✶
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊
P❘Ó✲❘❊■❚❖❘■❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊ P❊❙◗❯■❙❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆
❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ✲ P❘❖❋▼❆❚
❯♠❛ ❈♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞♦s ◆ú♠❡r♦s
❘❡❛✐s
❙✐♠♦♥❡ ❞❡ ❈❛r✈❛❧❤♦ ❙❛♥t♦s
❖r✐❡♥t❛❞♦r✿ Pr♦❢❡ss♦r ❉r✳ ❊✈✐❧s♦♥ ❞❛ ❙✐❧✈❛ ❱✐❡✐r❛
✷
❙✐♠♦♥❡ ❞❡ ❈❛r✈❛❧❤♦ ❙❛♥t♦s
❯♠❛ ❈♦♥str✉çã♦ ●❡♦♠étr✐❝❛ ❞♦s ◆ú♠❡r♦s
❘❡❛✐s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢❡ss♦r ❉r✳ ❊✈✐❧s♦♥ ❞❛ ❙✐❧✈❛ ❱✐❡✐r❛✳
FICHI CITILOGRÁFICI ELIBORIDI PELI BIBLIOTECI UNIOERSIDIDE FEDERIL DE SERGIPE
S237
Santos, Simone de Carvalho
Uma construção geométrica dos números reais / Simone de Carvalho Santos ; orientador Evilson da Silva Vieira. – São Cristóvão, 2015.
93 f. il.
Dissertação (mestrado em Matemática) – Universidade Federal de Sergipe, 2015.
1. Matemática. 2. Números reais. 3. Números irracionais. 4.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦
❆ ❉❡✉s✱ ♣♦r ♠❡ ❞❛r ❢♦rç❛s ♣❛r❛ s✉♣❡r❛r ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s ❡ ♠♦str❛r ♦ ❝❛♠✐♥❤♦ ♥❛s ❤♦r❛s ❞❡ ✐♥❝❡rt❡③❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❊✈✐❧s♦♥ ❞❛ ❙✐❧✈❛ ❱✐❡✐r❛✱ ♥ã♦ só ♣❡❧❛ ❝♦♥st❛♥t❡ ❞✐r❡çã♦ ♥❡st❡ tr❛❜❛✲ ❧❤♦✱ ♠❛s s♦❜r❡t✉❞♦ ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛✱ ❜♦❛ ✈♦♥t❛❞❡ ❡ ❝♦♠♣r❡❡♥sã♦ ❞❛s ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s✳ ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♠ã❡✱ ❈✐♦♥❡✱ ❡ ❛ ♠✐♥❤❛ ✐r♠ã✱ P❛✉❧❛✱ ♣♦r t❡r❡♠ ♠❡ ❛♣♦✐❛❞♦ ❡ ✐♥❝❡♥t✐✈❛❞♦ ❝♦♠ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✳
❆♦ ♠❡✉ ♥❛♠♦r❛❞♦ ❱✐♥í❝✐✉s✱ ♣❡❧♦ ❛♠♣❛r♦✱ ♣❡❧♦ ❛♣♦✐♦ ❞✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ t❡♥❞♦ s❡♠♣r❡ ♣❛❧❛✈r❛s ❞❡ ❛♠✐③❛❞❡✱ â♥✐♠♦ ❡ ✐♥❝❡♥t✐✈♦✱ s❡♠ ♦ q✉❛❧ ❡✉ ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦ ✈❡♥❝❡r ♠❛✐s ❡st❛ ❡t❛♣❛ ❡ ♣♦r ♠❡ ❢❛③❡r ❢❡❧✐③ t♦❞♦s ♦s ❞✐❛s✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❢♦r♠❛ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛ ♣❛r❛ ❛ r❡❛❧✐✲ ③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❡ ❝♦♦r❞❡♥❛❞♦r❡s ❞♦ P❘❖❋▼❆❚✲❯❋❙ ♣❡❧♦ ❡♠♣❡♥❤♦ ❡ ❞❡❞✐❝❛çã♦ ♠♦s✲ tr❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ ❝✉rs♦✳
➚ ❈❆P❊❙ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳
✹
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ✉♠❛ ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡✲ r♦s r❡❛✐s ❝❛r❛❝t❡r✐③❛♥❞♦✲♦s ❝♦♠♦ ♥ú♠❡r♦s q✉❡ ❡①♣r❡ss❛♠ ✉♠❛ ♠❡❞✐❞❛✳ ◆❡st❛ ❝♦♥str✉çã♦ ❝❛❞❛ ♣♦♥t♦ ❞❡ ✉♠❛ r❡t❛ ♦r✐❡♥t❛❞❛ r❡♣r❡s❡♥t❛ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ✭✉♠ ♥ú♠❡r♦ r❡❛❧✮✱ ❝♦♠ ❜❛s❡ ♥♦s ❝✐♥❝♦ ❛①✐♦♠❛s ❞❛ ❣❡♦♠❡tr✐❛ ❡✉❝❧✐❞✐❛♥❛ ❞❡✜♥✐✉✲s❡ ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ✉♠ ♠ét♦❞♦ ♣❛r❛ s♦♠❛r ❡ ♠✉❧t✐♣❧✐❝❛r ♣♦♥t♦s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❢♦ss❡ ♣♦ssí✈❡❧ ❞❡♠♦♥str❛r q✉❡ ❛ r❡t❛ ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ❝♦♠♣❧❡t♦ ❛ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ P❛r❛ t❛♥t♦✱ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ❡❧❡♠❡♥t♦s ❤✐stór✐❝♦s q✉❡ ♣❡r♠✐t❡♠ ❝♦♠♣r❡❡♥❞❡r ♦ s✉r❣✐♠❡♥t♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❝♦♠♦ s♦❧✉çã♦ ♣❛r❛ ❛ ✐♥s✉✜✲ ❝✐ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ♠❡❞✐❞❛✱ ❛ ❡✈♦❧✉çã♦ ❞♦ ♣ró♣r✐♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦✱ ❜❡♠ ❝♦♠♦ ❛ ✐♠♣♦rtâ♥❝✐❛ q✉❡ ❛ ❝♦♥str✉çã♦ r✐❣♦r♦s❛ ❞♦s ♥ú✲ ♠❡r♦s r❡❛✐s t✐✈❡r❛♠ ♣❛r❛ ♦s ❋✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛✳ ❊①✐❜✐♠♦s ✉♠❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❛ ♣❛rt✐r ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ❝♦♥str✉çõ❡s ❞❡ ❝♦♥✲ ❥✉♥t♦s ♥✉♠ér✐❝♦s✳ ❯s❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ♠❡❞✐❞❛ ♠♦str❛♠♦s ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❛ss♦❝✐❛♥❞♦✲♦s ❛♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ♦r✐❡♥t❛❞❛ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ❡❧❡s ❞❡✐①❛♠ ✏❜✉r❛❝♦s✑ ♥❛ r❡t❛ ❡ ❝♦♥❝❧✉✐r s♦❜r❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥str✉✐r ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥t❡♥❤❛ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡ q✉❡ ♣r❡❡♥❝❤❛♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛✳ ❖ t❡♠❛ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ✈✐st♦ q✉❡ ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ é ♣r♦♠♦✈❡r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ♥ú♠❡r♦s ❡ ❞❛s ♦♣❡r❛çõ❡s✱ ❞❡✲ s❡♥✈♦❧✈❡r ♦ s❡♥t✐❞♦ ❞❡ ♥ú♠❡r♦ ❡ ❞❡s❡♥✈♦❧✈❡r ❛ ✢✉ê♥❝✐❛ ♥♦ ❝á❧❝✉❧♦✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ♣❛r❛ t❛❧ ❛ss✐♠✐❧❛r ♦s ♥ú♠❡r♦s r❡❛✐s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ✐rr❛❝✐♦♥❛✐s✱ ♦s q✉❛✐s sã♦ tr❛t❛❞♦s ❛ ♣❛rt✐r ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✳
✺
❆❜str❛❝t
