Técnicas para Demonstração de Teoremas Parte IV

(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 ✖ ✭■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✻

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ↔◗✿

(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♦✈❛✿

(→) ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ P →◗❪✳

(5)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❱✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❍✐♣ót❡s❡s ✺

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P ↔◗✿

(6)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✺✿ ❙✉♣♦♥❤❛ ① ∈ Z✳ ❊♥tã♦ ① é ♣❛r

ss❡ ①✷ _{é ♣❛r✳}

❘❡♣r❡s❡♥t❛♥❞♦ ♦ t❡♦r❡♠❛ ♣♦r ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛

(7)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶

❍✐♣ót❡s❡s Pr♦✈❛r

① ∈ Z ① é ♣❛r → ①✷ é ♣❛r✳

①✷ _{é ♣❛r} _→ _{① é ♣❛r✳}

(8)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ①✷ é ♣❛r

① é ♣❛r

(9)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❱■■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ∃❦.❦ ∈ Z∧ ①

✷ ₌ _✷❦

∃♥.♥ ∈ Z∧① = ✷♥

(10)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭■❳✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✹

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ∃❦.❦ ∈ Z∧ ①

✷ ₌ _✷❦

∃♥.♥ ∈ Z∧① = ✷♥

(11)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✺

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ∃❦.❦ ∈ Z∧①

✷ ₌ _✷❦

♥✵ ∈ Z

① = ✷♥_✵

❙❡❥❛ ❦ = ✷♥✵✷✳ P♦✐s✱ ①✷ = ✹♥✵✷ = ✷.(✷♥✵

✷₎_✳

❈♦♠♦ ✷♥✵✷ ∈ Z✱ ❝♦♥❝❧✉í♠♦s ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛

(12)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■✮

❊①❡♠♣❧♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✶

❍✐♣ót❡s❡s Pr♦✈❛r

① ∈ Z ①✷ é ♣❛r → ① é ♣❛r✳

(14)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■■■✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✷

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ¬✭①✷ é ♣❛r✮ ¬✭① é ♣❛r✮

(15)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❱✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✸

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ∃❦.❦ ∈ Z∧ ①

✷ ₌ _✷❦ ₊_✶

∃♥.♥ ∈ Z ∧① = ✷♥+✶

(16)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱✮

❘❛s❝✉♥❤♦ ✖ P❛ss♦ ✹

❍✐♣ót❡s❡s Pr♦✈❛r ① ∈ Z ∃❦.❦ ∈ Z∧①

✷ ₌ _✷❦ ₊_✶

♥✵ ∈ Z

① = ✷♥✵ +✶

(17)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■✮

❊①❡r❝í❝✐♦s

❚❡♦r❡♠❛ ✺✿ P❛r❛ q✉❛✐sq✉❡r ❝♦♥❥✉♥t♦s ❆ ❡ ❇✱

(18)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■✮

❊str❛té❣✐❛ ♣❛r❛ ♦ ✉s♦ ❞❡ ❤✐♣ót❡s❡s ✻

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P ∨◗✿

(19)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❱■■■✮

❊str❛té❣✐❛ ♣❛r❛ ♦ ✉s♦ ❞❡ ❤✐♣ót❡s❡s ✻

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P ∨◗✿

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

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

P ...

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳■❳✮

❊str❛té❣✐❛ ♣❛r❛ ♦ ✉s♦ ❞❡ ❤✐♣ót❡s❡s ✻

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛ P ∨◗✿

❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿ ❈❛s♦ ✶✿ P é ✈❡r❞❛❞❡✐r♦✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ q✉❡ ✉s❛ P❪✳ ❈❛s♦ ✷✿ ◗ é ✈❡r❞❛❞❡✐r♦✳

❬❚❡①t♦ ❞❛ ♣r♦✈❛ q✉❡ ✉s❛ ◗❪✳

(21)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳✮

❊①❡♠♣❧♦

(22)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■✮

❊str❛té❣✐❛ ❞❡ ♣r♦✈❛ ✼

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∨◗

(23)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■✮

❊①❡♠♣❧♦

(24)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■■■✮

❊str❛té❣✐❛ ❞❡ ♣r♦✈❛ ✽

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∨◗

❙❡ P é ✈❡r❞❛❞❡✐r♦ ❡♥tã♦ P ∨◗ é ✈❡r❞❛❞❡✐r♦✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❱✮

❊str❛té❣✐❛ ❞❡ ♣r♦✈❛ ✽

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

... ◗

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱✮

❊str❛té❣✐❛ ❞❡ ♣r♦✈❛ ✽

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ P ∨◗✿

❚❡①t♦ ✜♥❛❧ ❞❛ ♣r♦✈❛✿

❙❡ P é ✈❡r❞❛❞❡✐r♦✱ é ó❜✈✐♦ q✉❡ P ∨◗ é

✈❡r❞❛❞❡✐r♦✳ ❆❣♦r❛ s✉♣♦♥❤❛ q✉❡ P é ❢❛❧s♦✳ ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ◗❪

