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FUNDAMENTOS DE LÓGICA, CONJUNTOS E NÚMEROS NATURAIS

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊

P❘Ó✲❘❊■❚❖❘■❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊ P❊❙◗❯■❙❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆

❘❊❉❊ ◆❆❈■❖◆❆▲ ✲ P❘❖❋▼❆❚

❘❆❋❆❊▲ ▼❊❙❙■❆❙ ❙❆◆❚❖❙

❋❯◆❉❆▼❊◆❚❖❙ ❉❊ ▲Ó●■❈❆✱ ❈❖◆❏❯◆❚❖❙ ❊

◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙

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❘❆❋❆❊▲ ▼❊❙❙■❆❙ ❙❆◆❚❖❙

❋❯◆❉❆▼❊◆❚❖❙ ❉❊ ▲Ó●■❈❆✱ ❈❖◆❏❯◆❚❖❙ ❊

◆Ú▼❊❘❖❙ ◆❆❚❯❘❆■❙

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦s❡ ❆♥❞❡rs♦♥ ❱❛✲ ❧❡♥ç❛ ❈❛r❞♦s♦

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❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r s❡♠♣r❡ ✐❧✉♠✐♥❛r ♠❡✉s ❝❛♠✐♥❤♦s ❡ ❞❡❝✐sõ❡s✳ ❙ó t❡♥❤♦ q✉❡ ❛❣r❛❞❡❝❡r✱ t♦❞♦s ♦s ❞✐❛s✳

➚ ♠✐♥❤❛ ❡s♣♦s❛ Pr✐s❝✐❧❛ ▼❡♥❞♦♥ç❛ ▼♦✉r❛✱ ❣r❛♥❞❡ ❝♦♠♣❛♥❤❡✐r❛ q✉❡ s♦✉❜❡ t♦❧❡r❛r ♠✐♥❤❛s ✐rr✐t❛çõ❡s ❡ ❡str❡ss❡s ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡✐♦ ❞❡ ♠❡str❛❞♦ ❡ ❝♦♥tr✐❜✉✐✉ ❜❛st❛♥t❡✱ ❛❥✉❞❛♥❞♦✲♠❡ ♠✉✐t♦✱ t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣s✐❝♦❧ó❣✐❝♦ q✉❛♥t♦ ✐♥t❡❧❡❝t✉❛❧ ❝♦♠ t♦❞♦ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ ▲❡tr❛s P♦rt✉❣✉ês✳

➚ ♠✐♥❤❛ ✜❧❤❛ ◆❛t❤á❧✐❛ ▼❡♥❞♦♥ç❛ ▼♦✉r❛ ❙❛♥t♦s✱ q✉❡ ♥❛s❝❡r❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❤á ✉♠ ❛♥♦✱ ♠✐♥❤❛ ✏❢♦rç❛✲♠♦tr✐③✑✱ r❛③ã♦ ♣❡❧❛ q✉❛❧ ♣❛ss❡✐ t✉❞♦ q✉❡ ♣❛ss❡✐ ♥❡ss❡ ❧♦♥❣♦ ♣❡rí♦❞♦ ❞♦ ♠❡str❛❞♦✳

❆♦s ♠❡✉s ♣❛✐s ▼❛♥♦❡❧ ▼❡ss✐❛s ❙❛♥t♦s ❡ ▼❛r✐❛ ❞❡ ▲♦✉r❞❡s ❙❛♥t♦s✱ q✉❡ ♠❡ ❞❡r❛♠✱ ❡ ♠✉✐t♦ ❜❡♠✱ ♦ ♠❡❧❤♦r ♣r❡s❡♥t❡ ❞❡ t♦❞♦s✿ ❛ ❡❞✉❝❛çã♦✱ ♣♦rt❛♥t♦✱ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ s❡r ❛❧❣✉é♠ ♥❛ ✈✐❞❛✳ ❚❡♥❤♦ ♠✉✐t♦ ♦r❣✉❧❤♦ ❞❡❧❡s✦

❆♦s ♠❡✉s ✐r♠ã♦s ❚❤✐❛❣♦ ▼❡ss✐❛s ❙❛♥t♦s ❡ ▲✉❝✐❛♥❛ ❙❛♥t♦s✱ q✉❡ s❡♠♣r❡ t♦r❝❡r❛♠ ❡ t♦r❝❡♠ ♣♦r ♠✐♠✳

➚ ♠✐♥❤❛ s♦❣r❛ ▼❛r✐❛ ❏♦sé ▼❡♥❞♦♥ç❛ ▼♦✉r❛✱ q✉❡ s❡♠♣r❡ ♠❡ ❛❥✉❞♦✉ ♥♦ q✉❡ ♣ô❞❡ ❡ s❡♠♣r❡ ❝♦♠❡♠♦r♦✉ ♠✐♥❤❛s ❝♦♥q✉✐st❛s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ❏♦s❡ ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦✱ ❛♦ q✉❛❧ s♦✉ ♠✉✐t♦ ❣r❛t♦✱ ♣♦✐s s❡♠ ❡❧❡ t♦❞♦ ❡st❡ tr❛❜❛❧❤♦ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧✳

❆♦s ♣r♦❢❡ss♦r❡s✱ ❈r✐s❧❡♥❡ ❙❛♥t♦s ❞❛ P❛✐①ã♦ ❡ ❆❧♠✐r ❘♦❣ér✐♦ ❙✐❧✈❛ ❙❛♥t♦s ♣♦r ❛❝❡✐✲ t❛r❡♠ ♦ ❝♦♥✈✐t❡ ❞❡ ♣❛rt✐❝✐♣❛r ❞❛ ♠✐♥❤❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❡ ♣❡❧❛s ✐♥t❡r❢❡rê♥❝✐❛s ♣❡rt✐♥❡♥t❡s ❛♦ ♠❡✉ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❆❧❧②s♦♥✱ ❆❧♠✐r ❘♦❣ér✐♦✱ ❉❛♥✐❧♦ ❉✐❛s✱ ❉❛♥✐❧♦ ❋❡❧✐③❛r❞♦✱ ❉é❜♦r❛ ▲♦♣❡s✱ ❊✈✐❧s♦♥✱ ❍✉♠❜❡rt♦ ❍❡♥r✐q✉❡✱ ❏♦s❡ ❆♥❞❡rs♦♥✱ ❑❛❧❛s❛s✱ ▲❡❛♥❞r♦ ❋❛✈❛❝❤♦✱ ▲✉❝❛s ❱❛❧❡r✐❛♥♦✱ ◆❛❧❞✐ss♦♥ ❡ ❘♦♠❡r♦ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ❡♠♣❡♥❤♦ ❡♠ ❝♦♠♣❛rt✐❧❤❛r❡♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s✳

❆ t♦❞♦s ♦s ❝♦❧❡❣❛s ❞❛ t✉r♠❛ ❞♦ P❘❖❋▼❆❚✲✷✵✶✸✱ ♣❡❧♦s ♠✉✐t♦s ♠♦♠❡♥t♦s q✉❡ ❡st✐✈❡♠♦s ❥✉♥t♦s ❡st✉❞❛♥❞♦ ❡ ♣❡❧❛s ❛♠✐③❛❞❡s ❝♦♥q✉✐st❛❞❛s✱ ❛✜♥❛❧✱ é ♦ q✉❡ ✈❛❧❡ ❛ ♣❡♥❛✦

➚ ❙❇▼ ✭❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✮✱ ♣❡❧❛ ❡①❝❡❧❡♥t❡ ✐♥✐❝✐❛t✐✈❛ ❞❡ ♣r♦♠♦✲ ✈❡r ✉♠ ♠❡str❛❞♦ ✈♦❧t❛❞♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛✳

➚ ❈❆P❊❙ ✭❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✮✱ ♣❡❧❛ ❝♦♥❝❡ssã♦ ❞❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s q✉❡ ❢♦✐ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞❡st❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧✐③❛♥t❡✳

❊ ❛ t♦❞♦s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❡ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

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❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛❜♦r❞❛r ♦s ❢✉♥❞❛♠❡♥t♦s ❞❡ ❧ó❣✐❝❛ ❡ ❛s ♥♦çõ❡s ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ♠❛♥❡✐r❛ ❡str❡✐t❛ ❡ ❡❧❡♠❡♥t❛r✱ ❝✉❧♠✐♥❛♥❞♦ ♥❛ ❝♦♥str✉✲ çã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❆♣r❡s❡♥t❛♠♦s✱ ❡ ♣r♦❣r❡❞✐♠♦s ♥❛ ♠❡❞✐❞❛ ❞♦ ♣♦ssí✈❡❧✱ ❞❡ ❢♦r♠❛ ♥❛t✉r❛❧ ❡✴♦✉ ✐♥t✉✐t✐✈❛✱ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣r♦♣♦s✐çõ❡s ❡ ♣r♦♣♦s✐çõ❡s ❛❜❡rt❛s✱ ❡ ♦ ✉s♦ ❞❡st❡s ♥❛s ❡s♣❡❝✐✜❝❛çõ❡s ❞❡ ❝♦♥❥✉♥t♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❛①✐♦♠❛ ❞❛ ❡s♣❡❝✐✜❝❛çã♦✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♦s ❝♦♥❡❝t✐✈♦s ❧ó❣✐❝♦s ❞❡ ♣r♦♣♦s✐çõ❡s ❛❜❡rt❛s ❡ ❛s ❡q✉✐✈❛❧ê♥✲ ❝✐❛s ❧ó❣✐❝❛s✱ r❡❧❛❝✐♦♥❛♥❞♦✲♦s ❛♦s ❝♦♥❥✉♥t♦s✳ ▼♦str❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❚❡♦r❡♠❛✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♠❛s ❢♦r♠❛s ❞❡ ❡s❝r✐t❛s ❡ ❞❡♠♦♥str❛çõ❡s ♥♦ â♠❜✐t♦ ❞♦s ❝♦♥❥✉♥t♦s✱ ❡ ✉t✐❧✐③❛✲ ♠♦s ♣r♦♣r✐❡❞❛❞❡s ❡ r❡❧❛çõ❡s ❞❡ ❝♦♥❥✉♥t♦s ♥❛s té❝♥✐❝❛s ❞❡ ❞❡♠♦♥str❛çã♦✳ ❊♥❝❡rr❛♠♦s ♥♦ss♦ ❡st✉❞♦ ❝♦♠ ❛ ❝♦♥str✉çã♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❡ ❛❧❣✉♠❛s ❞❛s s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❘❡❧❛çã♦ ❞❡ ❖r❞❡♠✳

P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ◆♦çõ❡s ❞❡ ▲ó❣✐❝❛✳ ❊q✉✐✈❛❧ê♥❝✐❛s ▲ó❣✐❝❛s✳ ◆♦çõ❡s ❞❡ ❈♦♥❥✉♥✲ t♦s✳ ❊s♣❡❝✐✜❝❛çõ❡s ❞❡ ❈♦♥❥✉♥t♦s✳ ❚❡♦r❡♠❛s✳ ❚é❝♥✐❝❛s ❞❡ Pr♦✈❛✳ ◆ú♠❡r♦s ◆❛t✉r❛✐s✳ ❆①✐♦♠❛s ❞❡ P❡❛♥♦✳

