Técnicas para Demonstração de Teoremas Parte II

(1)

❚é❝♥✐❝❛s ♣❛r❛ ❉❡♠♦♥str❛çã♦ ❞❡

❚❡♦r❡♠❛s ✖ P❛rt❡ ■■

▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ■

❘♦❞r✐❣♦ ❘✐❜❡✐r♦

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❆♣❧✐❝❛❞❛s ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦

(2)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■✮

❆♥t❡r✐♦r♠❡♥t❡✳✳✳

❋♦✐ ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❡str❛té❣✐❛ ❞❡ ♣r♦✈❛ ♣❛r❛ ❛✜r♠❛t✐✈❛s ❞♦ t✐♣♦ P →◗✳

❊①✐st❡ ♦✉tr❛ ❡str❛té❣✐❛ ♣❛r❛ ♣r♦✈❛s ❞❡

(3)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✷

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P →◗✿

❘❛s❝✉♥❤♦ ❛♥t❡s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r

... P →◗

❘❛s❝✉♥❤♦ ❞❡♣♦✐s ❞❡ ✉s❛r ❛ ❡str❛té❣✐❛✿ ❍✐♣ót❡s❡s Pr♦✈❛r

... ¬P

(4)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■■■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✷

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P →◗✿

❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿

❙✉♣♦♥❤❛ q✉❡ ◗ é ❢❛❧s♦✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ¬P❪✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❱✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✷✿ ❙✉♣♦♥❤❛ ❛,❜,❝ ∈ R ❡ ❛ > ❜✳ ❙❡

(6)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱✮

❍✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛

❛,❜,❝ ∈ R

❛ > ❜

❊①♣r❡ss❛♥❞♦ ♦ t❡♦r❡♠❛ ✉s❛♥❞♦ ❧ó❣✐❝❛

(7)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶

❍✐♣ót❡s❡s Pr♦✈❛r

❛,❜,❝ ∈ R ❛❝ ≤ ❜❝ →❝ ≤ ✵

❛ > ❜

P♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ❛ ❢ór♠✉❧❛ ♣♦ss✉✐ ❛ ❢♦r♠❛ P →◗✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s s✉♣♦r ❝♦♠♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷

❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜,❝ ∈ R ¬(❛❝ ≤ ❜❝)

❛ > ❜

¬(❝ ≤✵)

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸

❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜,❝ ∈ R ❛❝ > ❜❝

❛ > ❜

❝ > ✵

❈♦♠♦ ❝ > ✵✱ ♣♦❞❡♠♦s ♠✉❧t✐♣❧✐❝❛r ❛♠❜♦s ♦s

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❳✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸

❍✐♣ót❡s❡s Pr♦✈❛r ❛,❜,❝ ∈ R ❛❝ > ❜❝

❛ > ❜

❝ > ✵

❛❝ > ❜❝

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳✮

❖❜t❡♥❞♦ ♦ t❡①t♦ ✜♥❛❧ ❛ ♣❛rt✐r ❞♦ ❣❛❜❛r✐t♦✳✳✳

❙❡❣✉✐♥❞♦ ♦ ❣❛❜❛r✐t♦ ❞❛ ❡str❛té❣✐❛✳✳✳

❙✉♣♦♥❤❛ q✉❡ ❝ > ✵✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ¬P❪✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ❛ > ❜ ❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■✮

❖❜t❡♥❞♦ ♦ t❡①t♦ ✜♥❛❧ ❛ ♣❛rt✐r ❞♦ ❣❛❜❛r✐t♦✳✳✳

❈♦❧♦❝❛♥❞♦ ❛ ♣r♦✈❛ ❞❡ ¬P✳✳✳

❙✉♣♦♥❤❛ q✉❡ ❝ > ✵✳

❈♦♠♦ ❛ > ❜ ❡ ❝ > ✵✱ ♣♦❞❡♠♦s ♠✉t✐♣❧✐❝❛r ❛♠❜♦s

♦s ❧❛❞♦s ❞❡ ❛> ❜ ♣♦r ❝✱ ♦❜t❡♥❞♦ ❛❝ > ❜❝✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ❛ > ❜ ❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■✮

❚❡①t♦ ❋✐♥❛❧ ❞❛ Pr♦✈❛

❚❡♦r❡♠❛ ✷✿ ❙✉♣♦♥❤❛ ❛,❜,❝ ∈ R ❡ ❛> ❜✳ ❙❡

❛❝ ≤ ❜❝ ❡♥tã♦ ❝ ≤✵✳

Pr♦✈❛✿ ❙✉♣♦♥❤❛ q✉❡ ❝ > ✵✳ ❈♦♠♦ ❛ > ❜ ❡ ❝ > ✵✱

♣♦❞❡♠♦s ♠✉t✐♣❧✐❝❛r ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ ❛ > ❜ ♣♦r ❝

s❡♠ ❛❧t❡r❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♦r✐❣✐♥❛❧✱ ♦❜t❡♥❞♦ ❛❝ > ❜❝✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ❛ > ❜ ❡ ❛❝ ≤❜❝