❚❤✐s st✉❞② ❛✐♠s t♦ ♣r❡s❡♥t ❛ ❣❡♦♠❡tr✐❝ ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ♥✉♠❜❡rs ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡♠ ❛s ♥✉♠❜❡rs t❤❛t ❡①♣r❡ss ❛ ♠❡❛s✉r❡✳ ■♥ t❤✐s ❝♦♥str✉❝t✐♦♥✱ ❡❛❝❤ ♣♦✐♥t ✐♥ ❛♥ ♦r✐❡♥t❡❞ ❧✐♥❡ r❡♣r❡s❡♥ts t❤❡ ♠❡❛s✉r❡ ♦❢ ❛ s❡❣♠❡♥t ✭❛ r❡❛❧ ♥✉♠❜❡r✮✳ ❇❛s❡❞ ♦♥ ✜✈❡ ❛①✐♦♠s ♦❢ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ✐t ✇❛s ❞❡✜♥❡❞ ❛♥ ♦r❞❡r r❡❧❛t✐♦♥✱ ❛ ♠❡t❤♦❞ t♦ ❛❞❞ ❛♥❞ ♠✉❧t✐♣❧② ♣♦✐♥ts s♦ t❤❛t ✐t ✇❛s ♣♦ss✐❜❧❡ t♦ ❞❡♠♦♥str❛t❡ t❤❛t t❤❡ ❧✐♥❡ ❤❛s ❛ ❢✉❧❧ ♦r❞❡r❡❞ ❜♦❞② ♦❢ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ t❤❛t ✇❡ ❝❛❧❧ t❤❡ s❡t ♦❢ r❡❛❧ ♥✉♠❜❡rs✳ ❚♦ ❞♦ s♦✱ ✐t ✇❡r❡ ♣r❡s❡♥t❡❞ ❤✐st♦r✐❝❛❧ ❡❧❡♠❡♥ts t❤❛t ❛❧❧♦✇ ✉s t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❡♠❡r❣❡♥❝❡ ♦❢ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ✐♥s✉✣❝✐❡♥❝② ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♠❡❛s✉r✐♥❣ ♣r♦❜❧❡♠✱ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ ♥✉♠❜❡r✱ ❛s ✇❡❧❧ ❛s t❤❡ ✐♠♣♦rt❛♥❝❡ t❤❛t t❤❡ str✐❝t ❝♦♥str✉❝t✐♦♥ ♦❢ r❡❛❧ ♥✉♠❜❡rs ❤❛❞ t♦ t❤❡ ❋♦✉♥❞❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝s✳ ❲❡ ❞✐s♣❧❛② ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❢r♦♠ t❤❡ ✐♥t❡❣❡r♥✉♠❜❡rs ❛s ♠♦t✐✈❛t✐♦♥ ❢♦r ❝♦♥str✉❝t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ s❡ts✳ ❯s✐♥❣ t❤❡ ♥♦t✐♦♥ ♦❢ ♠❡❛s✉r❡✱✇❡ s❤♦✇ ❛ ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❧✐♥❦✐♥❣ t❤❡♠ t♦ t❤❡ ♣♦✐♥ts ♦❢ ❛♥ ♦r✐❡♥t❡❞ ❧✐♥❡ t♦ ❞❡♠♦♥str❛t❡ t❤❛t t❤❡② ❧❡❛✈❡ ✏❤♦❧❡s✑ ✐♥ t❤❡ ❧✐♥❡ ❛♥❞ ❝♦♥❝❧✉❞❡ ♦♥ t❤❡ ♥❡❡❞ t♦ ❜✉✐❧❞ ❛ s❡t t❤❛t ❝♦♥t❛✐♥s t❤❡ r❛t✐♦♥❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❛t ✜❧❧ ❛❧❧ t❤❡ ♣♦✐♥ts ♦❢ ❛ ❧✐♥❡✳ ❚❤❡ t❤❡♠❡ ✐s ♦❢ ✉t♠♦st ✐♠♣♦rt❛♥❝❡ t♦ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s ❜❡❝❛✉s❡ ♦♥❡ ♦❢ t❤❡ ♠❛❥♦r ❣♦❛❧ ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ✐s t♦ ♣r♦♠♦t❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ♥✉♠❜❡rs ❛♥❞ ♦♣❡r❛t✐♦♥s✱ t♦ ❞❡✈❡❧♦♣ ♥✉♠❜❡r s❡♥s❡ ❛♥❞ t♦ ❞❡✈❡❧♦♣ ✢✉❡♥❝② ✐♥ t❤❡ ❝❛❧❝✉❧❛t✐♦♥✳ ❚♦ ❛❝❤✐❡✈❡ t❤✐s✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❛ss✐♠✐❧❛t❡ t❤❡ r❡❛❧ ♥✉♠❜❡rs✱ ❡s♣❡❝✐❛❧❧② t❤❡ ✐rr❛t✐♦♥❛❧ ♦♥❡s✱ ✇❤✐❝❤ ❛r❡ t❛✉❣❤t s✐♥❝❡ t❤❡ ❡❧❡♠❡♥t❛r② s❝❤♦♦❧✳
❈♦♥t❡ú❞♦
✶ ■♥tr♦❞✉çã♦ ✾
✶✳✶ ❈♦♥t❡①t♦ ❤✐stór✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷ ❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s ✶✺
✷✳✶ ❈♦♥t❛r✿ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ✉♠✲❛✲✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❈♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✐♥✜♥✐t♦s ❡ ❡♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳✶ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳✷ ❈♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❘❡❧❛çõ❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✹✳✶ ❈❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺ ❈♦r♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺✳✶ ❈♦r♣♦ ♦r❞❡♥❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺✳✷ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ♥✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ K ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✻ ❱❛❧♦r ❛❜s♦❧✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✼ ❈♦r♣♦ ♦r❞❡♥❛❞♦ ❝♦♠♣❧❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✽ ❆ ❞❡s❝♦❜❡rt❛ ❞♦s s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✽✳✶ ❙❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✽✳✷ ❙❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸ ❈♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ✸✼
✸✳✶ ❖ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ✐♥❝♦♠♣❧❡t♦ ❞♦s ❘❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✷ ❆❞✐çã♦ ❡♠ Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ❛❞✐çã♦ ❡♠ Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸✳✸ ❙✉❜tr❛çã♦ ❡♠Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✽ ❈❖◆❚❊Ú❉❖ ✸✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ s✉❜tr❛çã♦ ❡♠Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸✳✹ ▼✉❧t✐♣❧✐❝❛çã♦ ❡♠ Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✸✳✹✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✺ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠ Q ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✸✳✻ ❆ ❡♥✉♠❡r❛❜✐❧✐❞❛❞❡ ❞♦s r❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✼ ■❞❡♥t✐✜❝❛çã♦ ❞❡ ♥ú♠❡r♦s ❛ ♣♦♥t♦s ❞❡ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✼✳✶ ◆ú♠❡r♦s s♦❜r❡ ✉♠❛ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✼✳✷ ❆ r❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹ ❈♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ✺✼
✹✳✶ ❖ q✉❡ é ✉♠ ♥ú♠❡r♦ r❡❛❧❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✷ ❖♣❡r❛çõ❡s ❡♠ R✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
✹✳✷✳✶ ❆❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✷✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✸ ❘❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
✹✳✹ ❆ ❝♦♠♣❧❡t✉❞❡ ❞♦s ♥ú♠❡r♦s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✺ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ é ❡♥✉♠❡rá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
✺ ❆❧❣✉♠❛s ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ✼✼
✺✳✶ ❈♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✺✳✶✳✶ ❘❡t❛s ♣❛r❛❧❡❧❛s ❡ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✺✳✶✳✷ ❉✐✈✐sã♦ ❞❡ s❡❣♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✺✳✷ ❈♦♠ ♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✺✳✷✳✶ ■♥t❡r❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✺✳✷✳✷ ❈♦♥str✉çõ❡s ♥♦ ●❡♦●❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❆ ❝♦♠♣r❡❡♥sã♦ ❞♦s ♥ú♠❡r♦s é ❛ ❜❛s❡ ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r t♦❞❛ ❛ ♠❛t❡♠át✐❝❛✳ ❆♣❡s❛r ❞❡ ♦ ✉t✐❧✐③❛♠♦s ❞✐❛r✐❛♠❡♥t❡ ❛ t♦❞♦ ✐♥st❛♥t❡✱ ❞❛r ✉♠❛ ❞❡✜♥✐çã♦ ♣❛r❛ ♥ú♠❡r♦ ♥ã♦ é ✉♠❛ t❛r❡❢❛ ❢á❝✐❧✳ ◆❡st❡ tr❛❜❛❧❤♦ ❞✐r❡♠♦s q✉❡ ♦s ♥ú♠❡r♦s sã♦ ❛❜str❛çõ❡s q✉❡ s✉r❣❡♠ ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s ❡ ❞❡ ♠❡❞✐çã♦ ❞❡ ✈ár✐❛s q✉❛♥t✐❞❛❞❡s✱ ❝♦♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ♣❡s♦ ❡ t❡♠♣♦✳ ❊ q✉❡ ❡①✐st❡ t✐♣♦s ❞✐❢❡r❡♥t❡s ❞❡ ♥ú♠❡r♦s✳