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❆❜str❛❝t

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ❤❛s ❛s ♠❛✐♥ ♦❜❥❡❝t✐✈❡ t♦ ❛♣♣r♦❛❝❤ t❤❡ ❢✉♥❞❛♠❡♥ts ♦❢ ❧♦❣✐❝ ❛♥❞ t❤❡ ♥♦t✐♦♥s ♦❢ s❡ts ✐♥ ❛ ♥❛rr♦✇ ❛♥❞ ❡❧❡♠❡♥t❛r② ✇❛②✱ ❝✉❧♠✐♥❛t✐♥❣ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ❲❡ ♣r❡s❡♥t ❛♥❞ ❛❞✈❛♥❝❡✱ ❛s ❢❛r ❛s ♣♦ss✐❜❧❡✱ ♥❛t✉r❛❧ ❛♥❞ ✐♥t✉✐✲ t✐✈❡❧②✱ t❤❡ ❝♦♥❝❡♣ts ♦❢ ♣r♦♣♦s✐t✐♦♥s ❛♥❞ ♦♣❡♥ ♣r♦♣♦s✐t✐♦♥s✱ ❛♥❞ t❤❡ ✉s❡ ♦❢ t❤❡s❡ ✐♥ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ s❡ts✱ ❛❝❝♦r❞✐♥❣ ✇✐t❤ t❤❡ ❛①✐♦♠ ♦❢ t❤❡ s♣❡❝✐✜❝❛t✐♦♥✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t t❤❡ ❧♦❣✐❝ ❝♦♥♥❡❝t✐✈❡s ♦❢ ♦♣❡♥ ♣r♦♣♦s✐t✐♦♥s ❛♥❞ ❧♦❣✐❝ ❡q✉✐✈❛❧❡♥❝❡s✱ r❡❧❛t✐♥❣ t❤❡♠ t♦ t❤❡ s❡ts✳ ❲❡ s❤♦✇❡❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ❚❤❡♦r❡♠✱ ❛s ✇❡❧❧ ❛s s♦♠❡ ❢♦r♠s ♦❢ ✇r✐t✐♥❣ ❛♥❞ ❞❡♠♦♥str❛t✐♦♥s ✐♥ t❤❡ s❝♦♣❡ ♦❢ t❤❡ s❡ts✱ ❛♥❞ ✇❡ ✉s❡❞ ♣r♦♣❡rt✐❡s ❛♥❞ r❡❧❛t✐♦♥s ♦❢ s❡ts ✐♥ t❤❡ ❞❡♠♦♥str❛t✐♦♥ t❡❝❤♥✐q✉❡s✳ ❖✉r st✉❞② ❡♥❞❡❞ ✇✐t❤ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❛♥❞ s♦♠❡ ♦❢ ✐ts ♣r♦♣❡rt✐❡s✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❘❡❧❛t✐♦♥ ❖r❞❡r✳

❑❡②✇♦r❞s✿ ▲♦❣✐❝ ◆♦t✐♦♥s✳ ❊q✉✐✈❛❧❡♥ts ▲♦❣✐❝✳ ◆♦t✐♦♥s ♦❢ ❙❡ts✳ ❙♣❡❝✐✜❝❛t✐♦♥s ♦❢ ❙❡ts✳ ❚❤❡♦r❡♠s✳ ❚❡❝❤♥✐q✉❡s ♦❢ Pr♦♦❢✳ ◆❛t✉r❛❧ ◆✉♠❜❡rs✳ ❆①✐♦♠s ♦❢ P❡❛♥♦✳

(8)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶✵

✶ ◆♦çõ❡s ❞❡ ▲ó❣✐❝❛ ❡ ❈♦♥❥✉♥t♦s ✶✷

✶✳✶ ❈♦♥❝❡✐t♦s ■♥tr♦❞✉tór✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✶ ◆♦çã♦ ❞❡ ❈♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✷ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷ Pr♦♣♦s✐çõ❡s ❡ Pr♦♣♦s✐çõ❡s ❆❜❡rt❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ❊s♣❡❝✐✜❝❛çõ❡s ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✹✳✶ ◗✉❛♥t✐✜❝❛❞♦r ❯♥✐✈❡rs❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✹✳✷ ◗✉❛♥t✐✜❝❛❞♦r ❊①✐st❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✹✳✸ ❈♦♠❜✐♥❛çã♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✺ ◆❡❣❛çã♦✱ ❈♦♥❡❝t✐✈♦s ▲ó❣✐❝♦s ❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✺✳✶ ◆❡❣❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✺✳✷ ◆❡❣❛çã♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✺✳✸ ❈♦♥❥✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✺✳✹ ■♥t❡rs❡çã♦ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✺✳✺ ❉✐s❥✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✶✳✺✳✻ ❯♥✐ã♦ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶✳✺✳✼ ❈♦♥❞✐❝✐♦♥❛❧ ❡ ■♠♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✳✺✳✽ ❙✉❜❝♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✶✳✺✳✾ ❇✐❝♦♥❞✐❝✐♦♥❛❧ ❡ ❇✐✲✐♠♣❧✐❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✶✳✻ ❊q✉✐✈❛❧ê♥❝✐❛s ▲ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✶✳✼ ❚❛✉t♦❧♦❣✐❛ ❡ ❈♦♥tr❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✷ ❚❡♦r❡♠❛s ❡ ❚é❝♥✐❝❛s ❞❡ Pr♦✈❛ ✺✻

✷✳✶ ❈♦♥❥❡❝t✉r❛s ❡ ❚❡♦r❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✳✶✳✶ ❚❡♦r❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✳✶✳✷ ❚❡♦r❡♠❛s ❞❛ ❋♦r♠❛ ✏s❡✱ ❡ s♦♠❡♥t❡ s❡✱✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✷✳✶✳✸ ●❡♥❡r❛❧✐③❛çã♦ ❞❡ ✉♠ ❚❡♦r❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✷✳✶✳✹ ▲❡♠❛s ❡ ❈♦r♦❧ár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✷✳✷ ▼ét♦❞♦s ❞❡ Pr♦✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✷✳✷✳✶ ▼ét♦❞♦ ❉✐r❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

(9)

✷✳✷✳✷ ▼ét♦❞♦ ❞❛ ❈♦♥tr❛♣♦s✐t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✷✳✸ ▼ét♦❞♦ ❞❛ ❈♦♥tr❛❞✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✷✳✷✳✹ Pr♦✈❛ ❞❡ ❆✜r♠❛çõ❡s ❞❛ ❋♦r♠❛ ✏s❡✱ ❡ s♦♠❡♥t❡ s❡✑ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✷✳✸ Pr♦✈❛ ♣❛r❛ ❆❧❣✉♥s ❚✐♣♦s ❞❡ ❆✜r♠❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✷✳✸✳✶ Pr♦✈❛ ❞❡ ■♠♣❧✐❝❛çõ❡s ❝♦♠ ❍✐♣ót❡s❡s ❈♦♥❥✉♥t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✷✳✸✳✷ Pr♦✈❛ ❞❡ ■♠♣❧✐❝❛çõ❡s ❝♦♠ ❚❡s❡s ❉✐s❥✉♥t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✷✳✸✳✸ Pr♦✈❛ ❞❡ ■♠♣❧✐❝❛çõ❡s ❝♦♠ ❍✐♣ót❡s❡s ❉✐s❥✉♥t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✷✳✸✳✹ Pr♦✈❛ ❞❡ ■♠♣❧✐❝❛çõ❡s ❝♦♠ ❚❡s❡s ❈♦♥❥✉♥t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✷✳✸✳✺ Pr♦✈❛ ❞❡ ❆✜r♠❛çõ❡s ❞❡ ❊①✐stê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✷✳✸✳✻ Pr♦✈❛ ❞❡ ❆✜r♠❛çõ❡s ❞❡ ❯♥✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✷✳✹ Pr♦✈❛ ❞❡ ❋❛❧s✐❞❛❞❡ ❞❡ ❆✜r♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✷✳✹✳✶ ❈♦♥tr❛✲❡①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✷✳✹✳✷ ❋❛❧s✐❞❛❞❡ ❞❡ ❆✜r♠❛çã♦ ❊①✐st❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✷✳✹✳✸ ❖ ❈♦♠♣❧❡♠❡♥t❛r ❞❡ ✉♠ ❈♦♥❥✉♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✸ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✽✷

✸✳✶ ❖s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✸✳✶✳✶ ❖s ❆①✐♦♠❛s ❞❡ P❡❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✸✳✷ ▼ét♦❞♦ ❞❡ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✸✳✷✳✶ ❙♦♠❛ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✸✳✷✳✷ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✸✳✷✳✸ ❘❡❧❛çã♦ ❞❡ ❖r❞❡♠ ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✸✳✷✳✹ ❙✉❜tr❛çã♦ ❡♠N ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼

✸✳✷✳✺ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✸✳✷✳✻ ❙♦♠❛tór✐♦s ❡ Pr♦❞✉tór✐♦s ❞❡ ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✸✳✸ ❉✐✈✐s♦r❡s ❡ ♠ú❧t✐♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✼ ✸✳✸✳✶ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾ ✸✳✸✳✷ ❙✐st❡♠❛ ❞❡ ◆✉♠❡r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✶✷✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✷✷

(10)

■♥tr♦❞✉çã♦

◆♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦ Pr♦❣r❛♠❛ ❈✉rr✐❝✉❧❛r ❞❡ ▼❛t❡♠át✐❝❛ ♥ã♦ ❝♦♥t❡♠♣❧❛ ♦s ❋✉♥✲ ❞❛♠❡♥t♦s ❞❡ ▲ó❣✐❝❛✱ ❛♣❡s❛r ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❛ ♠❡s♠❛ ❛♣❛r❡❝❡r❡♠ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s s❡♠ ✉♠❛ ♣ré✈✐❛ ❡①♣❧✐❝❛çã♦✳ P♦r ❡①❡♠♣❧♦✱ ✉♠ ❛❧✉♥♦ ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❞❡♣❛r❛✲s❡ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❞❡ ❧♦❣❛r✐t♠♦✿

✏❉❛❞♦s ♦s ♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s a ❡ b✱ ❝♦♠ a 6= 1✱ b = ac✱ ❡♥tã♦ ♦

❡①♣♦❡♥t❡ c ❝❤❛♠❛✲s❡ ❧♦❣❛r✐t♠♦ ❞❡ b ♥❛ ❜❛s❡ a✱ ♦✉ s❡❥❛✿ logab=c⇔ac =

b✱ ❝♦♠ a ❡ b ♣♦s✐t✐✈♦s ❡ a6= 1✑ ✭✈❡r ❬✷✱ ♣✳ ✷✻✶❪✮✳

❖ ❡st✉❞❛♥t❡ ♥ã♦ ♣❡r❝❡❜❡rá ❛ ♣r❡s❡♥ç❛ ❡①♣❧í❝✐t❛ ❞♦s ❝♦♥❡❝t✐✈♦s✿ ❝♦♥❞✐❝✐♦♥❛❧❀ ❡ s❡✱ ❡ s♦♠❡♥t❡ s❡❀ ♠✉✐t♦ ♠❡♥♦s ❡♥t❡♥❞❡rá ❛ ❡q✉✐✈❛❧ê♥❝✐❛ q✉❡ ❤á ❡♥tr❡ ♦ ❧♦❣❛r✐t♠♦ ❡ ❛ ❡①♣♦♥❡♥❝✐❛❧ ♣r❡s❡♥t❡s ♥❛ ❞❡✜♥✐çã♦✳ ❖✉tr♦s ♣r♦❜❧❡♠❛s ❡♥❝♦♥tr❛❞♦s ♥❛ ❛♣r❡♥❞✐③❛❣❡♠ ♠❛t❡♠át✐❝❛ sã♦ ♦ ✉s♦ ❞❡ sí♠❜♦❧♦s ❡ ♣❛❧❛✈r❛s q✉❡ ♥ã♦ sã♦ ❞♦ ❝♦t✐❞✐❛♥♦✳ ❊①♣r❡s✲ sõ❡s ❝♦♠♦✿ ❡stá ❝♦♥t✐❞♦✱ s✉❜❝♦♥❥✉♥t♦✱ ✐♥❝❧✉sã♦✱ ❡♥tr❡ ♦✉tr❛s✱ q✉❡ sã♦ ♣ró♣r✐❛s ❞❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ❝♦♥❥✉♥t♦s✱ ♥ã♦ sã♦ ✉t✐❧✐③❛❞❛s ♥♦ ❝♦t✐❞✐❛♥♦ ❞❛s ♣❡ss♦❛s✳ ❚❛❧ ❧✐♥❣✉❛❣❡♠✱ ✉♥✐✈❡rs❛❧♠❡♥t❡ ❛❞♦t❛❞❛ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞❛ ▼❛t❡♠át✐❝❛ ♥♦s ❞✐❛s ❛t✉❛✐s✱ ❣❛♥❤♦✉ ❡st❛ ♣♦s✐çã♦ ♣♦rq✉❡ ♣❡r♠✐t❡ ❞❛r ❛♦s ❝♦♥❝❡✐t♦s ❡ às ♣r♦♣♦s✐çõ❡s ❞❡st❛ ❝✐ê♥❝✐❛ ❛ ♣r❡❝✐sã♦ ❡ ❛ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ ❝♦♥st✐t✉❡♠ s✉❛ ❝❛r❛❝t❡ríst✐❝❛ ❜ás✐❝❛✳ ❖s ❝♦♥❥✉♥t♦s s✉❜st✐t✉❡♠ ❛s ✏♣r♦♣r✐❡❞❛❞❡s✑ ❡ ❛s ✏❝♦♥❞✐çõ❡s✑✳ ❆ss✐♠✱ ❡♠ ✈❡③ ❞❡ ❞✐③❡r♠♦s q✉❡ ✏♦ ❡❧❡♠❡♥t♦ x ❣♦③❛

