.
✷✳✽ Pr♦♣r✐❡❞❛❞❡s ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧
◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❞❡s❝r❡✈❡r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♣r♦❜❛❜✐❧✐✲ ❞❛❞❡ ❝♦♥❞✐❝✐♦♥❛❧✳
Pr♦♣♦s✐çã♦ ✷ ❈♦♥s✐❞❡r❡ ✉♠ ❡✈❡♥t♦ A ❞❡ ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧ Ω✳ ❚❡r❡♠♦s ❛s s❡✲
❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭❛✮ P(∅|A) = 0✳
✭❜✮ P(Ω|A) = 1✳
✭❝✮ 0≤P(A|B)≤1✳
❉❡♠♦♥str❛çã♦✿ ❉❡ ❢❛t♦✱ ♣❛r❛ ♦ ✐t❡♠ ✭❛✮✱ ♥♦t❡ ❞❡ ✭✷✳✷✮ q✉❡
P(∅|A) = P(∅ ∩A)
P(A) = 0
P(A) = 0.
❏á ♣❛r❛ ♦ ✐t❡♠ ✭❜✮✱ t❡♠♦s
P(Ω|A) = P(Ω∩A)
P(A) =
P(A)
P(A) = 1.
✷✳✽✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆ P❘❖❇❆❇■▲■❉❆❉❊ ❈❖◆❉■❈■❖◆❆▲
❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♦ A∩B ⊂A✱ s❡❣✉❡ q✉❡ P(A∩B)≤P(A)✱ ❞♦♥❞❡
0 = 0
P(A) ≤
P(A∩B)
P(A)
(2.2)
= P(A|B)≤ P(A)
P(A) = 1,
♦ q✉❡ ♣r♦✈❛ ♦ ú❧t✐♠♦ ✐t❡♠✳
Pr♦♣♦s✐çã♦ ✸ ✭❚❡♦r❡♠❛ ❞♦ Pr♦❞✉t♦✮ ❉❛❞♦s A1, A2, . . . , An ❡✈❡♥t♦s ❞❡ ✉♠ ❡s✲
♣❛ç♦ ❛♠♦str❛❧ Ω✱ t❡♠♦s q✉❡
P(A1∩A2∩. . .∩An)
=P(A1)·P(A2|A1)·P(A3|A1∩A2)· · ·P(An|A1∩A2∩. . .∩An−1).
❉❡♠♦♥str❛çã♦✿ ❆ ❞❡♠♦♥str❛çã♦ é ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳ ❖ ❝❛s♦ n = 1 é
✐♠❡❞✐❛t♦✳ ❙✉♣♦♥❤❛ q✉❡ n= 2✳ ❊♥tã♦✱ t❡♠♦s ♣♦r ✭✷✳✷✮ q✉❡
P(A2|A1) =
P(A2∩A1)
P(A1)
,
♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡
P(A2∩A1) = P(A1)·P(A2|A1).
❊st❛ ú❧t✐♠❛ ❡q✉❛çã♦ é ❛ ♣r♦♣r✐❡❞❛❞❡ r❡❢❡r❡♥t❡ à ♣r♦♣♦s✐çã♦ ❝♦♠ n = 2✳ ❙✉♣♦♥❤❛✱
❡♥tã♦✱ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ n=m✳ ❉❛í✱ ❞❡ P(Am+1|A1∩. . .∩Am) =
P(A1∩. . .∩Am∩Am+1)
P(A1∩. . .∩Am)
✈❡♠ q✉❡
P(A1∩. . .∩Am+1) = P(A1∩. . .∩Am)·P(Am+1|A1∩. . .∩Am)
= P(A1)·P(A2|A1)·P(A3|A1∩A2)· · ·P(Am+1|A1∩. . .∩Am),
♦♥❞❡ ❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡ é ❥✉st✐✜❝❛❞❛ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✳
✷✳✽✳ P❘❖P❘■❊❉❆❉❊❙ ❉❆ P❘❖❇❆❇■▲■❉❆❉❊ ❈❖◆❉■❈■❖◆❆▲
▲❡♠❛ ✷✳✶ ❯♠❛ ❛♣❧✐❝❛çã♦ ❞♦ t❡♦r❡♠❛ ❞♦ ♣r♦❞✉t♦ é ❛ s❡❣✉✐♥t❡✳ ❈♦♥s✐❞❡r❡A1, A2, . . . , An
♣❛rt✐çõ❡s ❞❡ ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧ Ω ❡ B ✉♠ ❡✈❡♥t♦ q✉❛❧q✉❡r ❞❡ss❡ ❡s♣❛ç♦ ❛♠♦str❛❧✳
❊♥tã♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
A1∪A2∪. . .∪An= Ω
❡ t❛♠❜é♠
B = (A1∩B)∪(A2∩B)∪. . .∪(An∩B).