❉♦s ♥ú♠❡r♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ♦s q✉❡ ❝❛✉s❛♠ ♠❛✐s ❡♠❜❛r❛ç♦ sã♦ ♦s ♥ú♠❡r♦s r❡❛✐s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ ❊ss❡ ❡♠❜❛r❛ç♦ ❝♦♥t✐♥✉❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ❛s ✈❡③❡s ♥♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✳ ❆ ❢❛❧t❛ ❞❡ ✉♠❛ ♥♦çã♦ ❝❧❛r❛ ❞♦ q✉❡ é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❡ ♦ q✉❡ ♦ ❞✐st✐♥❣✉❡ ❞❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✱ ♣❛r❛ ❛❧é♠ ❞❛ ♥♦t❛çã♦✱ ♠❛✐s ❛ ✐♠♣r❡ssã♦ ❞❡ q✉❡ sã♦ ✐♥út❡✐s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✱ ✐♠♣❧✐❝❛ ♥✉♠❛ sér✐❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❝♦♠♣r❡❡♥sã♦ ❞❡st❡s ♥ú♠❡r♦s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❡ ♦✉tr♦s ❝♦♥t❡ú❞♦s ❞❡♣❡♥❞❡♥t❡s ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ r❡❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❢✉♥çõ❡s✱ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ sér✐❡s✱ ❧✐♠✐t❡s✱ ❡t❝✳
P❡♥t❡❛❞♦✱ ❡♠ ❬✷✵❪✱ ❝✐t❛ ❛❧❣✉♠❛s ❝♦♥❝❧✉sõ❡s ❞❛s ♣❡sq✉✐s❛s r❡❛❧✐③❛❞❛s ♥♦ ❇r❛s✐❧ ❡ ♥♦ ❡①✲ t❡r✐♦r ❛❝❡r❝❛ ❞❛s ❝♦♥❝❡♣çõ❡s ❞♦s ❛❧✉♥♦s s♦❜r❡ ♦s ♥ú♠❡r♦s r❡❛✐s q✉❡ ❞❡st❛❝❛♠♦s ❛ s❡❣✉✐r✿ ❛ ❛ss♦❝✐❛çã♦ ❞❡ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❛ ✉♠ ♥ú♠❡r♦ q✉❡ ♥ã♦ é ❡①❛t♦❀ ❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú♠❡r♦ ✐rr❛✲ ❝✐♦♥❛❧ ❝♦♠♦ s❡♥❞♦ ✐♥✜♥✐t♦ ♦✉ ❛q✉❡❧❡ q✉❡ ❝♦♥té♠ ✐♥✜♥✐t♦s ❞í❣✐t♦s ❛♣ós ❛ ✈ír❣✉❧❛ ♦✉ ❛✐♥❞❛ ❛s r❛í③❡s❀ ❛ ❝♦♥❢✉sã♦ ❡♥tr❡ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❡ s✉❛ ❛♣r♦①✐♠❛çã♦ ❛tr✐❜✉✐♥❞♦ ♦ ♠❡s♠♦ s✐❣✲ ♥✐✜❝❛❞♦ ✭❝♦♠♦π= 3,14✮❀ ❛ ❝♦♥❝❡♣çã♦ ❞❡ q✉❡ ❞✉❛s ❣r❛♥❞❡③❛s sã♦ s❡♠♣r❡ ❝♦♠❡♥s✉rá✈❡✐s❀ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛tr✐❜✉í❞❛s à r❡t❛ r❡❛❧ ❝♦♥t✐♥✉❛✈❛♠ ✈á❧✐❞❛s ♠❡s♠♦ s❡♠ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❡ ♦ ❞❡s❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❝♦♠♣❧❡t✉❞❡ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
✶✵ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ❝♦♠ ✽✹ ❛❧✉♥♦s ❞♦s ❝✉rs♦s ❞❡ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋▼● ❡ ❯❋❙❈✱ ✈❡r ❬✾❪✱ ♠♦str♦✉ q✉❡ q✉❛s❡ ✺✵✪ ♥ã♦ ❝♦♠♣r❡❡♥❞❡♠ ♦ q✉❡ é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✱ ❞❡✜♥❡♠ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ♥ã♦ ❡①❛t♦❀ ❛ ❞✐st✐♥çã♦ ❡♥tr❡ r❛❝✐♦♥❛❧ ❡ ✐rr❛❝✐♦♥❛❧ é ❛♣❡♥❛s ♥❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ✭❢r❛çã♦ ① ♥ã♦ ❢r❛çã♦ ✲ ❞❡❝✐♠❛❧ ✜♥✐t♦ ♦✉ ♣❡r✐ó❞✐❝♦ ① ❞❡❝✐♠❛❧ ✐♥✜♥✐t♦ ♥ã♦ ♣❡r✐ó❞✐❝♦✮❀ ♥ã♦ ❛ss♦❝✐❛♠ ♦ s✐❣♥✐✜❝❛❞♦ ❞❛ ✐♥❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❞❡ ❞♦✐s s❡❣♠❡♥t♦s ❝♦♠ ♦ s❡♥t✐❞♦ ❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳
◆ã♦ é ❞❡ s❡ ❡s♣❛♥t❛r ❝♦♠ t❛✐s r❡s✉❧t❛❞♦s ✉♠❛ ✈❡③ q✉❡ ♦ tr❛t❛♠❡♥t♦ q✉❡ s❡ ❞á ❛♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦ é ✉♠ t❛♥t♦ s✉♣❡r✜❝✐❛❧✳ ❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❛♣♦♥t❛ q✉❡ ✉♠❛ ❞❛s ❝❛✉s❛s q✉❡ ❝♦♥tr✐❜✉✐ ♣❛r❛ q✉❡ ♦s ❛❧✉♥♦s ♥ã♦ ❞❡s❡♥✈♦❧✈❛♠ ❜❡♠ ♦ ❝♦♥❝❡✐t♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s é q✉❡ ♦ ❡st✉❞♦ ❞❡ss❡s ♥ú♠❡r♦s tê♠ s❡ ❧✐♠✐t❛❞♦ q✉❛s❡ q✉❡ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❛♦ ❡♥s✐♥♦ ❞♦ ❝á❧❝✉❧♦ ❝♦♠ r❛❞✐❝❛✐s✳ ◆♦ ❡♥t❛♥t♦✱ ❞❛s q✉❛tr♦s s✉❣❡stõ❡s q✉❡ ❢❛③ ♣❛r❛ tr❛❜❛❧❤❛r ❡ss❡ t❡♠❛ ❞✉❛s ❞❡❧❛s é ❝♦♠ ♦ ❝á❧❝✉❧♦ ❞❡ r❛❞✐❝❛✐s✳
❖ ❡st✉❞♦ ❞❡ss❡s ♥ú♠❡r♦s ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞♦ ♣♦r ♠❡✐♦ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ q✉❡ ❡✈✐❞❡♥❝✐❡♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♦✉tr♦s ♥ú♠❡✲ r♦s ❛❧é♠ ❞♦s r❛❝✐♦♥❛✐s✳ ❯♠❛ s✐t✉❛çã♦ é ❛ ❞❡ ❡♥❝♦♥tr❛r ♥ú♠❡r♦s q✉❡ t❡♥❤❛♠ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ✐♥✜♥✐t❛✱ ❡ ♥ã♦ ♣❡r✐ó❞✐❝❛✳ ❖✉✲ tr❛ é ♦ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❡♥❝♦♥tr❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦✱ t♦♠❛♥❞♦ ♦ ❧❛❞♦ ❝♦♠♦ ✉♥✐❞❛❞❡✱ q✉❡ ❝♦♥❞✉③ ❛♦ ♥ú♠❡r♦ √2✳ ◆❡ss❡ ❝❛s♦✱ ♣♦❞❡✲s❡ ✐♥❢♦r♠❛r ✭♦✉ ✐♥❞✐❝❛r ❛ ♣r♦✈❛✮
❞❛ ✐rr❛❝✐♦♥❛❧✐❞❛❞❡ ❞❡ √2✱ ♣♦r ♥ã♦ s❡r ✉♠❛ r❛③ã♦ ❞❡ ✐♥t❡✐r♦s✳ ❖
♣r♦❜❧❡♠❛ ❞❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s q✉❡ ♥ã♦ sã♦ q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s✱ √3✱√5✱ ❡t❝✳✱ ♣♦❞❡r✐❛ s❡❣✉✐r✲s❡ ❛♦ ❝❛s♦ ♣❛r✲
t✐❝✉❧❛r ❞❡√2✳
✳✳✳ ❡①♣❧♦r❛r ♥ú♠❡r♦s ♥❛ ❢♦r♠❛a+b√2✱ ❝♦♠a❡br❛❝✐♦♥❛✐s✱ ♣♦❞❡
❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ s✉♣❡r❛çã♦ ❞❛ ✐❞❡✐❛ ❡q✉✐✈♦❝❛❞❛ ❞❡ q✉❡ ❤á ♣♦✉❝♦s ✐rr❛❝✐♦♥❛✐s✳ ✭P❈◆✱ ❬✹❪✱ ♣á❣✳ ✶✵✻✱ ✶✵✼✮✳
❊ é ❡①❛t❛♠❡♥t❡ ✐ss♦ q✉❡ ♦s ❧✐✈r♦s ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ tr❛③❡♠✱ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦✲ ♥❛✐s sã♦ ❛❜♦r❞❛❞♦s ❛ ♣❛rt✐r ❞♦ ✽♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❣❡r❛❧♠❡♥t❡ ❝♦♠ ♣r♦❜❧❡♠❛s
✶✶ ♠❡r♦s r❡❛✐s ❝♦♠♦ ❛ ✉♥✐ã♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ss❛s ❞❡✜♥✐çõ❡s é ❛❞♠✐t✐r ♣r❡✈✐❛♠❡♥t❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ q✉❡ s❡❥❛ ♥ú♠❡r♦✱ ✐st♦ é✱ ♥ú✲ ♠❡r♦ r❡❛❧✳ ◆♦ ✶♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡st✉❞❛✲s❡ ♠❛✐s ✉♠❛ ✈❡③ ♦s ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s✱