❞❛ ♣r♦♣r✐❡❞❛❞❡ P✑ ♦✉ ✏♦ ❡❧❡♠❡♥t♦ y s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ C✑✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rx ∈A

❡ y ∈ B✱ ♦♥❞❡ A é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s q✉❡ ❣♦③❛♠ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P ❡ B é ♦

❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ C✱ ♣♦r ❡①❡♠♣❧♦ ✭✈❡r ❬✺❪✮✳

■♥❣r❡ss❛♥❞♦ ♥❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❛ ♠❛✐♦r✐❛ ❞❡ ♥♦ss♦s ❛❧✉♥♦s s❡ ❝❤♦❝❛ ❛♦ ❞❡♣❛r❛r✲s❡ ❝♦♠ ♦ ❢♦r♠❛❧✐s♠♦ ❡ ❛ ❛❜str❛çã♦✱ r❡q✉❡r✐❞❛s ♣♦r ❛❧❣✉♠❛s ❞❛s ♣r✐♠❡✐r❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ▼❛t❡♠át✐❝❛✳ ❖ ❝❤♦q✉❡ ❞❡❝♦rr❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❞❛ ❝❛rê♥❝✐❛ ♥❛ ❢♦r♠❛çã♦ ❞♦s ❛❧✉♥♦s✱ ❞♦s ♣r♦❢❡ss♦r❡s ❡ ❞❡ ✉♠ ❊♥s✐♥♦ ▼é❞✐♦ q✉❡✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ♥ã♦ ❧❤❡s ❢♦r♥❡❝❡ ✉♠ ♣r❡♣❛r♦ ❛❞❡q✉❛❞♦ ❡ ♥❡♠ ❧❤❡s tr❡✐♥❛ ♣❛r❛ ✉s❛r ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦✲❞❡❞✉t✐✈♦✱ ♦ q✉❛❧ ♣♦st❡r✐♦r♠❡♥t❡ s❡rá ❝♦❜r❛❞♦✳ ❈r❡♠♦s q✉❡ q✉❛♥t♦ ♠❛✐s ❝❡❞♦ ✉♠ ❡st✉❞❛♥t❡ t✐✈❡r ❛❝❡ss♦ ❛♦s ❝♦♥❝❡✐t♦s ❧ó❣✐❝♦s✱ ♠❛✐s ❢❛❝✐❧♠❡♥t❡ ❝♦♥str✉✐rá ♦ r❛❝✐♦❝í♥✐♦ ♠❛t❡♠át✐❝♦✳

❘❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s ❞❡✈❡♠ s❡r ❡①♣r❡ss♦s ❝♦♠ ❛ ❡①❛t✐❞ã♦ ♥❡❝❡ssár✐❛ q✉❡ ❡①✐✲ ❣❡♠✳ ◆❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥ã♦ ❤á ❧✉❣❛r ♣❛r❛ ❛♠❜✐❣✉✐❞❛❞❡s✱ ♣❛r❛ ✜❣✉r❛s ❞❡ ❧✐♥❣✉❛❣❡♠ ♦✉ ♣❛r❛ ♠❡tá❢♦r❛s✱ tã♦ ❝♦♠✉♥s ❡✱ ❛té ♠❡s♠♦✱ ❛♣r❡❝✐❛❞❛s ♥❛ ❧✐♥❣✉❛❣❡♠ ❝♦❧♦q✉✐❛❧ ♦✉ ❧✐t❡rár✐❛✳ ◆♦ ❞✐❛ ❛ ❞✐❛✱ q✉❛♥❞♦ ❛❧❣✉é♠ ❞✐③ ❛ ❢r❛s❡✿ ✏❈❤❡❣❛r❡✐ ❡♠ ✉♠ ♠✐♥✉t♦✦✑✱ s✐❣♥✐✜❝❛ q✉❡ ❝❤❡❣❛rá ❡♠ ♣♦✉❝♦ t❡♠♣♦✳ ❏á ♥❛ ▼❛t❡♠át✐❝❛✱ ✉♠ ♠✐♥✉t♦ r❡✲ ♣r❡s❡♥t❛ ✉♠ ♠✐♥✉t♦ ♠❡s♠♦✱ ✐st♦ é✱ s❡ss❡♥t❛ s❡❣✉♥❞♦s✳ ❙❡ ♣❡r❣✉♥t❛r♠♦s ❛♦s ❛❧✉♥♦s✿

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✏❱♦❝ês ✜③❡r❛♠ t♦❞♦ ♦ ❡①❡r❝í❝✐♦❄✑✱ ❣r❛♥❞❡ ♣❛rt❡ r❡s♣♦♥❞❡rá ✏❙✐♠✱ só ❢❛❧t❛ ✉♠❛ q✉❡s✲ tã♦✑✳ ❊ss❡ ❢❛t♦ ❡ ♠✉✐t♦s ♦✉tr♦s ❛tr❛♣❛❧❤❛♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❢♦r♠❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥❝❧✉sã♦ ✭♥♦t❡ q✉❡ ❛ ♣❛❧❛✈r❛ ✏t♦❞♦✑ ♥ã♦ ❡①❝❧✉✐ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦✮✳

❆ ♣❛rt✐r ❞♦s s✐♠♣❧❡s ❡①❡♠♣❧♦s ❝✐t❛❞♦s✱ ♣♦❞❡✲s❡ ♣❡r❝❡❜❡r q✉❡ ♥❛ ▼❛t❡♠át✐❝❛ ❛s ♠❡♥s❛❣❡♥s ❞❡✈❡♠ s❡r ❡①♣r❡ss❛s ❝♦♠ ❧✐♥❣✉❛❣❡♠ ❡ ❝✉✐❞❛❞♦s ❡s♣❡❝í✜❝♦s✱ ♠✉✐t❛s ✈❡③❡s✱ ❞✐❢❡r❡♥t❡s ❞❛q✉❡❧❡s ❛♦s q✉❛✐s ❝♦st✉♠❛♠♦s ✉t✐❧✐③❛r ❡ ✐♥t❡r♣r❡t❛r ♥❛ ❧✐♥❣✉❛❣❡♠ ❝♦❧♦✲ q✉✐❛❧✳ ◆ã♦ é à t♦❛ q✉❡ ❛ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❝✐ê♥❝✐❛ ❡①❛t❛ ❡ ❢✉♥❝✐♦♥❛ ❞❡ t❛❧ ❢♦r♠❛✳

❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❛♠♣❧✐❛r ❛ ❧✐t❡r❛t✉r❛ q✉❡ ✈❡rs❛ s♦❜r❡ ♦s ❢✉♥✲ ❞❛♠❡♥t♦s ❜ás✐❝♦s ❞❛ ❧ó❣✐❝❛ ❡ ❜✉s❝❛ ❛♣r❡s❡♥t❛r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❝♦♠♦ ❡❧❡s sã♦ r❡❛❧♠❡♥t❡ tr❛t❛❞♦s ♣♦r q✉❡♠ ❡st✉❞❛ ▼❛t❡♠át✐❝❛✳ P❛r❛ s✉❛ ❧❡✐t✉r❛✱ ❜❛s✐❝❛♠❡♥t❡ ♥ã♦ é ♥❡❝❡ssár✐♦ ♣ré✲r❡q✉✐s✐t♦ ♠❛t❡♠át✐❝♦✳ ❇✉s❝❛♠♦s ❡❧❛❜♦r❛r ✉♠ t❡①t♦ ♦ ♠❛✐s ❛✉t♦❝♦♥t✐❞♦ ♣♦ssí✈❡❧✳

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡✲ s❡♥t❛♠♦s ❝♦♥❝❡✐t♦s ✐♥tr♦❞✉tór✐♦s q✉❡ ❛❥✉❞❛♠ ❛ ❡♥t❡♥❞❡r ❝♦♠♦ sã♦ ♦r❣❛♥✐③❛❞❛s ❛s t❡♦r✐❛s ♠❛t❡♠át✐❝❛s✳ ❏✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ♥♦çõ❡s ❞❡ ❝♦♥❥✉♥t♦s✱ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❧ó❣✐❝♦s ✈ã♦ s❡♥❞♦ ❣r❛❞❛t✐✈❛♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦s ❡ ♦ ❧❡✐t♦r é ❧❡✈❛❞♦ ❛ ❝♦♠♣r❡❡♥❞❡r ❛ ❧ó❣✐❝❛ ❛♦ ♠❡s♠♦ t❡♠♣♦ q✉❡ é ❧❡✈❛❞♦ ❛ ❝♦♥❤❡❝❡r ♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❝♦♥❥✉♥t♦s✱ ❡✈✐t❛♥❞♦ ❛ss✐♠ ✉♠❛ ❞❡s❝♦♥❡①ã♦ ❞♦s ❛ss✉♥t♦s✳ ❊ss❡♥❝✐❛❧♠❡♥t❡✱ ❜✉s❝❛♠♦s ✏❝♦♥❤❡❝❡r ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝♦♥❥✉♥t♦s ♣❛r❛ ❡st✉❞❛r ❛ ❧ó❣✐❝❛ ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❝♦♥❤❡❝❡r ❛ ❧ó❣✐❝❛ ♣❛r❛ ❡st✉❞❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝♦♥❥✉♥t♦s✑✳ ❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ tr❛t❛ ❞♦s ♠ét♦❞♦s ❞❡ ♣r♦✈❛ ❝❧áss✐❝♦s q✉❡ ❛ ❧ó❣✐❝❛ ♥❛t✉r❛❧♠❡♥t❡ ♣r♦♣♦r❝✐♦♥❛✱ ❛ s❛❜❡r✱ ❡st✉❞❛♠♦s ♦s ♠ét♦❞♦s ❞✐r❡t♦✱ ❞❛ ❝♦♥tr❛♣♦s✐t✐✈❛✱ ❞❛ ❝♦♥tr❛❞✐çã♦✱ ❡t❝✳ ❊ss❡s ♠ét♦❞♦s ❝♦♠♣õ❡♠ ✉♠ ❢❡rr❛♠❡♥✲ t❛❧ ♥❡❝❡ssár✐♦ ❛ ❝♦♥str✉çã♦ ❡ ❡st✉❞♦ ❞❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❖ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ é ❞❡❞✐❝❛❞♦ à ❝♦♥str✉çã♦ ❞♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ❡ ❛♦ ❡st✉❞♦ ❞❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s✳