(27)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✽✿ ❙✉♣♦♥❤❛ ① ∈ R✳ ❙❡ ①✷ ≥ ① ❡♥tã♦

(28)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■✮

❊str❛té❣✐❛ ♣❛r❛ ♦ ✉s♦ ❞❡ ❤✐♣ót❡s❡s ✽

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ P ∨◗✿

❙❡ ✈♦❝ê ♣♦ss✉✐ ❛❧❣✉♠❛ ❞❡st❛s ❤✐♣ót❡s❡s ¬P ♦✉ ¬◗✱ ✈♦❝ê ♣♦❞❡ ✉s❛r ❛ s❡❣✉✐♥t❡ ❡q✉✐✈❛❧ê♥❝✐❛

(29)

❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❱■■■✮

❊①❡♠♣❧♦

❚❡♦r❡♠❛ ✾✿ ❙✉♣♦♥❤❛ ♠,♥ ∈ Z✳ ❙❡ ♠♥ é ♣❛r ❡♥tã♦

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳■❳✮

❘❡♣r❡s❡♥t❛♥❞♦ ❯♥✐❝✐❞❛❞❡

▼✉✐t❛s ✈❡③❡s ❡♠ ♠❛t❡♠át✐❝❛✱ ♣r❡❝✐s❛♠♦s ❞❡♠♦♥str❛r ❛✜r♠❛t✐✈❛s s✐♠✐❧❛r❡s ❛✿

❊①✐st❡ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ① t❛❧ q✉❡ ✳✳✳ ❘❡♣r❡s❡♥t❛♠♦s t❛✐s ❛✜r♠❛t✐✈❛s ❝♦♠♦✿

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳✮

❘❡♣r❡s❡♥t❛♥❞♦ ∃!①.♣(①)

P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ∃!①.♣(①) ✉s❛♥❞♦ ♦✉tr♦s

q✉❛♥t✐✜❝❛❞♦r❡s✿

∃①.(♣(①)∧ ¬∃②.(♣(②)∧② 6=①))

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✾

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃!①.♣(①)✿

❉✐✈✐❞❛ s✉❛ ♣r♦✈❛ ❡♠✿

∃①.♣(①)✿ Pr♦✈❛ ❞❛ ❡①✐stê♥❝✐❛

∀②.∀③.((♣(②)∧♣(③))→② =③)✿ Pr♦✈❛ ❞❛

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✾

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃!①.♣(①)✿

❚❡①t♦ ❞❛ ♣r♦✈❛✿

❊①✐stê♥❝✐❛✿ ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡ ∃①.♣(①)❪

❯♥✐❝✐❞❛❞❡✿ ❬❚❡①t♦ ❞❛ ♣r♦✈❛ ❞❡

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■■■✮

❊str❛té❣✐❛ ❞❡ Pr♦✈❛ ✶✵

P❛r❛ ♣r♦✈❛r ✉♠❛ ❛✜r♠❛t✐✈❛ ∃!①.♣(①)✿

Pr♦✈❡ q✉❡ ∃①.(♣(①)∧ ∀②.(♣(②) →② = ①))✱

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳■❱✮

❊str❛té❣✐❛ ♣❛r❛ ✉s♦ ❞❡ ❤✐♣ót❡s❡s ✽

P❛r❛ ✉s❛r ✉♠❛ ❤✐♣ót❡s❡ ❞❛ ❢♦r♠❛

∃!①.♣(①)✿

❉✐✈✐❞❛✲❛ ❡♠ ❞✉❛s ❤✐♣ót❡s❡s✿

∃①.♣(①)✳

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳❱✮

❊①❡♠♣❧♦s

❚❡♦r❡♠❛ ✶✵✿ ❊①✐st❡ ✉♠ ú♥✐❝♦ ❝♦♥❥✉♥t♦ ❆ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ ❇✱ ❆∪ ❇ = ❇✳

❚❡♦r❡♠❛ ✶✶✿ ❙❡ ① ∈ R✱ ① 6= ✷✱ ❡♥tã♦ ❡①✐st❡ ✉♠

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❉❡♠♦♥str❛çã♦ ❞❡ ❚❡♦r❡♠❛s ✖ ✭❳❳❳❱■✮

❈♦♥❝❧✉sã♦

❈♦♠ ❡st❛s ❡str❛té❣✐❛s ✈♦❝ê ❥á é ❝❛♣❛③ ❞❡ ♣r♦✈❛r ♠✉✐t♦s t❡♦r❡♠❛s✳