❛❣♦r❛ ❞❡ ❢♦r♠❛ ❛♣r❡ss❛❞❛✱ ✉♠❛ ❡s♣é❝✐❡ ❞❡ r❡✈✐sã♦ rá♣✐❞❛✱ q✉❛♥❞♦ ❞❡✈❡r✐❛ s❡r ❢❡✐t♦ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❣❛♥❤❛r ♠❛✐s ❝♦♥s✐stê♥❝✐❛ t❡ór✐❝❛✳ ▼❛✐s ✉♠❛ ✈❡③ ❡♥❢❛t✐③❛✲s❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ ✐♥✜♥✐t❛ ❡ ♥ã♦ ♣❡r✐ó❞✐❝❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛ ♣❡r✐ó❞✐❝❛ ❞♦ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ s❡♠ ❛♦ ♠❡♥♦s ❥✉st✐✜❝❛r ♣♦r q✉❡ ❛ r❡♣❡t✐çã♦ ❛❝♦♥t❡❝❡✱ ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠♦ ❛ ✉♥✐ã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✳
◆ã♦ ❡st❛♠♦s ❝♦♠ ✐ss♦ q✉❡r❡♥❞♦ ❞❡❢❡♥❞❡r ✉♠❛ ❛♣r❡s❡♥t❛çã♦ r✐❣♦r♦s❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❥✉❧❣❛♠♦s ✐♥❝❧✉s✐✈❡ ✐♥❛❞❡q✉❛❞♦✳ ▼❛s é ♣❡r❢❡✐t❛♠❡♥t❡ ♣♦ssí✈❡❧ ❞❡✜♥✐r ✉♠ ♥ú♠❡r♦ r❡❛❧ ❝♦♠♦ ♦ r❡s✉❧t❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛✱ ❝♦♥t❛r ♦ ♠♦t✐✈♦ ♣❡❧♦ q✉❛❧ ❡❧❡s ❡①✐st❡♠ ❡①♣❧✐❝❛♥❞♦ ♦ q✉❡ sã♦ ❣r❛♥❞❡③❛s ✐♥❝♦♠❡♥s✉rá✈❡✐s✱ ❝♦♥❝❡✐t♦ ❝✉❥❛ ❞✐s❝✉ssã♦ ♣♦❞❡ ❢❛✈♦r❡❝❡r ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ q✉❡ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s sã♦ ✐♥❞✐s♣❡♥sá✈❡✐s ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ♠❡❞✐❞❛ ❞❡ q✉❛✐sq✉❡r ❣r❛♥❞❡③❛s✳
❯t✐❧✐③❛♥❞♦✲s❡ ❞♦ ❢❛t♦ q✉❡ s❡♠♣r❡ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❝♦♠♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ❝♦♥str✉çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❞❡✲ ✜♥✐♥❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞❡ ✉♠ r❡t❛ ♦r✐❡♥t❛❞❛ q✉❡ r❡♣r❡s❡♥t❛ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦✱ ❞❡✜♥✐r❡♠♦s ✉♠ ♠ét♦❞♦ ♣❛r❛ s♦♠❛r ❡ ♠✉❧t✐♣❧✐❝❛r ♣♦♥t♦s ❞❛ r❡t❛ ✉s❛♥❞♦ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ♦✉ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✱ ❝♦♠♦ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❛ ✐♥tr♦❞✉çã♦ ❞❡ss❡s ♥ú♠❡r♦s ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ s♦❜ ✉♠❛ ♣❡rs♣❡❝t✐✈❛ ❣❡♦♠étr✐❝❛✳
■♥✐❝✐❛♠♦s ♥♦ss♦ ❡st✉❞♦ ❢❛③❡♥❞♦ ✉♠ ❛♣❛♥❤❛❞♦ ❤✐stór✐❝♦ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ❞❡s❞❡ ❛ ❞❡s❝♦❜❡rt❛ ❞♦s s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s ❛té ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣♦r ❉❡❞❡❦✐♥❣✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❡✈♦❧✉çã♦ ❤✐stór✐❝❛ ❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s é ✐♠♣♦rt❛♥t❡✱ ♣♦✐s ❛❥✉❞❛✲♥♦s ❛ r❡✢❡t✐r s♦❜r❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞❛r t❡♠♣♦ ♣❛r❛ ♦ ❛❞❡q✉❛❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❞❡t❡r♠✐♥❛❞♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❢❛❧❛♠♦s s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛❣❡♠✱ ❝♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✐♥✜♥✐t♦s✱ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s✱ r❡❧❛çõ❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❝♦r♣♦✱ ❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ s✉♣r❡♠♦ ❡ ❞❡ í♥✜♠♦✱ ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ❝♦♥str✉çõ❡s q✉❡ ❢❛③❡♠♦s✳ ❆✐♥❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ ❡①♣❧✐❝❛♠♦s ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ♠❡❞✐r ❡ ♦ q✉❡ sã♦ s❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ❡ s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥s✉rá✈❡✐s✳
✶✷ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ r♦s ✐♥t❡✐r♦s ♣❛r❛ ♠♦str❛r ❛♦ ❧❡✐t♦r ❝♦♠♦ ❝♦♥str✉✐r ❝♦♥❥✉♥t♦s✳ ❋❛③❡♠♦s ✉♠❛ ✐❞❡♥t✐✜❝❛çã♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❛ ♣♦♥t♦s ❞❡ ✉♠❛ r❡t❛ ♦r✐❡♥t❛❞❛✱ ♠♦str❛♠♦s q✉❡ ❡①✐st❡♠ ♣♦♥t♦s ♥❛ r❡t❛ q✉❡ ♥ã♦ ❝♦rr❡s♣♦♥❞❡♠ ❛ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳
❖ q✉❛rt♦ ❝❛♣ít✉❧♦ ❝♦♥té♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♣r♦♣♦st❛ ❞❡st❡ tr❛❜❛❧❤♦ q✉❡ é ❝♦♥str✉✐r ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ■♥✐❝✐❛♠♦s ❞❡✜♥✐♥❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ r❡t❛ ♦r✐❡♥t❛❞❛ q✉❡ r❡♣r❡s❡♥t❛ ♦ t❛♠❛♥❤♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✱ ✐st♦ é✱ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛✳ ❯s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞♦ ❝❛♣ít✉❧♦ ✷ ♠♦str❛♠♦s q✉❡ ❡ss❛ r❡t❛ ♣♦ss✉✐ ❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ❝♦♠♣❧❡t♦✳
◆♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ✉s❛♥❞♦ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❡ ✉t✐❧✐③❛♥❞♦ ♦ ●❡♦●❡❜r❛✳
✶✳✶ ❈♦♥t❡①t♦ ❤✐stór✐❝♦
❆ ♠❛t❡♠át✐❝❛ ❞❡♠♦♥str❛t✐✈❛ ♥❛s❝❡✉ ♥✉♠❛ ❛t♠♦s❢❡r❛ ❞❡ r❛❝✐♦♥❛❧✐s♠♦ ❡♠ q✉❡ ♦ ❤♦♠❡♠ ❝♦♠❡ç♦✉ ❛ ❢♦r♠✉❧❛r q✉❡stõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❡♥t❡♥❞❡r ♦ ♣♦r q✉ê ❞❛s ❝♦✐s❛s✳ ❖s ♣r✐✲ ♠❡✐r♦s ♣❡rs♦♥❛❣❡♥s ❝♦♥❤❡❝✐❞♦s ❛♦s q✉❛✐s s❡ ❛ss♦❝✐❛♠ ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s sã♦ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✶ ❡ P✐tá❣♦r❛s ✭✺✼✷ ❛✳❈✳ ✲ ✹✾✻ ❛✳❈✳✮✱ ❢✉♥❞❛❞♦r ❞❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛✱ q✉❡ ❛❧é♠ ❞❡
s❡r ✉♠ ❝❡♥tr♦ ❞❡ ❡st✉❞♦ ❞❡ ✜❧♦s♦✜❛✱ ♠❛t❡♠át✐❝❛ ❡ ❝✐ê♥❝✐❛s ♥❛t✉r❛✐s✱ ❡r❛ t❛♠❜é♠ ✉♠❛ ✐r♠❛♥❞❛❞❡ ❡str❡✐t❛♠❡♥t❡ ✉♥✐❞❛ ♣♦r r✐t♦s s❡❝r❡t♦s ❡ ❝❡r✐♠♦♥✐❛✐s✳
❆ ✜❧♦s♦✜❛ ♣✐t❛❣ór✐❝❛ ❜❛s❡❛✈❛✲s❡ ♥❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ t♦❞❛s ❛s ❝♦✐s❛s ❞♦ ✉♥✐✈❡rs♦ ♣♦❞✐❛♠ s❡r ❞❡s❝r✐t❛s ❛tr❛✈és ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ■ss♦ ❧❡✈❛✈❛ ❛ ✉♠❛ ❡①❛❧t❛çã♦ ❡ ❛♦s ❡st✉❞♦s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ♥ú♠❡r♦s ❡ ❞❛ ❛r✐t♠ét✐❝❛✱ ❥✉♥t♦ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛✱ ❛ ♠ús✐❝❛ ❡ ❛ ❛str♦♥♦♠✐❛✱ q✉❡ ❝♦♥st✐t✉í❛♠ ❛s ❛rt❡s ❧✐❜❡r❛✐s ❜ás✐❝❛s ❞♦ ♣r♦❣r❛♠❛ ♣✐t❛❣ór✐❝♦✳