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❈❛♣ít✉❧♦ ✶

◆♦çõ❡s ❞❡ ▲ó❣✐❝❛ ❡ ❈♦♥❥✉♥t♦s

P♦❞❡✲s❡ ❛✜r♠❛r q✉❡ ♣r❛t✐❝❛♠❡♥t❡ t♦❞❛ ❛ ▼❛t❡♠át✐❝❛ ❛t✉❛❧ é ❢♦r♠✉❧❛❞❛ ♥❛ ❧✐♥✲ ❣✉❛❣❡♠ ❞❡ ❝♦♥❥✉♥t♦s✳ ◆ã♦ ❤á r❡❝❡✐♦ ❡♠ ❞✐③❡r q✉❡ ❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦ é ✉♠❛ ❞❛s ♠❛✐s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♠❛t❡♠át✐❝❛✳ ❆ ♣❛rt✐r ❞❡❧❛✱ ♣r❛t✐❝❛♠❡♥t❡ t♦❞♦s ♦s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ♣♦❞❡♠ s❡r ❡①♣r❡ss♦s✳ ❍❡✉ríst✐❝❛ ❡ ✐♥t✉✐t✐✈❛♠❡♥t❡ ❢❛❧❛♥❞♦✱ ❡❧❛ é t❛♠❜é♠ ✉♠❛ ❞❛ ♠❛✐s s✐♠♣❧❡s ✐❞é✐❛s ❞❛ ♠❛t❡♠át✐❝❛✳

✶✳✶ ❈♦♥❝❡✐t♦s ■♥tr♦❞✉tór✐♦s

◆❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ t❡♦r✐❛ ♠❛t❡♠át✐❝❛✱ t♦❞❛ ❞❡✜♥✐çã♦ ❢❛③ ✉s♦ ❞❡ t❡r♠♦s ❡s♣❡❝í✜❝♦s✱ ♦s q✉❛✐s ❢♦r❛♠ ❞❡✜♥✐❞♦s ✉s❛♥❞♦ ♦✉tr♦s t❡r♠♦s✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❊st❡ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ t❡r♠✐♥❛ ♥✉♠❛ ♣❛❧❛✈r❛✱ ♦✉ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s✱ q✉❡ ♥ã♦ sã♦ ❞❡✜♥✐❞❛s✱ ✐st♦ é✱ q✉❡ sã♦ t♦♠❛❞❛s ❝♦♠♦ r❡♣r❡s❡♥t❛t✐✈❛s ❞❡ ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s✳

P❛r❛ ♣♦❞❡r ❡♠♣r❡❣❛r ♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ❛❞❡q✉❛❞❛♠❡♥t❡ é ♥❡❝❡ssár✐♦ ❞✐s♣♦r ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r✐♥❝í♣✐♦s ♦✉ r❡❣r❛s q✉❡ ❞✐s❝✐♣❧✐♥❡♠ s✉❛ ✉t✐❧✐③❛çã♦ ❡ ❡st❛❜❡❧❡ç❛♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❚❛✐s ♣r✐♥❝í♣✐♦s sã♦ ❝❤❛♠❛❞♦s ❛①✐♦♠❛s ♦✉ ♣♦st✉❧❛❞♦s✳ ❆ss✐♠✱ ❝♦♠♦ ♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✈♦s sã♦ ♦❜❥❡t♦s q✉❡ ♥ã♦ s❡ ❞❡✜♥❡♠✱ ♦s ❛①✐♦♠❛s sã♦ ❛✜r♠❛çõ❡s q✉❡ ♥ã♦ s❡ ❞❡♠♦♥str❛♠✳

❆♥t❡s ❞❡ ❝♦♠❡ç❛r♠♦s ♥♦ss♦ ❡st✉❞♦✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ♥❛ ♠❛t❡♠át✐❝❛✱ t❛✐s ❝♦♠♦✿ ❝♦♥❝❡✐t♦ ♣r✐♠✐t✐✈♦✱ ❛①✐♦♠❛ ❡ ❞❡✜✲ ♥✐çã♦✳

• ❯♠ ❈♦♥❝❡✐t♦ Pr✐♠✐t✐✈♦ é ✉♠ ❝♦♥❝❡✐t♦ ❛❞♦t❛❞♦ ❡ ♥ã♦✲❞❡✜♥✐❞♦✳ ❊①❡♠♣❧♦s✿ P♦♥t♦s✱ ❘❡t❛s ❡ P❧❛♥♦s✳

• ❯♠ ❆①✐♦♠❛ é ✉♠❛ ❛✜r♠❛çã♦ s✐♠♣❧❡s s♦❜r❡ ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s q✉❡ é ❛❝❡✐t❛ ❝♦♠♦ ✈❡r❞❛❞❡✐r❛✳

❊①❡♠♣❧♦✿ ❉❛❞♦s ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s✱ ♣♦❞❡✲s❡ tr❛ç❛r ✉♠❛ ú♥✐❝❛ r❡t❛ ♣❛ss❛♥❞♦ ♣♦r ❡❧❡s ✭Pr✐♠❡✐r♦ ❆①✐♦♠❛ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✮✳

• ❯♠❛ ❉❡✜♥✐çã♦ ▼❛t❡♠át✐❝❛ é ✉♠❛ ❝♦♥✈❡♥çã♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ✉s❛r ✉♠ ✏♥♦♠❡✑ ♣❛r❛ ❞❡s✐❣♥❛r ✉♠ ♦❜❥❡t♦ ♠❡❞✐❛♥t❡ ❞❡t❡r♠✐♥❛❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦ ❝❛r❛❝t❡r✐✲

(13)

③❡♠ ❡ ♦ ✐❞❡♥t✐✜q✉❡♠ ♣❧❡♥❛♠❡♥t❡✳

❊①❡♠♣❧♦✿ ❯♠ tr✐â♥❣✉❧♦ é ■sós❝❡❧❡s q✉❛♥❞♦ ♣♦ss✉✐ ❞♦✐s ❧❛❞♦s ❞❡ ♠❡s♠♦ ❝♦♠✲ ♣r✐♠❡♥t♦ ✭❉❡✜♥✐çã♦ ❞❡ ❚r✐â♥❣✉❧♦ ■só❝❡❧❡s✮✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✳ Pr❡❝✐s❛✲s❡ r❡ss❛❧t❛r q✉❡ ♦s ❚❡r♠♦s Pr✐♠✐t✐✈♦s ❡ ♦s ❆①✐♦♠❛s ♥ã♦ s❡ ✜r♠❛♠ ♣♦r ♦♣✐♥✐õ❡s ✐s♦❧❛❞❛s ♣❡ss♦❛✐s✳ ❊❧❡s sã♦ ❢r✉t♦s ❞❛ ❡①♣❡r✐ê♥❝✐❛✱ ❞❛ ♦❜s❡r✈❛✲ çã♦ ❡ ❞❡ ✉♠ ❝❡rt♦ ✏❝♦♥s❡♥s♦ ❝♦❧❡t✐✈♦✑✳

✶✳✶✳✶ ◆♦çã♦ ❞❡ ❈♦♥❥✉♥t♦

❖ ❝♦♥❝❡✐t♦ ♠❛t❡♠át✐❝♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r ✉s❛❞♦ ❝♦♠♦ ❢✉♥❞❛♠❡♥t♦ ♣❛r❛ t♦❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦✳ ❆ss✐♠ ❝♦♠♦ ♥ã♦ s♦♠♦s ❝❛♣❛③❡s ❞❡ ❞❡✜♥✐r ♣r❡❝✐s❛✲ ♠❡♥t❡ ♥❛ ❣❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣♦♥t♦✱ r❡t❛ ❡ ♣❧❛♥♦✱ ♣♦r ❡①❡♠♣❧♦✱ t❛♠❜é♠ ♥ã♦ s♦♠♦s ❝❛♣❛③❡s ❞❡ ❞❡✜♥✐r ♣r❡❝✐s❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❈♦♥❥✉♥t♦✳ ▼❡s♠♦ ♥ã♦ ❞❡✜♥✐❞♦ r✐❣♦r♦s❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ✐♥t✉✐t✐✈❛ ❞❛❞❛ ♣♦r ●❡♦r❣ ❈❛♥t♦r ✭✶✽✹✺✲✶✾✶✽✮✿

✂ ✁

❯♠ ❈♦♥❥✉♥t♦ é q✉❛❧q✉❡r ❝♦❧❡çã♦✱ ❞❡♥tr♦ ❞❡ ✉♠ t♦❞♦ ❞❡ ♦❜❥❡t♦s ❞❡✜♥✐❞♦s ❡ ❞✐st✐♥❣✉í✈❡✐s✱ ❝❤❛♠❛❞♦s ❊❧❡♠❡♥t♦s✱ ❞❡ ♥♦ss❛ ✐♥t✉✐çã♦ ♦✉ ♣❡♥s❛♠❡♥t♦✳

❊①❡♠♣❧♦ ✶✳✶✳✷✳ ❊①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s✿ ❛✮ ❖ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s✿ ❛✱ ❡✱ ✐✱ ♦ ❡ ✉❀

❜✮ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝❛❞❡✐r❛s ❞❡ ✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛❀ ❝✮ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡st✉❞❛♥t❡s ♣❛✉❧✐st❛s ❞❛ ❯❋❙❀

❈♦♠✉♠❡♥t❡ r❡♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❥✉♥t♦s ❛tr❛✈és ❞❡ ❧❡tr❛s ♠❛✐ús❝✉❧❛s✱

A, B, C, . . .

❡ ♦s ❡❧❡♠t❡♥t♦s ♣♦r ❧❡tr❛s ♠✐♥ús❝✉❧❛s

a, b, c, . . . .

❊♠ ❝❛s♦s ❡s♣❡❝í✜❝♦s ❝♦♠♦ ♥♦ ✐t❡♠ a)❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

V ={a, e, i, o, u},

♦♥❞❡ V r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s✳

❊①❡♠♣❧♦ ✶✳✶✳✸✳ ◆❛t✉r❛❧♠❡♥t❡ ♣♦❞❡♠♦s t❡r ❝♦♥❥✉♥t♦s q✉❡ ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s q✉❡ t❛♠❜é♠ sã♦ ❝♦♥❥✉♥t♦s✳ P♦r ❡①❡♠♣❧♦✱

D={a, i, o,{e, o, u},{o,{a, i}}, e}.

❖❜s❡r✈❡ q✉❡✱ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ié ❡❧❡♠❡♥t♦ ❞❡ D✱ t❡♠♦s {e, o, u} ✉♠ ❡❧❡♠❡♥t♦ ❞❡

D✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ u♥ã♦ é ❡❧❡♠❡♥t♦ ❞❡ D✱ ❛ss✐♠ ❝♦♠♦{a, i}t❛♠❜é♠ ♥ã♦ é ❡❧❡♠❡♥t♦ ❞❡ D✳

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❘❡❧❛çã♦ ❞❡ P❡rt✐♥ê♥❝✐❛

❯♠ ❞♦s ❝♦♥❝❡✐t♦s ♣r✐♠✐t✐✈♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♥♦ ❡st✉❞♦ ❞♦s ❝♦♥❥✉♥t♦s é ♦ ❝♦♥❝❡✐t♦ ❞❡ P❡rt✐♥ê♥❝✐❛✳ ◗✉❛♥❞♦ ✉♠ ✏♦❜❥❡t♦✑ a é ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ A✱ ❞✐③❡♠♦s q✉❡ a

♣❡♥t❡♥❝❡ ❛ A ❡ ❞❡♥♦t❛♠♦s t❛❧ ❝♦♥❝❡✐t♦ ♣♦r a ∈A.