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■■✮

❊①❡r❝í❝✐♦

❚❡♦r❡♠❛ ✷✿ ❙✉♣♦♥❤❛ ❛,❜,❝,❞ ∈ R t❛✐s q✉❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❱✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✸

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ¬P✿

❚❡♥t❡ r❡❡①♣r❡ss❛r ¬P ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛ ✭✉s❛♥❞♦

(16)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✸✿ ❙✉♣♦♥❤❛ ❆∩❈ ⊆ ❇ ❡ ❛ ∈ ❈✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■✮

❍✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛

❆∩❈ ⊆ ❇

❛ ∈ ❈

❊①♣r❡ss❛♥❞♦ ♦ t❡♦r❡♠❛ ✉s❛♥❞♦ ❧ó❣✐❝❛

(18)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■✮

❆♣❧✐❝❛♥❞♦ ❛ ❡str❛té❣✐❛

❈♦♠♦ t❡♠♦s q✉❡ ♣r♦✈❛r ¬(❛ ∈ ❆\❇) ❡ q✉❡

❡st❛ ♣♦ss✉✐ ❛ ❢♦r♠❛ ¬P✱ ♣♦❞❡♠♦s r❡❡①♣r❡ss❛r ¬(❛ ∈ ❆\❇) ❡♠ t❡r♠♦s ❞❡ ♦✉tr♦s ❝♦♥❡❝t✐✈♦s

❧ó❣✐❝♦s✳

(19)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■■✮

❆♣ós ✉♠ ♣♦✉❝♦ ❞❡ á❧❣❡❜r❛✳✳✳

❚r♦❝❛♠♦s ✉♠❛ ❞❡♠♦♥str❛çã♦ ❡♥✈♦❧✈❡♥❞♦

¬(❛ ∈ ❆\❇) ♣♦r ✉♠❛ ❡♥✈♦❧✈❡♥❞♦

❛ ∈ ❆ →❛ ∈ ❇ q✉❡ é ♠❛✐s s✐♠♣❧❡s✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❳✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ¬P✿

❆ss✉♠❛ q✉❡ P é ✈❡r❞❛❞❡✐r♦ ❡ t❡♥t❡ ♦❜t❡r ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❙❡ ✈♦❝ê ❝♦♥s❡❣✉✐r✱ ♣♦❞❡rá ❝♦♥❝❧✉✐r q✉❡ ¬P é ✈❡r❞❛❞❡✐r♦ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞♦ t❡r❝❡✐r♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ¬P✿

... ¬P

... ⊥

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✹

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ¬P✿

❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿

❙✉♣♦♥❤❛ q✉❡ P é ✈❡r❞❛❞❡✐r♦✳ ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ⊥❪✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✹✿ ❙❡ ①✷ ₊_② ₌ _{✶✸ ❡ ②} ₆₌ _{✹ ❡♥tã♦}

① 6= ✸✳

❊①♣r❡ss❛♥❞♦ ♦ t❡♦r❡♠❛ ✉s❛♥❞♦ ❧ó❣✐❝❛

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶

①✷ ₊ _② ₌ _✶✸_∧_② ₆₌ _✹ _→ _① ₆₌ _✸

❈♦♠♦ ❛ ❢ór♠✉❧❛ t❡♠ é ❞❛ ❢♦r♠❛ P → ◗✱

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❱✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷

①✷ ₊_② ₌ _✶✸_∧_② ₆₌ _✹ _① ₆₌ _✸

P♦❞❡♠♦s ❛❣♦r❛ ❞✐✈✐❞✐r ①✷ ₊_② ₌ _✶✸_∧ _② ₆₌ _{✹ ❡♠}

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸

❍✐♣ót❡s❡s Pr♦✈❛r ①✷ ₊_② ₌ _✶✸ _① ₆₌ _✸

② 6= ✹

❆❣♦r❛ t❡♠♦s q✉❡ ♣r♦✈❛r q✉❡ ① 6= ✸✳ P❛r❛ ✐ss♦

♣♦❞❡♠♦s ✉s❛r ❛ ❡str❛té❣✐❛ ❞❡ s✉♣♦r ① = ✸ ❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✹

❍✐♣ót❡s❡s Pr♦✈❛r ①✷ ₊_② ₌ _✶✸ _⊥

② 6= ✹

① = ✸

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✺

❍✐♣ót❡s❡s Pr♦✈❛r ✾+② = ✶✸ ⊥

② 6= ✹

① = ✸

▼❛s ♥❛ ❡q✉❛çã♦ ✾+② = ✶✸✱ t❡♠♦s q✉❡ ② = ✹✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■■✮