❆tr✐❜✉✐✲s❡ ❛ P✐tá❣♦r❛s ❛ ❞❡s❝♦❜❡rt❛ ❞♦ t❡♦r❡♠❛ s♦❜r❡ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✱ ❤♦❥❡ ✉♥✐✲ ✈❡rs❛❧♠❡♥t❡ ❝♦♥❤❡❝✐❞♦ ♣❡❧♦ s❡✉ ♥♦♠❡✿
✏❖ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛✲ ❞r❛❞♦s s♦❜r❡ ♦s ❝❛t❡t♦✳✑
❯♠❛ ❞❛s ♠❛✐♦r❡s r❡❛❧✐③❛çõ❡s ❞♦s ♣✐t❛❣ór✐❝♦s ❢♦✐ ❛ ❞❡s❝♦❜❡rt❛ ❞♦s s❡❣♠❡♥t♦s ✐♥❝♦♠❡♥✲ s✉rá✈❡✐s✱ ✉♠ ♠❛r❝♦ ♥❛ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳
✶❈♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s s❡t❡ sá❜✐♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡✳ ❈r❡❞✐t❛♠✲s❡ ❛ ❡❧❡✱ ❡♥tr❡ ♦✉tr❛s ❝♦✐s❛s✱ ♦s s❡❣✉✐♥t❡s
✶✳✶✳ ❈❖◆❚❊❳❚❖ ❍■❙❚Ó❘■❈❖ ✶✸ ❊❧❡s ♣r♦✈❛r❛♠ q✉❡ ♥ã♦ ❤á ♥❡♥❤✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✷✱ ♥❡♠ ♥❡♥❤✉♠❛ r❛③ã♦ ❡♥tr❡ ✐♥t❡✐r♦s
✭♥ú♠❡r♦ r❛❝✐♦♥❛❧✮ q✉❡ ♣♦ss❛ r❡♣r❡s❡♥t❛r ❛ ♠❡❞✐❞❛ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦s ❧❛❞♦s ♠❡❞❡♠ ✉♠❛ ✉♥✐❞❛❞❡✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♥♦✈♦s ♥ú♠❡r♦s t✐✈❡r❛♠ q✉❡ s❡r ✐♥✈❡♥t❛❞♦s ♣❛r❛ r❡♣r❡s❡♥t❛r❡♠ ❡ss❛s ♠❡❞✐❞❛s✱ s✉r❣❡♠ ❡♥tã♦ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s ✭s✐❣♥✐✜❝❛ ♥ú♠❡r♦s ♥ã♦✲ r❛❝✐♦♥❛✐s✮✳
❆ r❡✈❡❧❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ss❡s ♥♦✈♦s ♥ú♠❡r♦s ❢♦✐ ❝♦♥s✐❞❡r❛❞❛ ✉♠❛ ❝r✐s❡ ♥♦s ❢✉♥✲ ❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛✱ ♣r✐♠❡✐r♦ ♣♦rq✉❡ ❛❜❛❧♦✉ ❛ ❜❛s❡ ❞❛ ✜❧♦s♦✜❛ ♣✐t❛❣ór✐❝❛ ❞❡ q✉❡ t✉❞♦ ❞❡♣❡♥❞✐❛ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❧✐♠✐t♦✉ ❛ t❡♦r✐❛ ❞❛ ♣r♦♣♦rçõ❡s à ❣r❛♥❞❡③❛s ❝♦♠❡♥✲ s✉rá✈❡✐s ❡ ✐♥✈❛❧✐❞♦✉ ❛ t❡♦r✐❛ ❣❡r❛❧ ❞❛s ✜❣✉r❛s s❡♠❡❧❤❛♥t❡s ✸✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♥tr❛r✐❛✈❛ ♦
s❡♥s♦ ❝♦♠✉♠✱ ♣♦✐s ✐♥t✉✐t✐✈❛♠❡♥t❡ é r❛③♦á✈❡❧ ❛❞♠✐t✐r q✉❡ ❞❛❞♦s ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛ s❡♠♣r❡ s❡r✐❛ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ t❡r❝❡✐r♦ s❡❣♠❡♥t♦✱ t❛❧✈❡③ ♠✉✐t♦ ♣❡q✉❡♥♦✱ q✉❡ ❝♦✉❜❡ss❡ ❡①❛t❛♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s ❡♠ ❝❛❞❛ ✉♠ ❞♦s ❞♦✐s s❡❣♠❡♥t♦s ❞❛❞♦s✳ ❆❜r✐✉✲s❡ ❛s ♣♦rt❛s ♣❛r❛ ♣r♦❝❡ss♦s ✐♥✜♥✐t♦s✱ ✈✐st♦ q✉❡ ✉♠ s❡❣♠❡♥t♦ ♥ã♦ ♣♦❞✐❛ ♠❛✐s s❡r ❝♦♥s✐❞❡r❛❞♦ ✐♥❞✐✈✐sí✈❡❧✱ ♠❛s ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡❧✳
◆❡ss❛ ♠❡s♠❛ é♣♦❝❛✱ s✉r❣❡ ♦s ♣❛r❛❞♦①♦s ❞❡ ❩❡♥ã♦ ❞❡ ❊❧é✐❛✳ ❊❧❡s ♠♦str❛✈❛♠ q✉❡ ❝♦♥✲ s✐❞❡r❛r ❣r❛♥❞❡③❛s ✐♥❞✐✈✐sí✈❡✐s ♦✉ ✐♥✜♥✐t❛♠❡♥t❡ ❞✐✈✐sí✈❡✐s ❧❡✈❛✈❛ ❛ ❝♦♥tr❛❞✐çõ❡s✳ ❱❡❥❛♠♦s ✉♠ ❞❡❧❡s✿
✏❆ ❉✐❝♦t♦♠✐❛✿ ❙❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ♣♦❞❡ s❡r s✉❜❞✐✈✐❞✐❞♦ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❡♥tã♦ ♦ ♠♦✈✐♠❡♥t♦ é ✐♠♣♦ssí✈❡❧ ♣♦✐s✱ ♣❛r❛ ♣❡r❝♦rrê✲❧♦✱ é ♣r❡❝✐s♦ ❛♥t❡s ❛❧❝❛♥ç❛r s❡✉ ♣♦♥t♦ ♠é❞✐♦✱ ❛♥t❡s ❛✐♥❞❛ ❛❧❝❛♥ç❛r ♦ ♣♦♥t♦ q✉❡ ❡st❛❜❡❧❡❝❡ ❛ ♠❛r❝❛ ❞❡ ✉♠ q✉❛rt♦ ❞♦ s❡❣♠❡♥t♦✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❛❞ ✐♥✜♥✐t✉♠✳ ❙❡❣✉❡✲s❡ ❡♥tã♦ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ❥❛♠❛✐s ❝♦♠❡ç❛rá✳✑
❉✉r❛♥t❡ ♠✉✐t♦ t❡♠♣♦ ♦s ♠❛t❡♠át✐❝♦s ❜✉s❝❛r❛♠ ❡①♣❧✐❝❛r ❡ss❡s ♣❛r❛❞♦①♦s✱ s❡♠ s✉❝❡ss♦✱ ❞❡❝✐❞✐r❛♠ ❡✈✐t❛r ♦s ♣r♦❝❡ss♦s ✐♥✜♥✐t♦s✳
❆ s♦❧✉çã♦ ❞❡ss❡s ♣r♦❜❧❡♠❛s s❡ ❞❡✉✱ ♣♦r ✈♦❧t❛ ❞❡ ✸✼✵ ❛✳ ❈✳ ❝♦♠ ❛ ♥♦✈❛ ❞❡✜♥✐çã♦ ❞❡ ♣r♦♣♦rçã♦ ❢❡✐t❛ ♣♦r ❊✉❞♦①♦ q✉❡ s❡ ❛♣❧✐❝❛✈❛ t❛♥t♦ ❛ ❣r❛♥❞❡③❛s ❝♦♠❡♥s✉rá✈❡✐s q✉❛♥t♦ ❛ ❣r❛♥❞❡③❛s ✐♥❝♦♠❡♥s✉rá✈❡✐s✳ ❊ss❛ ❞❡✜♥✐çã♦ ❡stá ❡①♣♦st❛ ♥♦ ▲✐✈r♦ ❱ ❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ❡ ❞✐③ ♦ s❡❣✉✐♥t❡✿
✏❉✐③✲s❡ q✉❡ q✉❛tr♦ ❣r❛♥❞❡③❛s ❡stã♦ ♥❛ ♠❡s♠❛ r❛③ã♦✱ ❛ ♣r✐♠❡✐r❛ ♣❛r❛ s❡❣✉♥❞❛ ❡ ❛ t❡r❝❡✐r❛ ♣❛r❛ q✉❛rt❛✱ t♦♠❛♥❞♦✲s❡ ❡q✉✐♠ú❧t✐♣❧♦s q✉❛✐sq✉❡r ❞❛ ♣r✐♠❡✐r❛ ❡ ❞❛ t❡r❝❡✐r❛ ❡ ❡q✉✐✲ ♠ú❧t✐♣❧♦s q✉❛✐sq✉❡r ❞❛ s❡❣✉♥❞❛ ❡ ❞❛ q✉❛rt❛✱ ♦s ♣r✐♠❡✐r♦s ❡q✉✐♠ú❧t✐♣❧♦s sã♦ ❛♠❜♦s ♠❛✐♦r❡s q✉❡✱ ♦✉ ❛♠❜♦s ✐❣✉❛✐s ❛✱ ♦✉ ❛♠❜♦s ♠❡♥♦r❡s q✉❡ ♦s ú❧t✐♠♦s ❡q✉✐♠ú❧t✐♣❧♦s ❝♦♥s✐❞❡r❛❞♦s ❡♠
✶✹ ❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ♦r❞❡♠ ❝♦rr❡s♣♦♥❞❡♥t❡✳✑
❊♠ ❧✐♥❣✉❛❣❡♠ ❛t✉❛❧✱ ❞❛❞❛s q✉❛tr♦ ❣r❛♥❞❡③❛s ✭s❡❣♠❡♥t♦s ❞❡ r❡t❛✱ ár❡❛✱ ✈♦❧✉♠❡✱✳✳✳✮ ❞❛ ♠❡s♠❛ ❡s♣é❝✐❡ A, B, C ❡ D✱ ❞✐③✲s❡ q✉❡A ❡stá ♣❛r❛ B ❛ss✐♠ ❝♦♠♦ C ❡stá ♣❛r❛ D✱ ✐st♦ é✱
A
B =
D C
s❡ ♣❛r❛ ✐♥t❡✐r♦s m ❡ n ❛r❜✐trár✐♦s✱ ❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r ❡r❛♠ ✈❡r✐✜❝❛❞❛s✿
❙❡ m·A > n·B ❡♥tã♦ m·C > n·D ❙❡ m·A < n·B ❡♥tã♦ m·C < n·D ❙❡ m·A=n·B ❡♥tã♦ m·C =n·D
❊ss❛ t❡♦r✐❛ ❢♦r♥❡❝❡✉ ❛ ❢✉♥❞❛♠❡♥t❛çã♦ ♣❛r❛ ❝♦♥str✉çã♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ sé❝✉❧♦ ❳■❳✱ ♣♦r ❉❡❞❡❦✐♥❞✳
▼✉✐t❛ ♠❛t❡♠át✐❝❛ t✐♥❤❛ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞❛ ✉s❛♥❞♦ ✉♠❛ ✐❞❡✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❧✐♠✐t❡s✿ ❢✉♥✲ çõ❡s ❞❡ ✈❛r✐á✈❡✐s r❡❛✐s✱ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ sér✐❡s ✐♥✜♥✐t❛s✱ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧✳ ❊ ❛ ✐❞❡✐❛ ❞❡ ❧✐♠✐t❡ ❢♦r❛ ❝♦♥str✉í❞❛ s♦❜r❡ ✉♠❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ s✐♠♣❧❡s ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❆té q✉❡ ❡ss❛ ❢❛❧❤❛ ❢♦✐ ♣❡r❝❡❜✐❞❛ ❡ ♠✉✐t♦s ♠❛t❡♠át✐❝♦s ❞♦ sé❝✉❧♦ ❳■❳ s❡ ♣r❡♦❝✉♣❛r❛♠ ❡♠ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❞❡✜♥✐çã♦ r✐❣♦r♦s❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♣❛r❛ q✉❡ ❛ss✐♠ t✉❞♦ q✉❡ ❞❡❧❡ ❞❡✲ ❝♦rr❡ss❡ ♥❛ ❛♥á❧✐s❡ ✐♥s♣✐r❛ss❡ s❡❣✉r❛♥ç❛✱ ❡ss❡ ♣❡rí♦❞♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏❛r✐t♠❡t✐③❛çã♦ ❞❛ ❛♥á❧✐s❡✑✳ ▼❛t❡♠át✐❝♦s ❝♦♠♦ ❲❡✐❡rstr❛ss✱ ❈❛♥t♦r ❡ ❉❡❞❡❦✐♥❞ ❞❡❞✐❝❛r❛♠✲s❡ ❛ ❡st❛ t❛r❡❢❛✳ ❙❡♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ss❡ ✉❧t✐♠♦ ❛ ♠❛✐s ❞✐❢✉♥❞✐❞❛✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❚❡♦r✐❛ ❞♦s ❈♦rt❡s ❞❡ ❉❡❞❡❦✐♥❞✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❊✈❡s✱ ❡♠ ❬✽❪✱ ❛ ❛♥á❧✐s❡✱ ❛ ❣❡♦♠❡tr✐❛ ❡ ❛ á❧❣❡❜r❛ ♣♦❞❡♠ s❡r ❞❡❞✉③✐❞❛s ❧♦❣✐❝❛♠❡♥t❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦st✉❧❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❡♠ ♦ s✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡✲ ❛✐s✱ ♣♦rt❛♥t♦✱ ❛ ❝♦♥s✐stê♥❝✐❛ ❞❡ t♦❞❛ ❛ ♠❛t❡♠át✐❝❛ ❡①✐st❡♥t❡ ❞❡♣❡♥❞❡ ❞❛ ❝♦♥s✐stê♥❝✐❛ ❞♦ s✐st❡♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❉❛í ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❡s ♥ú♠❡r♦s✳