❈❛s♦ ❝♦♥trár✐♦✱ ❞❡♥♦t❛✲s❡

a6∈A

❡ ❞✐③✲s❡ q✉❡ a ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ A✳

❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❊①❡♠♣❧♦s ❞❡ ♣❡rt✐♥ê♥❝✐❛s✿

❛✮ ❈♦♠♦ V ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s✱ V ❂ ④❛✱ ❡✱ ✐✱ ♦✱ ✉⑥✱ t❡♠♦s✿

❛ ∈V✱ ❡ ∈V✱ ✐ ∈V✱ ♦ ∈V✱ ✉ ∈V❀

❥á

b 6∈V ❡ t6∈V✳

❜✮ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❛❧✉♥♦s ❞❛ ❯❋❙ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r

A={a1, a2, . . . , an}

♦♥❞❡ n é ♦ ♥ú♠❡r♦ ❞❡ ❛❧✉♥♦s ❞❛ ❯❋❙✳ ❊♥tã♦✱ t❡♠♦s✿ a1 ∈A✱ a2 ∈A✱ ✳ ✳ ✳ ✱ an−1 ∈A ❡ an∈A✳

❙❡ b r❡♣r❡s❡♥t❛ ♦ ❡①✲♣r❡s✐❞❡♥t❡ ●❡tú❧✐♦ ❱❛r❣❛s✱ ❡♥tã♦ b6∈A.

❝✮ ❉♦ ❡①❡♠♣❧♦ ✶✳✶✳✸ ♦♥❞❡

D={a, i, o,{e, o, u},{o,{a, i}}, e} t❡♠♦s✿

❛ ∈D✱ ✐ ∈D✱ {e, o, u} ∈D✱ {o,{a, i}} ∈D✱ ❡ ∈D❀

❥á

u6∈D ❡ {a, i} 6∈D✳

◆♦t❛çã♦ ✶✳✶✳✺✳ P♦❞❡♠♦s ❞❡♥♦t❛r

a1 ∈A, a2 ∈A, . . . , an−1 ∈A ❡ an∈A,

s✐♠♣❧❡s♠❡♥t❡ ♣♦r✿

a1, a2, . . . , an ∈A.

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■❣✉❛❧❞❛❞❡ ❞❡ ❈♦♥❥✉♥t♦s

❉✐③❡♠♦s q✉❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B sã♦ ✐❣✉❛✐s✶ q✉❛♥❞♦ ♣♦ss✉❡♠ ♦s ♠❡s♠♦s ❡❧❡✲

♠❡♥t♦s ❡ ❞❡♥♦t❛♠♦s

A=B. ✭✶✳✶✮

❊①❡♠♣❧♦ ✶✳✶✳✻✳ ❙❡♥❞♦ A={a, b, c, d} ❡ B ={d, b, b, b, c, c, a}✱ t❡♠♦s✿ A={a, b, c, d}={d, b, b, b, c, c, a}=B.

❖❜s❡r✈❛çã♦ ✶✳✶✳✼✳ ◆❛ ♠❛t❡♠át✐❝❛✱ ✉♠❛ ❝♦✐s❛ só é ✐❣✉❛❧ ❛ s✐ ♣ró♣r✐❛✳ ◗✉❛♥❞♦ s❡ ❡s❝r❡✈❡ a=b✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ a ❡ b sã♦ sí♠❜♦❧♦s ❞✐❢❡r❡♥t❡s✱ ✉s❛❞♦s ♣❛r❛ ❞❡s✐❣♥❛r ♦

♠❡s♠♦ ♦❜❥❡t♦✳

❆ ♦r❞❡♠ ❡♠ q✉❡ ❛♣❛r❡❝❡♠ ♦s ❡❧❡♠❡♥t♦s ♥♦ ❝♦♥❥✉♥t♦ ♥ã♦ t❡♠ ✐♠♣♦rtâ♥❝✐❛✳ ❆ss✐♠✱ ✉s❛♥❞♦ ♦ ❊①❡♠♣❧♦ ✶✳✶✳✻✱ t❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ A = {a, b, c, d} é ♦ ♠❡s♠♦ q✉❡ B =

{d, c, c, b, b, b, a}✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ sã♦ ✏❞✐st✐♥t♦s✑✱

♦ ❝♦♥❥✉♥t♦ {d, b, b, b, c, c, a} ♥ã♦ é ✉♠❛ ♥♦t❛çã♦ ❛♣r♦♣r✐❛❞❛✱ ❞❡✈❡♥❞♦ ❛ ♠❡s♠❛ s❡r s✉❜st✐t✉í❞❛ ♣♦r {a, b, c, d}✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✽✳ ◆❡st❡ ♣♦♥t♦ é út✐❧ ❢❛③❡r ❛s s❡❣✉✐♥t❡s ♦❜s❡r✈❛çõ❡s✿ ❛✮ ◗✉❛♥❞♦ ❡s❝r❡✈❡♠♦s

{a1, a2,· · ·, an},

❡st❛♠♦s ❛❞♠✐t✐♥❞♦ q✉❡ ♦s ❡❧❡♠❡♥t♦s a1, a2,· · · , an sã♦ t♦❞♦s ❞✐st✐♥t♦s✳ ❈♦♠

✐ss♦✱ ❡✈✐t❛♠♦s ❡s❝r❡✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ {a, a, i, o}❀ ❡✱ ♣♦rt❛♥t♦✱ ♥❡ss❡ ❝❛s♦ ❞❡✈❡✲ ♠♦s ❡s❝r❡✈❡r {a, i, o}✱ ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ♦❜s❡r✈❛❞♦✳

❜✮ ❙❡ a é ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦✱ ♥♦t❡ q✉❡ a ❡ {a} sã♦ ✏♦❜❥❡t♦s✑ ❞✐st✐♥t♦s ♣♦✐s✱ ❡♥q✉❛♥t♦ aé ✉♠ ❡❧❡♠❡♥t♦✱ {a} é ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ♣♦ss✉✐ a ❝♦♠♦ ❡❧❡♠❡♥t♦✳ ❉❛

♠❡s♠❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ a✱ {a} ❡ {{a}} sã♦ t♦❞♦s ✏♦❜❥❡t♦s✑ ❞✐st✐♥t♦s✱ ♣♦✐s {a} ❛♣❡s❛r ❞❡ s❡r ✉♠ ❝♦♥❥✉♥t♦✱ ❡❧❡ é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ {{a}}✳

◆♦t❛çã♦ ✶✳✶✳✾✳ ❙❡ ♦s ❝♦♥❥✉♥t♦sA❡B ♥ã♦ sã♦ ✐❣✉❛✐s✱ ❞✐③❡♠♦s q✉❡ ❡❧❡s sã♦ ❞✐❢❡r❡♥t❡s

❡ ❞❡♥♦t❛♠♦s ♣♦r

A6=B.

❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ ❈♦♥s✐❞❡r❡ ❛s ✈♦❣❛✐s ❛✱ ❡✱ ✐✱ ♦ ❡ ✉✳ ◆♦t❡ q✉❡ ♦s ❝♦♥❥✉♥t♦s

A={❡✱ ✐} e B ={❡✱ ✐✱ ✉}

♥ã♦ sã♦ ✐❣✉❛✐s✳ ❆ss✐♠✱ A6=B✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✶✳ ❖❜s❡r✈❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ A s❡r ❞✐❢❡r❡♥t❡ ❞♦ ❝♦♥❥✉♥t♦B s✐❣♥✐✜❝❛

q✉❡ ❤á ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❡♠ A q✉❡ é ❞✐❢❡r❡♥t❡ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ B✱ ♦✉ q✉❡

❡①✐st❡ ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❡♠ B q✉❡ é ❞✐❢❡r❡♥t❡ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ A✳ ◆♦t❡ q✉❡

♥♦ ❝❛s♦ ❞♦ ❊①❡♠♣❧♦ ✶✳✶✳✶✵✱ ❛ ✈♦❣❛❧ ✉ ❞♦ ❝♦♥❥✉♥t♦B é ❞✐❢❡r❡♥t❡ ❞❡ t♦❞❛s ❛s ✈♦❣❛✐s ❞♦

❝♦♥❥✉♥t♦ A✳

◆❛ t❡♦r✐❛ ❛①✐♦♠át✐❝❛ ❞❡ ❝♦♥❥✉♥t♦s✱ ❡ss❡ ❝♦♥❝❡✐t♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❆①✐♦♠❛ ❞❛ ❊①t❡♥sã♦

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✶✳✶✳✷ Pr♦❞✉t♦ ❈❛rt❡s✐❛♥♦ ❞❡ ❈♦♥❥✉♥t♦s

❈♦♠ ✉♠ ♦✉ ♠❛✐s ❡❧❡♠❡♥t♦s ♥♦ ❝♦♥❥✉♥t♦✱ ❡①✐st❡♠ ✈ár✐❛s ❢♦r♠❛s ❞❡ ❝r✐❛r ♥♦✈♦s ✏♦❜❥❡t♦s✑ ✭❡❧❡♠❡♥t♦s✮ q✉❡ ❞ã♦ ♦r✐❣❡♠ ❛ ♦✉tr♦s ❝♦♥❥✉♥t♦s✳ ❯♠❛ ❞❡ss❛s ❢♦r♠❛s é ✉s❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ P❛r ❖r❞❡♥❛❞♦✱ q✉❡ ♣♦❞❡♠♦s ✐♥tr♦❞✉③✐r ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦ ❤❡✉ríst✐❝♦✿

❞❛❞♦s ❞♦✐s ❡❧❡♠❡♥t♦s q✉❛✐sq✉❡r x ❡ y ❡♠ ❝♦♥❥✉♥t♦s✱ ♣♦❞❡♠♦s ❢♦r♠❛r ✉♠

♥♦✈♦ ❡❧❡♠❡♥t♦ ✭✏♦❜❥❡t♦✑✮ ❝❤❛♠❛❞♦ P❛r ❖r❞❡♥❛❞♦ ✏xy✑✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r

(x, y)✳

❆ ❢♦r♠❛ ♣r❡❝✐s❛ ❞❡ ❞❡✜♥✐r ✉♠ ♣❛r ♦r❞❡♥❛❞♦✷ ❢♦❣❡ ❞♦ ❝♦♥t❡①t♦ ❞❡ ✐♥t❡r❡ss❡ ♥❡st❡

♠♦♠❡♥t♦✳ ➱ ♥❡❝❡ssár✐♦ ♥ã♦ ❝♦♥❢✉♥❞✐r ♦ ♣❛r ♦r❞❡♥❛❞♦(a, b)❝♦♠ ♦ ❝♦♥❥✉♥t♦{a, b}✳ ❖ ❛❞❥❡t✐✈♦ ✏♦r❞❡♥❛❞♦✑ ❡♥❢❛t✐③❛ ❛q✉✐ q✉❡ ❛ ♦r❞❡♠ ♥❛ q✉❛❧ ♦s ❡❧❡♠❡♥t♦s a ❡ b ❛♣❛r❡❝❡♠

❡♥tr❡ ♦s ♣❛rê♥t❡s❡s é ❡ss❡♥❝✐❛❧✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✷✳ ◆❡st❡ ♠♦♠❡♥t♦ é s✉✜❝✐❡♥t❡ s❛❜❡r ❛♣❡♥❛s q✉❡ ❞♦✐s ♣❛r❡s ♦r❞❡✲ ♥❛❞♦s (x, y) ❡ (a, b) sã♦ ✐❣✉❛✐s q✉❛♥❞♦✱ ❡ s♦♠❡♥t❡ q✉❛♥❞♦✱ t❡♠♦s x=a ❡ y=b.