P♦rt❛♥t♦✱
P(B) =P[(A1∩B)∪(A2∩B)∪. . .∪(An∩B)]
❡✱ ❡♥tã♦✱
P(B) =P(A1∩B) +P(A2 ∩B) +. . .+P(An∩B).
❆ss✐♠✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ♣r♦❞✉t♦✱ t❡♠♦s q✉❡
P(B) =P(A1)·P(B|A1) +P(A2)·P(B|A2) +. . .+P(An)·P(B|An).
❊st❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❚❡♦r❡♠❛ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡ ❚♦t❛❧✳
❚❡♦r❡♠❛ ✶ ✭❚❡♦r❡♠❛ ❞❡ ❇❛②❡s✮ ❈♦♥s✐❞❡r❡ A1, A2, . . . , An ♣❛rt✐çõ❡s ❞❡ ✉♠ ❡s✲
♣❛ç♦ ❛♠♦str❛❧ Ω ❡ B ✉♠ ❡✈❡♥t♦ q✉❛❧q✉❡r ❞❡ss❡ ❡s♣❛ç♦ ❛♠♦str❛❧✳ ❊♥tã♦✱ P(Ai|B) =
P(Ai)P(B|Ai)
Pn
j=1P(Aj)P(B|Aj)
.
❉❡ss❛ ❢♦r♠❛✱ ❡st❛♠♦s ❝❛❧❝✉❧❛♥❞♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ P(Ai|B) ❝♦♠♦ s❡♥❞♦ ❛ ♣r♦❜❛✲
❜✐❧✐❞❛❞❡ ❞❡ ♦❝♦rr❡r ♦ ❡✈❡♥t♦ Ai✱ s❛❜❡♥❞♦ q✉❡ ♦ ❡✈❡♥t♦ B ❥á ♦❝♦rr❡✉✳ ❖ ♥ú♠❡r♦
P(Ai) é ❞❡♥♦♠✐♥❛❞♦ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣r✐♦r✐ ❞♦ ❡✈❡♥t♦ Ai ❡ P(Ai|B) é ❞❡♥♦♠✐♥❛❞♦
♣r♦❜❛❜✐❧✐❞❛❞❡ ❛ ♣♦st❡r✐♦r✐ ❞♦ ❡✈❡♥t♦ Ai✳
✷✳✾✳ ❊❱❊◆❚❖❙ ■◆❉❊P❊◆❉❊◆❚❊❙
❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡
P(Ai|B) =
P(Ai∩B)
P(B) .
❉❛í✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ♣r♦❞✉t♦ ❡ ❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳
✷✳✾ ❊✈❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s
❙❡❥❛♠ A ❡ B ❞♦✐s ❡✈❡♥t♦s ❞♦ ❡s♣❛ç♦ ❛♠♦str❛❧ Ω✳ ❖s ❡✈❡♥t♦s A ❡ B sã♦ ❞✐t♦s
✐♥❞❡♣❡♥❞❡♥t❡s q✉❛♥❞♦ P(A|B) = P(A) ❡ P(B|A) = P(B)✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱
t❡r ❛❝♦♥t❡❝✐❞♦ ✉♠ ❞♦s ❡✈❡♥t♦s ♥ã♦ ❡①❡r❝❡ ♥❡♥❤✉♠❛ ✐♥✢✉ê♥❝✐❛ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❛❝♦♥t❡❝❡r ♦ ♦✉tr♦✳
▲❡♠❛ ✷✳✷ ❙❡ A ❡ B sã♦ ❡✈❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❡♠♦s q✉❡ P(A∩B) =P(A)·P(B).