❈❛♣ít✉❧♦ ✷
❈♦♥s✐❞❡r❛çõ❡s ✐♥✐❝✐❛✐s
◆❡st❡ ❝❛♣ít✉❧♦ ✈❡r❡♠♦s ❛❧❣✉♥s tó♣✐❝♦s q✉❡ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ ❝♦♠♣r❡❡♥sã♦ ❞❛s ❝♦♥s✲ tr✉çõ❡s q✉❡ ❢❛③❡♠♦s ♥♦ t❡r❝❡✐r♦ ❡ q✉❛rt♦ ❝❛♣ít✉❧♦✳ ■♥✐❝✐❛♠♦s ❢❛❧❛♥❞♦ s♦❜r❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛❣❡♠ ❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡st❛❜❡❧❡❝❡♠♦s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❡♥tr❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ♣❛r❛ tr❛❜❛❧❤❛r ❛ ✐❞❡✐❛ ❞❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❡♥✉♠❡rá✈❡✐s✱ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦✱ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ❝♦r♣♦ ❡ ❝♦r♣♦ ♦r❞❡♥❛❞♦✳ ❊①♣❧✐❝❛♠♦s ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ♠❡❞✐r ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s s♦❜r❡ ♦s t❡♠❛s ❛q✉✐ ❛❜♦r❞❛❞♦s ♦✉ ♠❡s♠♦ ♣❛r❛ ❡♥❝♦♥tr❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦ r❡❝♦♠❡♥❞❛♠♦s ❛ ❧❡✐t✉r❛ ❞❡ ❬✶❪✱ ❬✺❪✱ ❬✼❪✱ ❬✶✺❪✱ ❬✶✻❪ ❡ ❬✶✼❪✳
✷✳✶ ❈♦♥t❛r✿ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ✉♠✲❛✲✉♠
❆ ♥♦çã♦ ❞❡ ♥ú♠❡r♦ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡stã♦ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦s à ❤✐stór✐❛ ❞❛ ❤✉♠❛♥✐❞❛❞❡✳ ❘❡s✉❧t❛ ❞❛s ♥❡❝❡ss✐❞❛❞❡s ❡ ♣r❡♦❝✉♣❛çõ❡s ❞❛s ❝✉❧t✉r❛s ❡ ❞♦s ❣r✉♣♦s s♦❝✐❛✐s ♦s ♠❛✐s ❞✐✈❡rs♦s✱ ♣r♦❝✉r❛♥❞♦ ♣♦r ✈❡③❡s ❝♦♥t❛r ♦s ❞✐❛s ❞♦ ❛♥♦✱ ❝♦♥❝❧✉✐r tr♦❝❛s ❡ tr❛♥s❛çõ❡s✱ ❡♥✉♠❡r❛r s❡✉s ♠❡♠❜r♦s✱ ❡s♣♦s❛s✱ ♠♦rt♦s✱ ❜❡♥s✱ r❡❜❛♥❤♦s✱ s♦❧❞❛❞♦s✱ ♣❡r❞❛s✱ ♣r♦❝✉r❛♥❞♦ ❞❛t❛r ♣♦r ✈❡③❡s ❛ ❢✉♥❞❛çã♦ ❞❡ s✉❛s ❝✐❞❛❞❡s ♦✉ ✉♠❛ ❞❡ s✉❛s ✈✐tór✐❛s✳ ✭❱❡r ❬✶✸❪✱ ♣❛❣✐♥❛ ①✈✐✐✮✳
P❛r❛ ●❡♦r❣❡s ❬✶✸❪✱ ✏é ❛ ❤✐stór✐❛ ❞❡ ✉♠❛ ❤✉♠❛♥✐❞❛❞❡ q✉❡✱ ❣r❛ç❛s à ✐♥t❡❧✐❣ê♥❝✐❛ ❞❡ s✉❛ ❛çã♦ ❡ r❡✢❡①ã♦ ❡ t❛♠❜é♠ ♣❡❧❛ ❢♦rç❛ ❞❛s ❝♦✐s❛s✱ ❢♦✐ ❝♦♥❞✉③✐❞❛ ❛ ❝♦♥s✐❞❡r❛r t✉❞♦ ♦ q✉❡ ❡①✐❣❡ ✉♠❛ ❛✈❛❧✐❛çã♦ ♥✉♠ér✐❝❛✑✳
◆ã♦ s❡ s❛❜❡ ❛♦ ❝❡rt♦ q✉❛♥❞♦ ♦ ❤♦♠❡♠ ❝♦♠❡ç♦✉ ❛ ❝♦♥t❛r✱ ♠❛s ♠❡s♠♦ ❡♠ é♣♦❝❛s ♠❛✐s ♣r✐♠✐t✐✈❛s ❛❝r❡❞✐t❛✲s❡ q✉❡ tí♥❤❛♠♦s ❛❧❣✉♠ s❡♥s♦ ♥✉♠ér✐❝♦ ❛♦ ♠❡♥♦s ❛♦ ♣♦♥t♦ ❞❡ ♥♦t❛r
✶✻ ❈❆P❮❚❯▲❖ ✷✳ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ■◆■❈■❆■❙ q✉❛♥❞♦ s❡ ❛❝r❡s❝❡♥t❛✈❛♠ ♦✉ r❡t✐r❛✈❛♠ ❛❧❣✉♠ ♦❜❥❡t♦ ❞❡ ✉♠❛ ❝♦❧❡çã♦ ♣❡q✉❡♥❛✳ ❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ✈✐❞❛ ❡♠ s♦❝✐❡❞❛❞❡✱ t♦r♥❛r❛♠✲s❡ ✐♥❡✈✐tá✈❡✐s ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♥t❛❣❡♠ ❡ ❞❡ r❡❣✐str♦✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♦s q✉❡ ❣✉❛r❞❛✈❛♠ ♦✈❡❧❤❛s ❡r❛ ✐♠♣♦rt❛♥t❡ ✈❡r✐✜❝❛r s❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛♥✐♠❛✐s q✉❡ s❛✐❛♠ ♣❡❧❛ ♠❛♥❤ã ❡r❛ ❛ ♠❡s♠❛ q✉❛♥❞♦ r❡t♦r♥❛✈❛♠ ❡ ❛ss✐♠ ♣❡r❝❡❜❡r ❛ ❢❛❧t❛ ❞❡ ❛❧❣✉♠❛ q✉❡ ✜❝❛ss❡ ♣❡r❞✐❞❛ ♦✉ ❞❡ ❛❧❣✉♠❛ q✉❡ s❡ ❛❝r❡s❝❡♥t❛ss❡ ❛♦ r❡❜❛♥❤♦✳ ❖ ❤♦♠❡♠ ❝♦♠❡ç♦✉ ❛ ❝♦♥t❛r ✉t✐❧✐③❛♥❞♦ ♦s r❡❝✉rs♦s ❞✐s♣♦♥í✈❡✐s ❡♠ s❡✉ ❡♥t♦r♥♦ ❡ ❛té ♣❛rt❡s ❞❡ s❡✉ ❝♦r♣♦✳
❆❝r❡❞✐t❛✲s❡ q✉❡ ❛ ♠❛♥❡✐r❛ ♠❛✐s ❛♥t✐❣❛ ❞❡ ❝♦♥t❛❣❡♠ s❡ ❜❛s❡❛ss❡ ❡♠ ❛❧❣✉♠ ♠ét♦❞♦ ❞❡ r❡❣✐str♦ s✐♠♣❧❡s✱ ❡♠♣r❡❣❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♠✲❛✲✉♠✳ P❛r❛ ❝♦♥t❛❣❡♠ ❞❡ ♦✈❡❧❤❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣♦❞✐❛✲s❡ ❞♦❜r❛r ✉♠ ❞❡❞♦ ♣❛r❛ ❝❛❞❛ ❛♥✐♠❛❧ ♦✉ ❛ss♦❝✐❛r ✉♠❛ ♣❡❞r❛ ♣❛r❛ ❝❛❞❛ ♦✈❡❧❤❛ ♦✉ ❢❛③❡r ✉♠❛ ♠❛r❝❛ ❡♠ ✉♠ ♦ss♦✳
P❡r❝❡❜❡♠♦s ❡ss❡ t✐♣♦ ❞❡ ❝♦♥t❛❣❡♠ q✉❛♥❞♦ ♣❡❞✐♠♦s ❛ ✉♠❛ ❝r✐❛♥ç❛✱ ♠❡s♠♦ q✉❡ ❛✐♥❞❛ ♥ã♦ s❛✐❜❛ ♦ ♣r♦❝❡ss♦ ❢♦r♠❛❧ ❞❡ ❝♦♥t❛❣❡♠✱ ♣❛r❛ ❝♦♠♣❛r❛r ❞✉❛s ❝♦❧❡çõ❡s ❞✐❢❡r❡♥t❡s✳ ❇❛st❛ ❧❤❡ ♠♦str❛r ✉♠ ♣❛❝♦t❡ ❝♦♠ três ❜♦rr❛❝❤❛s ❡ ✉♠ ❝♦♠ três ❧á♣✐s q✉❡ ❡❧❛ ♣❡r❝❡❜❡rá q✉❡ ♣♦❞❡ ❝♦rr❡s♣♦♥❞❡r ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ♣❛❝♦t❡ ❝♦♠ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ♦✉tr♦✳
P♦❞❡♠♦s ❞✐③❡r q✉❡ ❛ ❝♦♥t❛❣❡♠ s❡ ❞❡✉ ❢❛③❡♥❞♦ ❝♦rr❡s♣♦♥❞❡r s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛ ❝❛❞❛ ❜♦rr❛❝❤❛ ❞♦ ❝♦♥❥✉♥t♦ ✉♠ ❧á♣✐s✳ ❊ss❡ ♣r♦❝❡ss♦ ❞❡ ❛ss♦❝✐❛r ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ♦✉tr♦✱ ❛té q✉❡ ✉♠ ♦✉ ❛♠❜♦s ♦s ❝♦♥❥✉♥t♦s s❡ ❡s❣♦t❡♠✱ é ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♥í✈♦❝❛ ♦✉ ✉♠✲❛✲✉♠✳ ❖ ❛t♦ ❞❡ ❛ss♦❝✐❛r ❞♦✐s ❡❧❡♠❡♥t♦s é ✉♠❛ ❞❛s ♦♣❡r❛çõ❡s ♠❡♥t❛✐s ♠❛✐s ✐♠♣♦rt❛♥t❡s q✉❡ ✉t✐❧✐③❛♠♦s ❝♦♥st❛♥t❡♠❡♥t❡ t♦❞♦s ♦s ❞✐❛s✳
❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s s❡rá ❝❤❛♠❛❞❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ♦✉ ❜✐❥❡t✐✈❛ q✉❛♥❞♦ ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ♣r✐♠❡✐r♦ ❝♦rr❡s♣♦♥❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ s❡❣✉♥❞♦ ❡ r❡❝✐♣r♦❝❛♠❡♥t❡✳
❊①❡♠♣❧♦ ✶ ◆✉♠❛ t✉r♠❛ ❞❡ ❋✉♥❞❛♠❡♥t♦s ❞❡ ▼❛t❡♠át✐❝❛✱ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❛❧✉♥♦ ✴ ♥ú✲ ♠❡r♦ ❞❡ ❝❤❛♠❛❞❛ ❞❛ ❝❛❞❡r♥❡t❛ é ✉♥í✈♦❝❛✱ ❛ss✐♠ ❝♦♠♦ s✉❛ r❡❝í♣r♦❝❛✱ ♥ú♠❡r♦ ❞❡ ❝❤❛♠❛❞❛ ❞❛ ❝❛❞❡r♥❡t❛ ✴ ❛❧✉♥♦✳ ❉✐③❡♠♦s ❡♥tã♦ q✉❡ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❛❧✉♥♦ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❝❤❛♠❛❞❛ ❞❛ ❝❛❞❡r♥❡t❛ é ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✳