❖❜t❡♥❞♦ ♦ t❡①t♦ ✜♥❛❧ ❛ ♣❛rt✐r ❞♦ ❣❛❜❛r✐t♦✳✳✳

Pr✐♠❡✐r♦ ♦ ❣❛❜❛r✐t♦ ❞❛ ❡str❛té❣✐❛ P → ◗

❙✉♣♦♥❤❛ q✉❡ ①✷ ₊ _② ₌ _{✶✸ ❡ ②} ₆₌ _✹✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ① 6= ✸❪✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ①✷ ₊ _② ₌ _✶✸

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❳✮

❖❜t❡♥❞♦ ♦ t❡①t♦ ✜♥❛❧ ❛ ♣❛rt✐r ❞♦ ❣❛❜❛r✐t♦✳✳✳

❆❣♦r❛ ♦ ❣❛❜❛r✐t♦ ❞❛ ❡str❛té❣✐❛ ¬P

❙✉♣♦♥❤❛ q✉❡ ①✷ ₊ _② ₌ _{✶✸ ❡ ②} ₆₌ _✹✳

❙✉♣♦♥❤❛ q✉❡ ① = ✸✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ⊥❪

P♦rt❛♥t♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ① 6= ✸

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ①✷ ₊ _② ₌ _✶✸

(31)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳✮

❚❡①t♦ ❞❛ Pr♦✈❛

❚❡♦r❡♠❛ ✹✿ ❙❡ ①✷ ₊_② ₌ _{✶✸ ❡ ②} ₆₌ _{✹ ❡♥tã♦ ①} ₆₌ _✸✳

Pr♦✈❛✿ ❙✉♣♦♥❤❛ q✉❡ ①✷ ₊_② ₌ _{✶✸ ❡ ②} ₆₌ _✹✳

❙✉♣♦♥❤❛✱ ❛❣♦r❛ q✉❡ ① = ✸✳ ❙✉❜st✐t✉✐♥❞♦ ① = ✸ ❡♠

①✷ ₊_② ₌ _{✶✸✿ t❡♠♦s q✉❡ ✾}₊_② ₌ _{✶✸ ❡ ✜♥❛❧♠❡♥t❡}

② = ✹✳ ▼❛s s❛❜❡♠♦s q✉❡ ② 6= ✹✱ ♦ q✉❡ ❝♦♥st✐t✉✐ ✉♠❛

❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦ ❛ s✉♣♦s✐çã♦ ❞❡ q✉❡ ① = ✸ ♥ã♦

♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r❛ ❡ ♣♦rt❛♥t♦ t❡♠♦s q✉❡ ① 6= ✸✳

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ ①✷ ₊_② ₌ _{✶✸ ❡}

(32)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■✮

❊str❛té❣✐❛s ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s

❯♠❛ ✐♠♣♦rt❛♥t❡ tát✐❝❛ ❞✉r❛♥t❡ ❛ ❞❡♠♦♥str❛çã♦ é s❛❜❡r ✉t✐❧✐③❛r ❤✐♣ót❡s❡s✳

P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❝♦♥❤❡❝❡r ❡str❛té❣✐❛s ♣❛r❛ ❧✐❞❛r ❝♦♠ ❤✐♣ót❡s❡s q✉❡ ♣♦ss✉❛♠ ✉♠

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✶

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ ¬P✿

❊♠ ✉♠❛ ♣r♦✈❛ ♣♦r ❝♦♥tr❛❞✐çã♦ ❛❞✐❝✐♦♥❡ P às ❢ór♠✉❧❛s ❛ s❡r❡♠ ♣r♦✈❛❞❛s✳ ❙❡ ✈♦❝ê ♣r♦✈❛r q✉❡ P é ✈❡r❞❛❞❡✐r♦✱ ❝♦♠♦ ¬P é ✉♠❛ ❤✐♣ót❡s❡✱ ✐ss♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■■✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✶

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ ¬P✿

¬P ⊥

...

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■❱✮

❊①❡r❝í❝✐♦

❚❡♦r❡♠❛ ✹✿ ❙✉♣♦♥❤❛ ❆,❇ ❡ ❈ ❝♦♥❥✉♥t♦s t❛✐s q✉❡

(36)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳❱✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✷

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ ¬P✿

❙❡ ♣♦ssí✈❡❧✱ t❡♥t❡ r❡❡①♣r❡ss❛r ¬P ❡♠ t❡r♠♦s ❞❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳❱■✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✸

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P →◗✿

❯s❡ ❡st❛ ❤✐♣ót❡s❡ ♣❛r❛ ❞❡❞✉③✐r ◗ ♦✉ ¬P ✉s❛♥❞♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳❱■■✮

❊①❡r❝í❝✐♦

❚❡♦r❡♠❛ ✹✿ ❙✉♣♦♥❤❛ ❆ ⊆❇✱ ❛ ∈ ❆✱ ❛ 6∈ ❇ \❈✳