❊①❡♠♣❧♦ ✶✳✶✳✶✸✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s✱ ♣♦r ❡①❡♠♣❧♦✱ t❡♠♦s q✉❡(a, a)

❡ (i, a) sã♦ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞✐st✐♥t♦s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (a, o) ❡ (o, a) sã♦ t❛♠❜é♠ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞✐st✐♥t♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✹✳ ❙❡❥❛♠ A ❡ B ❞♦✐s ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s

♣❛r❡s ♦r❞❡♥❛❞♦s (x1, x2)✱ ❝♦♠ x1 ∈ A ❡ x2 ∈B✱ é ❝❤❛♠❛❞♦ ♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡

A ❡ B✱ ❡ é ❞❡♥♦t❛❞♦ ♣♦r A×B✳

❊①❡♠♣❧♦ ✶✳✶✳✶✺✳ ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s A={a, b, c} ❡ B ={x, y}✱ ♦s ♣r♦❞✉t♦s ❝❛rt❡✲ s✐❛♥♦s A×B ❡ B×A sã♦✿

A×B ={(a, x),(a, y),(b, x),(b, y),(c, x),(c, y)} ❡

B×A={(x, a),(x, b),(x, c),(y, a),(y, b),(y, c)}.

❖❜s❡r✈❡ q✉❡✱ ❞❡✈✐❞♦ à ❖❜s❡r✈❛çã♦ ✶✳✶✳✶✷✱ ♦s ❝♦♥❥✉♥t♦sA×B ❡B×Asã♦ ❝♦♥❥✉♥t♦s

❞✐st✐♥t♦s✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✻✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ♣❛r ♦r❞❡♥❛❞♦ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛ tr✐♣❧❛ ♦r❞❡♥❛❞❛ (x1, x2, x3), q✉á❞r✉♣❧❛ ♦r❞❡♥❛❞❛ (x1, x2, x3, x4), ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ❡♥✉♣❧❛

♦r❞❡♥❛❞❛

(x1, x2,· · · , xn).

❈♦♠ ✐ss♦✱ t❡♠♦s ♣r♦❞✉t♦s ❝❛t❡s✐❛♥♦s trí♣❧♦ A1×A2×A3✱ q✉á❞r✉♣❧♦ A1×A2×A3×A4

❡ ❡♥✉♣❧♦

A1×A2× · · · ×An.

❆ ❢♦r♠❛ ♣r❡❝✐s❛ ❞❡ ❞❡✜♥✐r ✉♠ ♣❛r ♦r❞❡♥❛❞♦ é✿ (x, y) ={x,{x, y}}✭✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✸✱ ❙❡çã♦

✻❪✮✳ ❈♦♠ ✐ss♦✱ é ♣♦ssí✈❡❧ ❣❛r❛♥t✐r ❛ ♣r♦♣r✐❡❞❡❞❡ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ♣❛r ♦r❞❡♥❛❞♦✱ ❛ s❛❜❡r✱ ❛ ❖❜s❡r✈❛çã♦ ✶✳✶✳✶✷✳

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✶✳✷ Pr♦♣♦s✐çõ❡s ❡ Pr♦♣♦s✐çõ❡s ❆❜❡rt❛s

❉❡✜♥✐çã♦ ✶✳✷✳✶ ✭Pr♦♣♦s✐çã♦✮✳ ❈❤❛♠❛✲s❡ Pr♦♣♦s✐çã♦ t♦❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s ♦✉ sí♠❜♦❧♦s q✉❡ ❝♦♥st✐t✉✐ ✉♠❛ ❛✜r♠❛çã♦ q✉❡ ❞❡✈❡ s❡r ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✱ ❞❡ ♠❛♥❡✐r❛ ❡①❝❧✉s✐✈❛✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❊①❡♠♣❧♦s ❞❡ ♣r♦♣♦s✐çõ❡s✿

P1. ❖ ❡①✲♣r❡s✐❞❡♥t❡ ●❡tú❧✐♦ ❱❛r❣❛s é ❛❧✉♥♦ ❞❛ ❯❋❙❀

♣r♦♣♦s✐çã♦ ❢❛❧s❛ ♣♦✐s✱ ❝♦♠♦ ✈✐st♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✳✹ ✐t❡♠ ❜✮✱ s❡♥❞♦A ♦ ❝♦♥❥✉♥t♦

❞❡ t♦❞♦s ♦s ❛❧✉♥♦s ❞❛ ❯❋❙ ❡ ✏b✑ ♦ ❡①✲♣r❡s✐❞❡♥t❡ ●❡tú❧✐♦ ❱❛r❣❛s✱ b6∈A✳ P2. ❆ ❖❝❡❛♥✐❛ é ♦ ♠❛✐♦r ❝♦♥t✐♥❡♥t❡ q✉❡ ❡①✐st❡❀

♣r♦♣♦s✐çã♦ ❢❛❧s❛✱ ♣♦✐s ♦ ❝♦♥t✐♥❡♥t❡ ✏❖❝❡❛♥✐❛✑ é ♠❡♥♦r t❡rr✐t♦r✐❛❧♠❡♥t❡ ❞♦ q✉❡ ♦ ❝♦♥t✐♥❡♥t❡ ❛s✐át✐❝♦✳

P3. ❆ ❧❡tr❛ ✏a✑ é ✉♠❛ ✈♦❣❛❧❀

♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛✱ ♣♦✐s a ♣❡rt❡♥❝❡ ❛ V✱ q✉❡ é ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s ❝♦♠♦

✈✐st♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✳✹ ✐t❡♠ ❛✮✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✸✳ ◆♦t❡ q✉❡ ♣❛r❛ s❡ t❡r ✉♠❛ s❡♥t❡♥ç❛ ♦✉ ♣r♦♣♦s✐çã♦✱ ❢❛③✲s❡ ♥❡❝❡s✲ sár✐♦✿

✭✶✮ ❯♠❛ ❡str✉t✉r❛ ❞❡ ♦r❛çã♦✱ ❝♦♠ s✉❥❡✐t♦✱ ✈❡r❜♦ ❡ ♣r❡❞✐❝❛❞♦❀

✭✷✮ ❙❡r ❞❡❝❧❛r❛t✐✈❛✱ ♥ã♦ ♣♦❞❡♥❞♦ s❡r ❡①❝❧❛♠❛t✐✈❛✱ ✐♠♣❡r❛t✐✈❛ ❡ ♥❡♠ ✐♥t❡rr♦❣❛t✐✈❛❀ ✭✸✮ ❙❛t✐s❢❛③❡r ♦s s❡❣✉✐♥t❡s ♣r✐♥❝í♣✐♦s✿

✭✸✳✶✮ Pr✐♥❝í♣✐♦ ❞♦ ❚❡r❝❡✐r♦ ❊①❝❧✉í❞♦✳ ❚♦❞❛ ♣r♦♣♦s✐çã♦ ♦✉ é ✈❡r❞❛❞❡✐r❛ ♦✉ é ❢❛❧s❛✱ ✐st♦ é✱ ✈❡r✐✜❝❛✲s❡ s❡♠♣r❡ ✉♠ ❞❡st❡s ❝❛s♦s ❡ ♥✉♥❝❛ ✉♠ t❡r❝❡✐r♦❀ ✭✸✳✷✮ Pr✐♥❝í♣✐♦ ❞❛ ◆ã♦✲❝♦♥tr❛❞✐çã♦✳❯♠❛ ♣r♦♣♦s✐çã♦ ♥ã♦ ♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r❛

❡ ❢❛❧s❛ ❛♦ ♠❡s♠♦ t❡♠♣♦✳

❊①❡♠♣❧♦ ✶✳✷✳✹✳ ❊①❡♠♣❧♦s ❞❡ ♥ã♦ ♣r♦♣♦s✐çõ❡s✿

a) ✏◗✉❡ ❤♦r❛s sã♦❄✑

❆ ❢r❛s❡ ♥ã♦ é ♣r♦♣♦s✐çã♦ ♣♦r s❡r ✐♥t❡rr♦❣❛t✐✈❛ ❡ ♥ã♦ ❞❡❝❧❛r❛t✐✈❛✳

b) ✏❆ ❜❡❧❛ ♠♦ç❛✳✑

❆ ❢r❛s❡ ♥ã♦ é ♣r♦♣♦s✐çã♦ ♣♦r ♥ã♦ s❡r ✉♠❛ ♦r❛çã♦✱ ♣♦s✉✐♥❞♦ ❛♣❡♥❛s s✉❥❡✐t♦✱ ❢❛❧t❛♥❞♦ ✈❡r❜♦ ❡ ♣r❡❞✐❝❛❞♦✳

c) ✏❊st❛ ❛✜r♠❛çã♦ é ❢❛❧s❛✳✑

❆ ❢r❛s❡ ♥ã♦ é ♣r♦♣♦s✐çã♦ ♣♦rq✉❡ ❢❛③ r❡❢❡rê♥❝✐❛ ❛ s✐ ♠❡s♠❛✱ t♦r♥❛♥❞♦ ✐♠♣♦ssí✈❡❧ ❛tr✐❜✉✐r✲❧❤❡ ✉♠ ✈❛❧♦r ❧ó❣✐❝♦✿ s❡ ❛ss✉♠✐r♠♦s ❝♦♠♦ ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ ❛ ♣r♦♣♦s✐çã♦ ❛✜r♠❛ q✉❡ é ❢❛❧s❛✱ ❡ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛♦ ❛ss✉♠✐r♠♦s ❝♦♠♦ ❢❛❧s❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❡❧❛ é ✈❡r❞❛❞❡✐r❛✱ ♥ã♦ s❛t✐s❢❛③❡♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❛ ◆ã♦✲❝♦♥tr❛❞✐çã♦✳

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❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❈❤❛♠❛♠♦s ❞❡ Pr♦♣♦s✐çã♦ ❆❜❡rt❛✱ ❛ ✉♠❛ ❛✜r♠❛çã♦ q✉❡ ❡stá s✉✲ ❜♦r❞✐♥❛❞❛ ❛ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ✈❛r✐á✈❡❧ q✉❡ ✜❝❛ ❧✐✈r❡✱ ❛ q✉❛❧ ♥❛❞❛ ♣♦❞❡♠♦s ❛✜r♠❛r✱ ✐♠♣♦ss✐❜✐❧✐t❛♥❞♦ ❛ ❛tr✐❜✉✐çã♦ ❞❡ ✉♠ ✈❛❧♦r ❧ó❣✐❝♦ ✭✈❡r❞❛❞❡✐r♦ ♦✉ ❢❛❧s♦✮ ♣❛r❛ ❛ ❛✜r♠❛✲ çã♦✳

❊①❡♠♣❧♦ ✶✳✷✳✻✳ ❊①❡♠♣❧♦ ❞❡ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛✿

✏❖ ♣♦❧í❣♦♥♦ t❡♠ ❡①❛t❛♠❡♥t❡ q✉❛tr♦ ❧❛❞♦s✱ t♦❞♦s ♣❛r❛❧❡❧♦s✑

❖❜s❡r✈❡ q✉❡ ❡ss❛ ❢r❛s❡ é ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ q✉❡ t❡♠ ❝♦♠♦ ✈❛r✐á✈❡❧ ❧✐✈r❡ ✏♣♦❧í❣♦♥♦✑✱ ♣♦✐s ❤á ♣♦❧í❣♦♥♦s✱ ❛ ❡①❡♠♣❧♦ ❞♦ q✉❛❞r❛❞♦✱ ❞♦ r❡tâ♥❣✉❧♦✱ ❞♦ ❧♦s❛♥❣♦ ❡ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❡ ❛ t♦r♥❛♠ ✉♠❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛❀

❡ ♦✉tr♦s q✉❡ ❛ t♦r♥❛♠ ❢❛❧s❛✱ ❛ ❡①❡♠♣❧♦ ❞♦ tr✐❛♥❣✉❧♦✱ ❞♦ tr❛♣é③✐♦ ❡ ❞♦ ❤❡①á❣♦♥♦✳

❉❡♥♦t❛♠♦s ✉♠❛ Pr♦♣♦s✐çã♦ ❆❜❡rt❛ ♥✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡ x♣♦r P(x).