❉❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥t❛r s✉r❣❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✶ ✭♦✉ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✮✳ ▼❛s
✶❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦❞❡ s❡r ❝♦♥str✉í❞♦ ♣♦r ♠❡✐♦ ❞❛ t❡♦r✐❛ ❛①✐♦♠át✐❝❛ ❞♦s ❝♦♥❥✉♥t♦s ❡♠
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❋■◆■❚❖❙✱ ■◆❋■◆■❚❖❙ ❊ ❊◆❯▼❊❘➪❱❊■❙ ✶✼ ❛s ♥❡❝❡ss✐❞❛❞❡s ❞❛ ✈✐❞❛ ❞✐ár✐❛ ❞❡♠❛♥❞❛♠✱ ❛❧é♠ ❞❛ ❝♦♥t❛❣❡♠ ❞❡ ♦❜❥❡t♦s ✐♥❞✐✈✐❞✉❛✐s✱ ❛ ♠❡❞✐çã♦ ❞❡ ✈ár✐❛s q✉❛♥t✐❞❛❞❡s✱ ❝♦♠♦ ❝♦♠♣r✐♠❡♥t♦✱ ♣❡s♦ ❡ t❡♠♣♦✳ ❊ss❛s ♠❡❞✐çõ❡s ❡r❛♠ r❡♣r❡s❡♥t❛❞❛s ♣♦r r❛③õ❡s ❡♥tr❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ♦ q✉❡ t❛❧✈❡③ t❡♥❤❛ ❧❡✈❛❞♦ ♦s ♠❡♠❜r♦s ❞❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❛ ♣♦st✉❧❛r❡♠ q✉❡ ♥❛ ♥❛t✉r❡③❛ t✉❞♦ é ♥ú♠❡r♦✱ ❞❡✈✐❞♦ ❛❝r❡❞✐t❛r❡♠ q✉❡ t✉❞♦ ♣♦❞✐❛ s❡r ❝♦♥t❛❞♦✱ ❧♦❣♦ ❛tr✐❜✉í❞♦ ✉♠ ♥ú♠❡r♦✱ ❡ q✉❡ ❛ q✉❛❧q✉❡r ♠❡❞✐❞❛ t❛♠❜é♠ s❡ ♣♦❞❡r✐❛ ❛tr✐❜✉✐r ✉♠ ♥ú♠❡r♦ ♦✉ ✉♠❛ r❛③ã♦ ❡♥tr❡ ♥ú♠❡r♦s✳
◗✉❛♥❞♦ q✉❡r❡♠♦s ❝♦♥t❛r ✉♠❛ ❝♦❧❡çã♦ ❞❡ ♦❜❥❡t♦s ❡st❛❜❡❧❡❝❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✉♠✲❛✲✉♠ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ❝♦❧❡çã♦ ❡ ❛ s❡q✉ê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❆ s❡❣✉✐r ❢❛r❡♠♦s ✉♠❛ ❞❡✜♥✐çã♦ ♠❛✐s ❝✉✐❞❛❞♦s❛ ♣❛r❛ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥t❛❣❡♠ ❡ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳
✷✳✷ ❈♦♥❥✉♥t♦s ✜♥✐t♦s✱ ✐♥✜♥✐t♦s ❡ ❡♥✉♠❡rá✈❡✐s
❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦s ❝♦♥❥✉♥t♦s✿
N={1,2,3,4,5, ...} ❂ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
In={1,2,3,4, ..., n} ❂ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ ✶ ❛té n✳
❈♦♠ ❜❛s❡ ♥❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s ❞❡✜♥✐r❡♠♦s ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❡♥✉♠❡✲ rá✈❡❧✳
❆ ✐❞❡✐❛ q✉❡ t❡♠♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ é q✉❛♥❞♦ ❝♦♥s❡❣✉✐♠♦s ❝♦♥t❛r s❡✉s ❡❧❡♠❡♥t♦s✳
❉❡✜♥✐çã♦ ✶ ❉✐③❡♠♦s q✉❡ A é ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ s❡ é ✈❛③✐♦ ♦✉ s❡✱ ♣❛r❛ ❛❧❣✉♠ n ∈ N✱
❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ f :In→A✳ ❖ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❝❤❛♠❛✲s❡ ♥ú♠❡r♦
❝❛r❞✐♥❛❧ ❞♦ ❝♦♥❥✉♥t♦ A✳
❆ ✐♥❥❡çã♦ f : In → A ❝❤❛♠❛✲s❡ ✉♠❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ A✱ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r
n(A) ♦ s❡✉ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✳
❆♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❛ss♦❝✐❛r❡♠♦s ♦ ❝❛r❞✐♥❛❧ ③❡r♦✳
❊①❡♠♣❧♦ ✷ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❧❡tr❛s ❞❛ ♣❛❧❛✈r❛ ❈■❖◆❊ é ✜♥✐t♦ ❡ ♣♦ss✉✐ ✺ ❡❧❡♠❡♥t♦s✳
C, O O I, O O O, O O N, O O
EOO
1, 2, 3, 4, 5
✶✽ ❈❆P❮❚❯▲❖ ✷✳ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ■◆■❈■❆■❙ ❉❡✜♥✐çã♦ ✷ ❯♠ ❝♦♥❥✉♥t♦ A s❡rá ✐♥✜♥✐t♦ q✉❛♥❞♦ ♥ã♦ ❢♦r ✜♥✐t♦✳ ❖✉ s❡❥❛✱ A ♥ã♦ é ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ ♥ã♦ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ f :In →A q✉❛❧q✉❡r q✉❡ s❡❥❛
n ∈N✳
❊①❡♠♣❧♦ ✸ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ✐♥✜♥✐t♦✳
◆ã♦ ❡①✐st❡ ✉♠ n ∈ N t❛❧ q✉❡ In → N✳ ■st♦ ❞❡❝♦rr❡ ❞♦ ❢❛t♦ q✉❡ ♣❛r❛ t♦❞♦ n ∈ N ❡①✐st❡
✉♠ s✉❝❡ss♦r n+ 1 ∈N✳ ❆ ♦♣❡r❛çã♦ ✏s♦♠❛r ✉♠ ♣♦❞❡✑ s❡r r❡♣❡t✐❞❛ ✐♥✜♥✐t❛♠❡♥t❡✳
✷✳✷✳✶ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s
✏◆✐♥❣✉é♠ ♣♦❞❡rá ♥♦s ❡①♣✉❧s❛r ❞♦ P❛r❛ís♦ q✉❡ ❈❛♥t♦r ❝r✐♦✉✑✳ ❉❛✈✐❞ ❍✐❧❜❡rt✳
❈♦✐s❛s ✐♥t❡r❡ss❛♥t❡s ❛❝♦♥t❡❝❡♠ q✉❛♥❞♦ ❡st❛❜❡❧❡❝❡♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳ ❆♥❛❧✐s❡♠♦s ♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✿
❊①❡♠♣❧♦ ✹ ❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦ ❝♦♥❥✉♥t♦ N ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ♦ ❝♦♥❥✉♥t♦
P ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣❛r❡s✱ ❛♠❜♦s ✐♥✜♥✐t♦s✳ ❆ ♣r✐♥❝í♣✐♦ ♦ ❧❡✐t♦r ♣♦❞❡ ♣❡♥s❛r q✉❡ ♦
❝♦♥❥✉♥t♦ P é ♠❡♥♦r ❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ N✱ t❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ♦ s❡❣✉♥❞♦ é ❢♦r♠❛❞♦ ♣♦r
t♦❞♦s ♦s ♥ú♠❡r♦s ❞❡P ♠❛✐s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s í♠♣❛r❡s ✭q✉❡ t❛♠❜é♠ sã♦ ✐♥✜♥✐t♦s✮✳ ▼❛s
♥ã♦ é ✈❡r❞❛❞❡✳ P♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡
N ❡ ❞❡ P✱ ❜❛st❛ ♣❛r❛ ✐ss♦ ❡st❛❜❡❧❡❝❡r♠♦s ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s
x ∈ N ❝♦♠ ♦s ❡❧❡♠❡♥t♦s y ∈ P✱ t❛❧ q✉❡ y = 2x✳ ❊ s✉❛ r❡❝í♣r♦❝❛✱ ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ y∈ P ❝♦rr❡s♣♦♥❞❡ ✉♠ ❡❧❡♠❡♥t♦ x∈N t❛❧ q✉❡ x= y
2✳ ❈♦♠♦ ♠♦str❛ ♦ ❡sq✉❡♠❛ ❛❜❛✐①♦✿
N: 1,
O
O
2,
O
O
3,
O
O
· · · n,
O
O
· · ·
P : 2, 4, 6, · · · 2n, · · ·
❖✉tr❛s ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❡♥tr❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♠ ✉♠ ❞❡ s❡✉s s✉❜✲ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ♣♦❞❡♠ s❡r ❡st❛❜❡❧❡❝✐❞❛s✳ ❈♦♠♦ ❛ ❞❡ N ❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s
♥❛t✉r❛✐s í♠♣❛r❡s✱ ♦✉ ❞❡ N❝♦♠ ♦ ❝♦♥❥✉♥t♦ ❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s ✷✳
✷●❛❧✐❧❡✉ ●❛❧✐❧❡✐ ✭✶✺✻✹✲✶✻✹✷✮ ❡♠ s✉❛ ♦❜r❛ ❉✐s❝♦rs✐✱ ❛✜r♠❛ q✉❡ ✏♥❡♠ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s é ♠❡♥♦r