❈♦♠ ❡❢❡✐t♦✱ ❝♦♠♦ P(A|B) = P(A) ❡ P(A|B) =P(A∩B)/P(B)✱ s❡❣✉❡ q✉❡
P(A) = P(A∩B)
P(B)
❡✱ ♣♦rt❛♥t♦✱ P(A∩B) =P(A)·P(B)✳
❊①❡♠♣❧♦ ✷ ◗✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ❝❛s❛❧ t❡r ❞✉❛s ❝r✐❛♥ç❛s✱ s❡♥❞♦ q✉❡ ❛ ♠❛✐s ✈❡❧❤❛ é ✉♠❛ ♠❡♥✐♥❛ ❡ ❛ ❝r✐❛♥ç❛ ♠❛✐s ♥♦✈❛ é ✉♠ ♠❡♥✐♥♦❄
❙♦❧✉çã♦✿ ❖❜s❡r✈❡ q✉❡ ♦ r❡s✉❧t❛❞♦ ❞♦ s❡①♦ ❞❛ ♣r✐♠❡✐r❛ ❝r✐❛♥ç❛ ♥ã♦ ❡①❡r❝❡ ✐♥✢✉ê♥❝✐❛ ♥❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ r❡s✉❧t❛❞♦ ❞♦ s❡①♦ ❞❛ s❡❣✉♥❞❛✳ ❙ã♦✱ ♣♦rt❛♥t♦✱ ❡✈❡♥t♦s ✐♥❞❡♣❡♥✲ ❞❡♥t❡s✳ ❈♦♥s✐❞❡r❡ ♦ ❡✈❡♥t♦ A ❝♦♠♦ s❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ❝r✐❛♥ç❛ s❡r ♠❡♥✐♥❛ ❡ ♦ ❡✈❡♥t♦ B ❝♦♠♦ s❡♥❞♦ ❛ s❡❣✉♥❞❛ ❝r✐❛♥ç❛ s❡r ♠❡♥✐♥♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❝❛s❛❧
✷✳✾✳ ❊❱❊◆❚❖❙ ■◆❉❊P❊◆❉❊◆❚❊❙
t❡r ❞✉❛s ❝r✐❛♥ç❛s✱ s❡♥❞♦ q✉❡ ❛ ♠❛✐s ✈❡❧❤❛ é ✉♠❛ ♠❡♥✐♥❛ ❡ ❛ ❝r✐❛♥ç❛ ♠❛✐s ♥♦✈❛ é ✉♠ ♠❡♥✐♥♦ é ❞❛❞❛ ♣♦r
P(A∩B) = P(A)·P(B) = 1 2 ·
1 2 =
1 4.
❖❜s❡r✈❡ ❛✐♥❞❛ q✉❡ ❡ss❡ ❡①❡♠♣❧♦ ♥♦s r❡♠❡t❡ ❛ r❡s♦❧✈❡r ✉♠❛ ❝♦♥❢✉sã♦ ♠✉✐t♦ ❝♦✲ ♠✉♠✱ q✉❡ é ♦ ❞❡ ❝♦♥❢✉♥❞✐r ❡✈❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❝♦♠ ❡✈❡♥t♦s ♠✉t✉❛♠❡♥t❡ ❡①✲ ❝❧✉❞❡♥t❡s ♦✉ ♠✉t✉❛♠❡♥t❡ ❡①❝❧✉s✐✈♦s✱ q✉❡ sã♦ ❛q✉❡❧❡s ❡♠ q✉❡ ❛ ♦❝♦rrê♥❝✐❛ ❞❡ ✉♠✱ ❛✉t♦♠❛t✐❝❛♠❡♥t❡✱ ❡①❝❧✉✐ ❛ ♦❝♦rrê♥❝✐❛ ❞♦ ♦✉tr♦✳ ❙❡ ♦ ❝❛s❛❧ q✉✐s❡ss❡ t❡r ❛♣❡♥❛s ✉♠❛ ❝r✐❛♥ç❛✱ ❡ ♦ ❡✈❡♥t♦A ❢♦ss❡ ♥❛s❝❡r ♠❡♥✐♥❛ ❡ ♦ ❡✈❡♥t♦ B ❢♦ss❡ ♥❛s❝❡r ♠❡♥✐♥♦✱ é ó❜✈✐♦
q✉❡ A∩B =∅✳ P♦rt❛♥t♦✱ P(A∩B) =P(∅) = 0✳
❈❛♣ít✉❧♦ ✸
Pr♦❜❛❜✐❧✐❞❛❞❡ ●❡♦♠étr✐❝❛
✸✳✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛
❊♠ ✶✼✵✼ ❡♠ ▼♦♥t❜❛r❞✱ ✉♠❛ ♣❡q✉❡♥❛ ❝✐❞❛❞❡ ❞♦ s✉❧ ❞❛ ❋r❛♥ç❛✱ ♥❛s❝❡✉ ●❡♦r❣❡s✲ ▲♦✉✐s ▲❡❝❧❡r❝ q✉❡ ♠❛✐s t❛r❞❡ s❡r✐❛ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❈♦♥❞❡ ❞❡ ❇✉✛♦♥✳ ❋✐❧❤♦ ❞❡ ❇❡♥❥❛♠✐♥✲❋r❛♥ç♦✐s ▲❡❝❧❡r❝ ❡ ❞❡ ❆♥♥❡✲❈r✐st✐♥❡ ▼❛r❧✐♥✱ ●❡♦r❣❡s ♥❛s❝❡✉ ❡♠ ✉♠❛ ❢❛✲ ♠í❧✐❛ r✐❝❛ ❡ ❡r❛ ♦ ♠❛✐s ✈❡❧❤♦ ❞❡ ❝✐♥❝♦ ✜❧❤♦s✳ ◗✉❛♥❞♦ ●❡♦r❣❡s t✐♥❤❛10❛♥♦s✱ s✉❛ ♠ã❡
❤❡r❞♦✉ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❛ ❡♠ ❞✐♥❤❡✐r♦✱ ♦ q✉❡ ♣❡r♠✐t✐✉ q✉❡ s❡✉ ♠❛r✐❞♦ s❡ t♦r♥❛ss❡ ♦ ❈♦♥❞❡ ❞❡ ▼♦♥t❜❛r❞✳ ❙❡❣✉✐♥❞♦ ❛s ♦r✐❡♥t❛çõ❡s ❞♦ ♣❛✐✱ ●❡♦r❣❡s s❡ ♠✉❞♦✉ ♣❛r❛ ❉✐✲ ❥♦♥ ❡ ❢♦✐ ❡st✉❞❛r ❉✐r❡✐t♦ ♥♦ ❈♦❧é❣✐♦ ❏❡s✉ít❛ ❞❡ ●♦r❞❛♥s✱ ♠❛s ❧♦❣♦ ❛ ✐♥❝❧✐♥❛çã♦ ♣♦r ▼❛t❡♠át✐❝❛ ❡ ❈✐ê♥❝✐❛s ❝♦♠❡ç♦✉ ❛ ❛✢♦r❛r✳
❊♠ ✶✼✸✷ ❤❡r❞♦✉ ✉♠❛ ❝♦♥s✐❞❡rá✈❡❧ q✉❛♥t✐❛ ❡♠ ❞✐♥❤❡✐r♦ ❡ ♣♦✉❝♦s ❛♥♦s ❞❡♣♦✐s✱ ❥á ❝♦♠♦ ❈♦♥❞❡ ❞❡ ❇✉✛♦♥✱ ✐♥❣r❡ss♦✉ ♥❛ ❆❝❛❞❡♠✐❛ ❋r❛♥❝❡s❛ ❞❡ ❈✐ê♥❝✐❛s✱ ♦ q✉❡ ♣♦s✲ s✐❜✐❧✐t♦✉ q✉❡ ❡❧❡ s❡ ❞❡❞✐❝❛ss❡ ❛ ❡s❝r❡✈❡r ✉♠❛ ♦❜r❛ ❝♦♠ 44 ✈♦❧✉♠❡s s♦❜r❡ ❍✐stór✐❛
◆❛t✉r❛❧ ✭✈✐❞❡ ❬✸❪✮✳ ❖ ❈♦♥❞❡ ❞❡ ❇✉✛♦♥ ❢♦✐ ♣r❡❝✉rs♦r ❞❡ ❉❛r✇✐♥ ❡ ❞❡ ▲❛♠❛r❝❦✱ ❡ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s ♠❛✐♦r❡s ❜✐ó❧♦❣♦s ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳
❖ ✐♥í❝✐♦ ❞♦ ❡st✉❞♦ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❣❡♦♠étr✐❝❛ s❡ ❞❡✉ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ❳❱■■■ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥✳ ◆❡ss❡
✸✳✷✳ ❖ P❘❖❇▲❊▼❆ ❉❆ ❆●❯▲❍❆ ❉❊ ❇❯❋❋❖◆
♣r♦❜❧❡♠❛✱ ♦ ❈♦♥❞❡ ❞❡ ❇✉✛♦♥ ❡st❛✈❛ ✐♥t❡r❡ss❛❞♦ ❡♠ ❞❡t❡r♠✐♥❛r ❛❧❣❡❜r✐❝❛♠❡♥t❡ q✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❛❣✉❧❤❛ ❧❛♥ç❛❞❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♠ ✉♠ ❛ss♦❛❧❤♦ ❝♦♠ ❧✐♥❤❛s ♣❛r❛❧❡❧❛s ❝❛✐r s❡♠ ❤❛✈❡r ✐♥t❡rs❡çã♦ ❝♦♠ ❡ss❛s ❧✐♥❤❛s✳