❉❡ss❛ ❢♦r♠❛ ♥ã♦ ❤á ❝♦♠♦ ❛tr✐❜✉✐r ❞✐r❡t❛♠❡♥t❡ ✉♠ ✈❛❧♦r ❧ó❣✐❝♦ ♣❛r❛ ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛✳

❉✐③✲s❡ q✉❡ P(x) é ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ♥✉♠ ❝♦♥❥✉♥t♦ A q✉❛♥❞♦ P(x) t♦r♥❛✲s❡

✉♠❛ ♣r♦♣♦s✐çã♦ ✭✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✮ t♦❞❛s ❛s ✈❡③❡s q✉❡ s❡ s✉❜st✐t✉✐ ❛ ✈❛r✐á✈❡❧ ❧✐✈r❡

x ♣♦r q✉❛❧q✉❡r ❡❧❡♠❡♥t♦a ❞❡ A✳

❊①❡♠♣❧♦ ✶✳✷✳✼✳ ❈♦♥s✐❞❡r❡ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦❧í❣♦♥♦s✳ ◆❡ss❡ ❝❛s♦✱ ♣♦r

❡①❡♠♣❧♦✱ ♦ q✉❛❞r❛❞♦✱ ♦ r❡tâ♥❣✉❧♦✱ ♦ tr✐â♥❣✉❧♦ ♣❡rt❡♥❝❡♠ ❛ A✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛

♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❡♠ A✳

✏P(x)✿ ♦ x t❡♠ ❡①❛t❛♠❡♥t❡ q✉❛tr♦ ❧❛❞♦s✱ t♦❞♦s ♣❛r❛❧❡❧♦s✳✑

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◆♦t❡ q✉❡✿

• P✭q✉❛❞r❛❞♦✮ é ✉♠❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛❀

• P✭tr✐â♥❣✉❧♦✮ é ✉♠❛ ♣r♦♣♦s✐çã♦ ❢❛❧s❛❀

❖ ❝♦♥❥✉♥t♦ A r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❉♦♠í♥✐♦✸ ❞❛ ✈❛r✐á✈❡❧ ❧✐✈r❡x❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ a ∈ A ❞✐③✲s❡ ✉♠ ✈❛❧♦r ❞❛ ✈❛r✐á✈❡❧ x✳ ❙❡ a ∈ A é t❛❧ q✉❡ P(a) é ✉♠❛ ♣r♦♣♦s✐çã♦

✈❡r❞❛❞❡✐r❛✱ ❞✐③✲s❡ q✉❡ a s❛t✐s❢❛③ ♦✉ ✈❡r✐✜❝❛ P(x)✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ❞✐③❡♠♦s q✉❡ P(x1, x2) é ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ♥✉♠ ❝♦♥❥✉♥t♦

A×B q✉❛♥❞♦P(x1, x2)t♦r♥❛✲s❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ✭✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✮ t♦❞❛s ❛s ✈❡③❡s

q✉❡ s❡ s✉❜st✐t✉✐ ❛s ✈❛r✐á✈❡✐s ❧✐✈r❡sx1 ♣♦r q✉❛❧q✉❡r ❡❧❡♠❡♥t♦a ❞❡A❡x2 ♣♦r q✉❛❧q✉❡r

❡❧❡♠❡♥t♦ b ❞❡B ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱❬✶❪✮✳

❊①❡♠♣❧♦ ✶✳✷✳✽✳ ❈♦♥s✐❞❡r❡A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛ís❡s ❞❛ ❆♠ér✐❝❛ ❞♦ ❙✉❧ ❡B ♦

❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛ís❡s ❞❛ ➪s✐❛✳ ◆❡ss❡ ❝❛s♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❇r❛s✐❧✱ ❛ ❆r❣❡♥t✐♥❛✱ ❛ ❇♦❧✐✈✐❛ ♣❡rt❡♥❝❡♠ ❛ A✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ❏❛♣ã♦✱ ❛ ❈❤✐♥❛✱ ❛ ❈♦r❡✐❛ ❞♦ ❙✉❧ ♣❡rt❡♥❝❡♠

❛ B✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ s❡♥t❡♥ç❛ ❛❜❡rt❛ ❡♠ A×B

✏P(x1, x2)✿ x1 é ♠❛✐s ♣♦♣✉❧♦s♦ q✉❡ x2✑✳

◆♦t❡ q✉❡✿

• P(❇r❛s✐❧,❏❛♣ã♦) é ✉♠❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛❀

• P(❇r❛s✐❧,❈❤✐♥❛) é ✉♠❛ ♣r♦♣♦s✐çã♦ ❢❛❧s❛✳

❆♥á❧♦❣♦ ❛♦ ❝❛s♦ ❞❡ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❝♦♠ ✉♠❛ ✈❛r✐á✈❡❧✱ ♦ ❝♦♥❥✉♥t♦ A ×B ♦

❉♦♠í♥✐♦ ❞❛s ✈❛r✐á✈❡✐s x1 ❡ x2✱ ❡ q✉❛✐sq✉❡r ❡❧❡♠❡♥t♦s a∈A ❡b∈B sã♦ ❞✐t♦s ✈❛❧♦r❡s

❞❛s ✈❛r✐á✈❡✐s x1 ❡ x2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ (a, b) ∈ A×B é t❛❧ q✉❡ P(a, b) é ✉♠❛

♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛✱ ❞✐③✲s❡ q✉❡ (a, b) s❛t✐s❢❛③ ♦✉ ✈❡r✐✜❝❛ P(x1, x2)✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✾✳ P♦❞❡✲s❡ r❡♣r❡s❡♥t❛r ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ♣r♦♣♦s✐çõ❡s ❛❜❡rt❛s ❝♦♠ n

✈❛r✐á✈❡✐s ❧✐✈r❡s ♥✉♠ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ A1×A2× · · · ×An ♣♦r

P(x1, x2, . . . , xn).

❖❜s❡r✈❛çã♦ ✶✳✷✳✶✵✳ ◆✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ P(x) ❡♠ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ A✱ três

❝❛s♦s ♣♦❞❡♠ ♦❝♦rr❡r✿

■✮ P(x) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ t♦❞♦ x∈ A✱ ♦✉ s❡❥❛✱ P(x) ❡①♣r✐♠❡ ✉♠❛ ❝♦♥❞✐çã♦ ✉♥✐✲

✈❡rs❛❧ ♦✉ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ♥♦ ❝♦♥❥✉♥t♦ A✳

❊①❡♠♣❧♦✿ ✏P(x)✿ x é ♠♦rt❛❧✑✱ s❡♥❞♦ A ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡r❡s ❤✉♠❛♥♦s✳

■■✮ P(x) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♥s x ∈ A✱ ♦✉ s❡❥❛✱ P(x) ❡①♣r✐♠❡ ✉♠❛ ❝♦♥❞✐çã♦

♣♦ssí✈❡❧ ♦✉ ♣r♦♣r✐❡❞❛❞❡ ♣♦ssí✈❡❧ ♥♦ ❝♦♥❥✉♥t♦ A✳

❊①❡♠♣❧♦✿ ✏P(x)✿ x ♣♦ss✉✐ ❝✉rs♦ s✉♣❡r✐♦r✑❀ s❡♥❞♦ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s

♣❡ss♦❛s✳

◆❛ ❧✐t❡r❛t✉r❛ é ❝♦♠✉♠ ♦ ✉s♦ ❞❛ ❡①♣r❡sã♦ ❯♥✐✈❡rs♦ ❞♦ ❉✐s❝✉rs♦✳

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■■■✮ P(x) é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ♥❡♥❤✉♠ x ∈ A✱ ♦✉ s❡❥❛✱ P(x) ❡①♣r✐♠❡ ✉♠❛ ❝♦♥❞✐çã♦

✐♠♣♦ssí✈❡❧ ♦✉ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦ssí✈❡❧ ♥♦ ❝♦♥❥✉♥t♦ A✳

❊①❡♠♣❧♦✿ ✏P(x)✿ x é ✐♠♦rt❛❧✑❀ s❡♥❞♦ A ♦ ❝♦♥❥✉♥t♦ ❞♦s s❡r❡s ❤✉♠❛♥♦s✳

❯s❛♠♦s ♥❡st❛ ♦❜s❡r✈❛çã♦ ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❝♦♠ ✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡ ❛♣❡♥❛s ♣♦r s✐♠♣❧✐❝✐❞❛❞❡❀ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ♣♦❞❡ s❡r ❡♠♣r❡❣❛❞♦ ♣❛r❛ ♣r♦♣♦s✐çõ❡s ❛❜❡rt❛s ❝♦♠ ♠❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡✳

✶✳✸ ❊s♣❡❝✐✜❝❛çõ❡s ❞❡ ❈♦♥❥✉♥t♦s

Pr❛t✐❝❛♠❡♥t❡ t♦❞♦s ♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ sã♦ ❡st❛❜❡❧❡❝✐❞♦s ♥❛ ❧✐♥✲ ❣✉❛❣❡♠ ❞❡ ❝♦♥❥✉♥t♦s✳ ❯♠ ❞♦s ♣r✐♥❝í♣✐♦s ❜ás✐❝♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♥❡ss❡ ❝♦♥t❡①t♦ é ❛ ❝♦♥str✉çã♦ ❞❡ ♥♦✈♦s ❝♦♥❥✉♥t♦s ❛ ♣❛rt✐r ❞❡ ♦✉tr♦s✳ ❆♥t❡s ❞❛ ❢♦r♠✉❧❛çã♦ ❞❡ss❡ ♣r✐♥❝í♣✐♦✱ ✈❡❥❛♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦ ❤❡✉ríst✐❝♦✿

❊①❡♠♣❧♦ ✶✳✸✳✶✳ ❙❡❥❛ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❤♦♠❡♥s ❡ ❝♦♥s✐❞❡r❡ ❛ ♣r♦♣♦s✐çã♦

❛❜❡rt❛ ❡♠ A✿

✏P(x)✿ x é ❝❛s❛❞♦✑✳

❖❜s❡r✈❡ q✉❡ ❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛P(x)t♦r♥❛✲s❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♥s

❡❧❡♠❡♥t♦sx❞❡A❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ❢❛❧s❛ ♣❛r❛ ♦✉tr♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ✉s❛♠♦s ❛ ♥♦t❛çã♦

{x∈A:x é ❝❛s❛❞♦}

♣❛r❛ ❡s♣❡❝✐✜❝❛r ♦ ❝♦♥❥✉♥t♦ ❞♦s ❤♦♠❡♥s ❝❛s❛❞♦s ❡ ❡ss❛ ♥♦t❛çã♦ é ❝♦♠✉♠❡♥t❡ ❧✐❞❛ ❝♦♠♦ ✏♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s x ❡♠ A t❛✐s q✉❡ x é ❝❛s❛❞♦✑✳

❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❤♦♠❡♥s s♦❧t❡✐r♦s✿ {x∈A:x ♥ã♦ é ❝❛s❛❞♦}.