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❋■◆■❚❖❙✱ ■◆❋■◆■❚❖❙ ❊ ❊◆❯▼❊❘➪❱❊■❙ ✶✾ ❊①❡♠♣❧♦ ✺ ❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❞❡ ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛✳ ❍á ✐♥✜♥✐t♦s ♣♦♥t♦s ♥✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✳ ❈♦♠♦ ❜❡♠ ♦❜s❡r✈♦✉ ❑✉❜r✉s❧② ❡♠ ❬✶✹❪✱ é ♥❡❝❡ssár✐♦ q✉❡ s❡ ❢❛ç❛ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♣♦♥t♦s ❢ís✐❝♦s q✉❡ sã♦ ♣❡❞❛ç♦s ❞❡ ♠❛tér✐❛ q✉❡ ❝♦♠♣õ❡ ✉♠ t♦❞♦ ❡✱ ♥❡ss❡ ❝❛s♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❞❡ q✉❛❧q✉❡r ♦❜❥❡t♦ s❡rá s❡♠♣r❡ ✜♥✐t♦ ❡ ♠❡♥♦r ❞♦ q✉❡ ♦ ♥ú♠❡r♦ ❛❝✐♠❛✱ ❡ ♣♦♥t♦s ♠❛t❡♠át✐❝♦s✳ P♦♥t♦s✱ r❡t❛s✱ ♣❧❛♥♦s sã♦ ❡♥t❡♥❞✐❞♦s ❝♦♠♦ ❡♥t✐❞❛❞❡s ♣✉r❛♠❡♥t❡ ♠❛t❡♠át✐❝❛s ❡ ❛❜str❛t❛s✳ ❙ã♦ ❝♦♥s✐❞❡r❛❞❛s ❝♦♥tí♥✉❛s ❡ ♣♦rt❛♥t♦ ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡ s❡r❡♠ ❞✐✈✐❞✐❞❛s ❡t❡r♥❛ ❡ ✐♥✜♥✐t❛♠❡♥t❡✳ ◗✉❛❧q✉❡r t❛♠❛♥❤♦ ♣♦❞❡ s❡r s❡♠♣r❡ ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ♠❡t❛❞❡s ❡ ❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❞✐✈✐❞✐r ♣♦r ♠❡t❛❞❡s ♥✉♥❝❛ t❡rá ✜♠✳ ◆❛ ❞❡✜♥✐çã♦ ❞❡ ❊✉❝❧✐❞❡s✱ ♦ ♣♦♥t♦ é ♦ q✉❡ ♥ã♦ t❡♠ ♣❛rt❡s ❡ ♣♦rt❛♥t♦ ♥ã♦ s❡rá ❞✐✈✐sí✈❡❧ ❡♠ ♠❡t❛❞❡s✳ ◆ã♦ t❡rá t❛♠❛♥❤♦ ♦✉ ♠❡❞✐❞❛ ❡ ♣♦r ♥ã♦ ♦❝✉♣❛r ❡s♣❛ç♦ s❡rá ♣♦ssí✈❡❧ ❛♠♦♥t♦❛r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡❧❡s ❡♠ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛✳
❈♦♥s✐❞❡r❡ ♦ tr✐â♥❣✉❧♦ ABC✱ ✈❡r ❋✐❣✉r❛ ✷✳✶✱ tr❛❝❡♠♦s ✉♠❛ r❡t❛ ♣❡❧♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ AC ♣❛r❛❧❡❧❛ ❛♦ s❡❣♠❡♥t♦ AB✳ ❙❛❜❡♠♦s q✉❡ AB = 2M N✳ ❆♣❡s❛r ❞♦ s❡❣♠❡♥t♦ M N t❡r ❝♦♠♣r✐♠❡♥t♦ ✐❣✉❛❧ ❛ ♠❡t❛❞❡ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ AB✱ ♦ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡ ♣♦♥t♦s ❞❡ M N é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡ ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ AB✳ P❛r❛ ✈❡r✐✜❝❛r ✐ss♦✱ ❜❛st❛ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ❡ss❡s ❞♦✐s ❝♦♥❥✉♥t♦s✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❛ ❝❛❞❛ ♣♦♥t♦ P ❞❡ M N ❢❛③✲s❡ ❝♦rr❡s♣♦♥❞❡r ♦ ♣♦♥t♦ D✭ú♥✐❝♦✮ ❞❡ AB ❡♠ q✉❡ D é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ CP ❝♦♠ ♦ s❡❣♠❡♥t♦ AB❀ ❛ ❝❛❞❛ ♣♦♥t♦ E ❞❡ AB ❢❛③✲s❡ ❝♦rr❡s♣♦♥❞❡r ♦ ♣♦♥t♦ Q✭ú♥✐❝♦✮ ❞❡ M N ❡♠ q✉❡ Q é ❛ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ EC ❝♦♠ ♦ s❡❣♠❡♥t♦ M N✳
❋✐❣✉r❛ ✷✳✶✿ ❈♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ ❆❇ ❡ ▼◆✳
✷✵ ❈❆P❮❚❯▲❖ ✷✳ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ■◆■❈■❆■❙ ■ss♦ é ❜❛st❛♥t❡ ❝✉r✐♦s♦ ❡ ❢♦❣❡ ❞❛ r❡❛❧✐❞❛❞❡ ✐♠❡❞✐❛t❛ ❞♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦ ❡♠ q✉❡ ♦ t♦❞♦ é s❡♠♣r❡ ♠❛✐♦r ❞♦ q✉❡ s✉❛ ♣❛rt❡✱ ♥♦ ❝❛♠♣♦ ✐♥✜♥✐t♦ ♦ t♦❞♦ ♣♦❞❡ s❡r ❡q✉✐✈❛❧❡♥t❡ ❛ s✉❛ ♣❛rt❡✳
❈♦♠♦ ❞✐ss❡♠♦s ♥♦ ❈❛♣ít✉❧♦ ✶✱ ❤♦✉✈❡ ♠✉✐t❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❡ ❛❝❡✐t❛r ❣r❛♥❞❡③❛s ✐♥✜♥✐t❛s✳ ❯♠ ❞♦s ♠❛t❡♠át✐❝♦s q✉❡ ❡st✉❞❛r❛♠ ❣r❛♥❞❡③❛s ✐♥✜♥✐t❛s ❢♦✐ ❈❛♥t♦r✱ r❡s♣♦♥sá✈❡❧ ♣♦r ✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❡ ♠♦str❛r q✉❡ ❤á ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s✳ ❙✉r❣✐r❛♠ ❛ss✐♠✱ ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ❡ ❞❡ ❝♦♥❥✉♥t♦ ♥ã♦ ❡♥✉♠❡rá✈❡❧✳
✷✳✷✳✷ ❈♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s
■♥t✉✐t✐✈❛♠❡♥t❡✱ ✉♠ ❝♦♥❥✉♥t♦ é ❡♥✉♠❡rá✈❡❧ s❡ s❡✉s ❡❧❡♠❡♥t♦s ♣♦❞❡♠ s❡r ❝♦❧♦❝❛❞♦s ♥✉♠❛ ❧✐st❛ ❞❡ ♠♦❞♦ q✉❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r ❛❧❝❛♥ç❛❞♦ s❡ ❛✈❛♥ç❛r♠♦s ♦ s✉✜❝✐❡♥t❡ ♥❛ ❧✐st❛✳
❉❡✜♥✐çã♦ ✸ ❯♠ ❝♦♥❥✉♥t♦ Aé ❞✐t♦ ❡♥✉♠❡rá✈❡❧✱ s❡ ❡❧❡ ❢♦r ✜♥✐t♦ ♦✉ s❡ ❡①✐st✐r ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ A ❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s N✳
❊①❡♠♣❧♦ ✻ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s í♠♣❛r❡s é ❡♥✉♠❡rá✈❡❧✳ ❖ ❡sq✉❡♠❛ ❛❜❛✐①♦ ✐♥❞✐❝❛ ✉♠❛ ❢♦r♠❛ ❞❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✳
1, O O 3, O O 5, O O
· · · , 2n−1,
O
O
· · ·
1, 2, 3, · · · , n, · · ·
❊①❡♠♣❧♦ ✼ ❖ ❝♦♥❥✉♥t♦ Z = {0,±1,±2,±3,· · · } ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s é ❡♥✉♠❡rá✈❡❧✳ ❖ ❡sq✉❡♠❛ ❛❜❛✐①♦ ✐♥❞✐❝❛ ❝♦♠♦ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ Z ❡ ♦s
❡❧❡♠❡♥t♦s ❞❡ N✳
0, O O 1, O O
−1,
O O 2, O O
−2,
O O 3, O O
−3,
O
O
· · · n,
O O −n, O O · · ·
1, 2, 3, 4, 5, 6, 7, · · · 2n, 2n+ 1, · · ·
✷✳✷✳ ❈❖◆❏❯◆❚❖❙ ❋■◆■❚❖❙✱ ■◆❋■◆■❚❖❙ ❊ ❊◆❯▼❊❘➪❱❊■❙ ✷✶ Pr♦♣♦s✐çã♦ ✶ ❙❡ ❞♦✐s ❝♦♥❥✉♥t♦s✱ A ❡ B✱ sã♦ ❡♥✉♠❡rá✈❡✐s✱ ❡♥tã♦ ❛ ✉♥✐ã♦ ❞❡ A ❡ B é ❡♥✉♠❡rá✈❡❧✳
Pr♦✈❛✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦sA={a1, a2, a3,· · · , an,· · · }❡B ={b1, b2, b3,· · · , bn,· · · }
❡♥tã♦ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡A∪B ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡N ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
a1,
O
O
b1,
O
O
a2,
O
O
b2,
O
O
· · · an,
O
O
bn,
O
O
· · ·
1, 2, 3, 4 · · · 2n−1, 2n, · · ·
❆ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❡✜♥✐❞❛ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ❝❛❞❛an∈A∪B❝♦rr❡s♣♦♥❞❡ ❛2n−1∈N
❡ ❝❛❞❛bn∈A∪B ❝♦rr❡s♣♦♥❞❡ ❛ 2n∈N✱ ❝♦♠ n ♥❛t✉r❛❧✳
⋄
Pr♦♣♦s✐çã♦ ✷ ❆ ✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s é ❡♥✉♠❡rá✈❡❧✳
Pr♦✈❛✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦sA1 ={a11, a12, a13,· · · }✱A2 ={b21, b22, b23,· · · },· · · , An =
{an1, an2,· · · }✳ P♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❜✐✉♥í✈♦❝❛ ❞❡ B =A1∪A2∪ · · · ∪An ❡
N❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❉✐s♣♦r ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ B ❡♠ ✉♠❛ t❛❜❡❧❛ ❡♠ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ An s❡rã♦ ❞✐s♣♦st♦s ♥❛ ❧✐♥❤❛n✱ s❡❣✉✐♥❞♦ ❛ ♦r❞❡♠ ❝r❡s❝❡♥t❡✳ ❖ ❡sq✉❡♠❛ ❛❜❛✐①♦ ♥♦s s✉❣❡r❡
✉♠❛ ❢♦r♠❛ ♣❛r❛ ❡♥✉♠❡r❛r✿
a11 a12 a13 · · · a21 a22 a23 · · · ✳✳✳ ✳✳✳ ✳✳✳ . . .
an1
I
I
an2
I
I
an3
I
I
· · ·
⋄