❏á ❡♠ ✶✼✸✸✱ ♦ ❈♦♥❞❡ ❞❡ ❇✉✛♦♥ ❛♣r❡s❡♥t♦✉ ✉♠ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ❞❡ ✏❥♦❣♦ ❞❡ ❋r❛♥❝ ❈❛rr❡❛✉✑✱ ♥♦ q✉❛❧ ❡❧❡ ❞❡s❝r❡✈✐❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ♠♦❡❞❛ ❧❛♥ç❛❞❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡♠ ✉♠ ♣✐s♦ ❧❛❞r✐❧❤❛❞♦ ❝♦♠ ❧❛❥♦t❛s ❝♦♥❣r✉❡♥t❡s✱ ❝❛✐r ❝♦♠♣❧❡t❛♠❡♥t❡ ❞❡♥tr♦ ❞❡ ✉♠ ❞♦s ❧❛❞r✐❧❤♦s✱ ♦✉ s❡❥❛✱ s❡♠ ❝♦rt❛r q✉❛❧q✉❡r ❧✐♥❤❛ ❞♦s ❧❛❞r✐❧❤♦s✳ ❊ss❡s ❧❛❞r✐❧❤♦s ♣♦❞❡r✐❛♠ s❡r tr✐❛♥❣✉❧❛r❡s✱ q✉❛❞r❛♥❣✉❧❛r❡s✱ ♣❡♥t❛❣♦♥❛✐s✱ ❡t❝✳
✸✳✷ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❛❣✉❧❤❛ ❞❡ ❇✉✛♦♥
❯♠ ❛ss♦❛❧❤♦ ❝♦♠ ❧✐♥❤❛s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐ ❡ ✉♠❛ ❛❣✉❧❤❛ ❡①tr❡♠❛♠❡♥t❡ ✜♥❛ ❢♦r❛♠ ♦s ♠❛t❡r✐❛✐s q✉❡ ❇✉✛♦♥ ♣♦ss✉í❛ ❛♦ ✐♥✐❝✐❛r s✉❛s ♦❜s❡r✈❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❡ ♣r♦❜❛❜✐❧✐✲ ❞❛❞❡ ❣❡♦♠étr✐❝❛✳
❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✳ ❯♠ ♣❧❛♥♦ é ♠❛r❝❛❞♦ ♣♦r ❧✐♥❤❛s ♣❛r❛❧❡❧❛s ❡q✉✐✲ ❞✐st❛♥t❡s ❡♥tr❡ s✐✳ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ✉♠❛ ❧✐♥❤❛ ❡ ♦✉tr❛ ♠❡❞❡ d ❡ ✉♠❛ ❛❣✉❧❤❛ ❞❡
❝♦♠♣r✐♠❡♥t♦ l ❡ ❞❡ ❡s♣❡ss✉r❛ ❞❡s♣r❡③í✈❡❧ s❡rá ❧❛♥ç❛❞❛ ❛❧❡❛t♦r✐❛♠❡♥t❡ s♦❜r❡ ❡ss❡
♣❧❛♥♦✳ ◗✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ ❛❣✉❧❤❛ ❝r✉③❛r ✉♠❛ ❞❛s ❧✐♥❤❛s ❞♦ ♣❧❛♥♦❄ P❛r❛ r❡s✲ ♣♦♥❞❡r ❛ ❡st❛ ♣❡r❣✉♥t❛✱ ✈❛♠♦s ❞✐✈✐❞✐r ♦ ♣r♦❜❧❡♠❛ ❡♠ três ♣❛rt❡s✳
Pr✐♠❡✐r♦ ❝❛s♦✿ ✭❈❛s♦ ❝❧áss✐❝♦✮ l = d✳ ❆♥❛❧✐s❛♥❞♦ ❛s ❋✐❣✉r❛s ✸✳✶ ❡ ✸✳✷✱
♦❜s❡r✈❛♠♦s q✉❡ ♦s ❡✈❡♥t♦s ❞❡ ✐♥t❡r❡ss❡ sã♦ ❛q✉❡❧❡s ❡♠ q✉❡ x é ♠❡♥♦r q✉❡ ♦ ❝❛t❡t♦
❛❞❥❛❝❡♥t❡ ❞♦ tr✐â♥❣✉❧♦ OP A ❞❡ ❛r❣✉♠❡♥t♦ θ✱ ♦♥❞❡ x ❞❡♥♦t❛ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦
❝❡♥tr♦ ❞❛ ❛❣✉❧❤❛ ❡ ❛ ❧✐♥❤❛ ❞♦ ♣❧❛♥♦ ♠❛✐s ♣ró①✐♠❛✳ ❙❛❜❡♠♦s q✉❡ OP =l/2 ❡✱ ♣❡❧❛
❞❡✜♥✐çã♦ ❞❡ ❝♦ss❡♥♦✱ q✉❡
cosθ= OP
OA
♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡
OP = l
2 ·cosθ.