❊s♣❡❝✐✜❝❛çã♦ ❞❡ ❈♦♥❥✉♥t♦✿✹ P❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦A ❡ ❝❛❞❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛

P(x)❝♦♠ ❞♦♠í♥✐♦ A✱ ♣♦❞❡✲s❡ ♦❜t❡r ✉♠ ❝♦♥❥✉♥t♦ B ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❡①❛t❛♠❡♥t❡

❛q✉❡❧❡s ❡❧❡♠❡♥t♦s a ❞❡ A t❛✐s q✉❡ P(a) é ✈❡r❞❛❞❡✐r❛✳ ❖ ❝♦♥❥✉♥t♦ B ♥❡ss❡ ❝❛s♦

r❡♣r❡s❡♥t❛♠♦s ♣♦r

B ={x∈A :P(x)} ♦✉ B ={x:P(x)}.

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❆♥á❧♦❣♦ ❛♦ ❊①❡♠♣❧♦ ✶✳✷✳✻✱ ❝♦♥s✐❞❡r❡ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s

♣♦❧í❣♦♥♦s ❡ ❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛

✏P(x)✿ ❖ x t❡♠ ❡①❛t❛♠❡♥t❡ q✉❛tr♦ ❧❛❞♦s✱ t♦❞♦s ♣❛r❛❧❡❧♦s✑✳

◆❛ t❡♦r✐❛ ❛①✐♦♠át✐❝❛ ❞❡ ❝♦♥❥✉♥t♦s✱ ❛ ❊s♣❡❝✐✜❝❛çã♦ ❞❡ ❈♦♥❥✉♥t♦ é ✉♠ ❝♦♥❝❡✐t♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦

❆①✐♦♠❛ ❞❛ ❊s♣❡❝✐✜❝❛çã♦ ✭❱❡r ❬✸✱ ♣✳ ✶✵❪✮✳

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❖ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ✏♣♦❧í❣♦♥♦sx✑ t❛✐s q✉❡ P(x)é ✈❡r❞❛❞❡✱ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦

♣♦r✿

{x∈A:P(x)}. ✭✶✳✷✮

◆♦t❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ ♦ P❛r❛❧❡❧♦❣r❛♠♦ ❡ ♦ ▲♦s❛♥❣♦ sã♦ ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ r❡✲ ♣❡s❡♥t❛❞♦ ❡♠ ✭✶✳✷✮✳

❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❙❡❥❛ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡st✉❞❛♥t❡s ❞❛ ❯❋❙✳ ❆ ♣r♦♣♦s✐çã♦

❛❜❡rt❛

✏P(x)✿ x é s❡r❣✐♣❛♥♦✑✱

é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛❧❣✉♥s ❡❧❡♠❡♥t♦s x ❞❡ A ❡ ❢❛❧s❛ ♣❛r❛ ♦✉tr♦s✳ ◆❡st❡ ❝❛s♦ ♣♦❞❡♠♦s

✉s❛r ❛ ♥♦t❛çã♦

{x∈A:P(x)}

♣❛r❛ ❡s♣❡❝✐✜❝❛r ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡st✉❞❛♥t❡s s❡r❣✐♣❛♥♦s ❞❛ ❯❋❙✳

❖❜s❡r✈❛çã♦ ✶✳✸✳✹✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣❛r❛ ♣r♦♣♦s✐çõ❡s ❛❜❡rt❛s ❝♦♠ ♥ ✈❛r✐á✈❡✐s ❧✐✈r❡s

P(x1, x2, . . . , xn) ❡ ❞♦♠í♥✐♦ A1 ×A2 × · · · ×An✱ ♣♦❞❡✲s❡ ♦❜t❡r t❛♠❜é♠ ✉♠ ❝♦♥❥✉♥t♦

❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❡①❛t❛♠❡♥t❡ ❛q✉❡❧❡s ❡❧❡♠❡♥t♦s(a1, a2, . . . , an)❞❡A1×A2×· · ·×An

t❛✐s q✉❡ P(a1, a2, . . . , an) é ✈❡r❞❛❞❡✐r❛✳ ◆❡ss❡ ❝❛s♦✱ r❡♣r❡s❡♥t❛♠♦s t❛❧ ❝♦♥❥✉♥t♦ ♣♦r

{(x1, x2, . . . , xn)∈A1×A2× · · · ×An :P(x1, x2, . . . , xn)}.

✶✳✹ ◗✉❛♥t✐✜❝❛❞♦r❡s

❯♠❛ ❢♦r♠❛ ❞❡ tr❛♥s❢♦r♠❛r ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❡♠ ♣r♦♣♦s✐çã♦ é s✉❜st✐t✉✐r ❛ ✈❛r✐á✈❡❧ ❧✐✈r❡ ♣♦r q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞♦ ❞♦♠í♥✐♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠♦ ✈✐st♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✷✳✼✳ ❯♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ tr❛♥s❢♦r♠❛r ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❡♠ ✉♠❛ ♣r♦♣♦s✐çã♦ é ❛tr❛✈és ❞♦ ✉s♦ ❞♦s ❝❤❛♠❛❞♦s ◗✉❛♥t✐✜❝❛❞♦r❡s✳

✶✳✹✳✶ ◗✉❛♥t✐✜❝❛❞♦r ❯♥✐✈❡rs❛❧

P❛r❛ ✐♥tr♦❞✉③✐r ♦ q✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✱ ❝♦♥s✐❞❡r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ ❡♠A ✭❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣❡ss♦❛s✮✿

“P(x) :x ❞❡♣❡♥❞❡ ❞❛ á❣✉❛ ♣❛r❛ s♦❜r❡✈✐✈❡r✑✱

♦♥❞❡ x r❡♣r❡s❡♥t❛ ❝❛❞❛ ♣❡ss♦❛ ❞♦ ❝♦♥❥✉♥t♦A✳ ❆❣♦r❛ ♦❜s❡r✈❡ ❛ ❛✜r♠❛çã♦

✏P❛r❛ t♦❞♦ x ❞❡A✱ x ❞❡♣❡♥❞❡ ❞❛ á❣✉❛ ♣❛r❛ s♦❜r❡✈✐✈❡r✑✳ ✭✶✳✸✮

◆♦t❡ q✉❡ ❛ ❛✜r♠❛çã♦ é ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠ ✈❛❧♦r ❧ó❣✐❝♦ ✈❡r❞❛❞❡✐r♦✳ ■st♦ ♠♦t✐✈❛ ❛ ✐♥tr♦❞✉çã♦ ❞♦ q✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✳

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❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❉❛❞❛ ✉♠❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛ P(x)✱ ❝♦♠ ✈❛r✐á✈❡❧ x ❡ ❞♦♠í♥✐♦ A✱ ❛

❛✜r♠❛çã♦

✏P❛r❛ t♦❞♦ x ♥♦ ❝♦♥❥✉♥t♦ A✱ P(x)✑✱

q✉❡ ❞❡♥♦t❛♠♦s s✐♠❜♦❧✐❝❛♠❡♥t♦ ♣♦r

∀ x∈A;P(x),

é ✈❡r❞❛❞❡✐r❛ q✉❛♥❞♦ P(x) ❢♦r s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛ ❛♦ s✉❜st✐t✉✐r♠♦s ❛ ✈❛r✐á✈❡❧ x ♣♦r

q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❞❡ A❀ ❡ ❢❛❧s❛ s❡ ❡①✐st✐r ❛♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ❡♠ A q✉❡ t♦r♥❛ P(x) ❢❛❧s❛ q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ é s✉❜st✐t✉✐❞❛ ♣♦r ❡ss❡ ❡❧❡♠❡♥t♦✳

◆♦t❛çã♦ ✶✳✹✳✷✳ ❖ sí♠❜♦❧♦∀s✐❣♥✐✜❝❛ ✏♣❛r❛ t♦❞♦✑ ♦✉ ✏q✉❛❧q✉❡r q✉❡ s❡❥❛✑❀ ❡ é ❝❤❛♠❛❞♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r ❯♥✐✈❡rs❛❧✳

❊①❡♠♣❧♦ ✶✳✹✳✸✳ ❉♦ ❡①❡♠♣❧♦ ✶✳✷✳✼✱ t❡♠♦s A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦❧í❣♦♥♦s ❡

✏P(x)✿ ♦ x t❡♠ ❡①❛t❛♠❡♥t❡ q✉❛tr♦ ❧❛❞♦s✱ t♦❞♦s ♣❛r❛❧❡❧♦s✑✱

❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛✳ ❯t✐❧✐③❛♥❞♦ ♦ q✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✱ t❡♠♦s ❛ ♣r♦♣♦s✐çã♦✿ ✏P❛r❛ t♦❞♦ x ♥♦ ❝♦♥❥✉♥t♦ A✱ ♦ x t❡♠ ❡①❛t❛♠❡♥t❡ q✉❛tr♦ ❧❛❞♦s✱ t♦❞♦s ♣❛r❛❧❡❧♦s✑✳

❙✐♠❜♦❧✐❝❛♠❡♥t❡ ❛ ♣r♦♣♦s✐çã♦ é

∀ x∈A;P(x).

❖❜s❡r✈❡ q✉❡ ❡ss❛ é ✉♠❛ ♣r♦♣♦s✐çã♦ ❢❛❧s❛✱ ♣♦✐s s✉❜st✐t✉✐♥❞♦ x ♣❡❧♦ ❡❧❡♠❡♥t♦ ✏❚r✐â♥✲

❣✉❧♦✑✱ t❡♠♦s P(❚r✐â♥❣✉❧♦) ❝♦♠ ✈❛❧♦r ❧ó❣✐❝♦ ❢❛❧s♦✳

❖❜s❡r✈❛çã♦ ✶✳✹✳✹✳ ❆ ❛✜r♠❛çã♦ ✏∀ x∈A;P(x)✑ t❛♠❜é♠ é ❡s❝r✐t❛ ❝♦♠♦

✏∀ x∈A, P(x)✑

❡ ❛♠❜❛s ♣♦❞❡♠ s❡r ❧✐❞❛s ❝♦♠♦ ✏♣❛r❛ t♦❞♦x ❡♠A✱ P(x)✑ ♦✉ ✏♣❛r❛ t♦❞♦x ❡♠ A✱ t❡♠♦s P(x)✑✱ ❛❧é♠ ❞❡ ♦✉tr❛s ❢♦r♠❛s ❞❡ ❧❡✐t✉r❛ q✉❡ ❛♣❡r❡❝❡♠ ♥❛ ❧✐t❡r❛t✉r❛✳

❊①❡♠♣❧♦ ✶✳✹✳✺✳ ❙❡♥❞♦ A ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡st❛❞♦s ❞♦ ♥♦r❞❡st❡ ❜r❛s✐❧❡✐r♦ ❡

✏P(x)✿ x é ✉♠ ❡st❛❞♦ ❞♦ ❇r❛s✐❧✑

❛ ♣r♦♣♦s✐çã♦ ❛❜❡rt❛✳ ❯t✐❧✐③❛♥❞♦ ♦ q✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✱ t❡♠♦s ❛ ♣r♦♣♦s✐çã♦✿ ✏P❛r❛ t♦❞♦ x ♥♦ ❝♦♥❥✉♥t♦ A✱ x é ✉♠ ❡st❛❞♦ ❞♦ ❇r❛s✐❧✑✳

❙✐♠❜♦❧✐❝❛♠❡♥t❡ ❛ ♣r♦♣♦s✐çã♦ é

∀ x∈A;P(x).

❖❜s❡r✈❡ q✉❡ ❡ss❛ é ✉♠❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛✱ ♣♦✐s s✉❜st✐t✉✐♥❞♦x♣♦r q✉❛❧q✉❡r ❡st❛❞♦

❞♦ ♥♦r❞❡st❡ ❜r❛s✐❧❡✐r♦✱ t❡♠♦s P(x) ❝♦♠ ✈❛❧♦r ❧ó❣✐❝♦ ✈❡r❞❛❞❡✐r♦✳

Referências

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