❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❊①♣❧♦r❛♥❞♦ ♦ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ♦✉tr♦s
❢r❛❝t❛✐s ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦
❆♠❛✉r✐ ❋❡r♥❛♥❞❡s ❋r❡✐t❛s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦
Freitas, Amauri Fernandes
Explorando o conjunto de cantor e outros fractais no ensino básico / Amauri Fernandes Freitas. - Rio Claro, 2014 42 f. : il., figs., gráfs., fots.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Suzete Maria Silva Afonso
1. Matemática - Estudo e ensino. 2. Conjuntos. 3. Funções. 4. Fractais. 5. Ensino médio. I. Título.
510.07 F865e
Ficha Catalográfica elaborada pela STATI - Biblioteca da UNESP Campus de Rio Claro/SP
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❆♠❛✉r✐ ❋❡r♥❛♥❞❡s ❋r❡✐t❛s
❊①♣❧♦r❛♥❞♦ ♦ ❈♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ♦✉tr♦s ❢r❛❝t❛✐s ♥♦
❊♥s✐♥♦ ❇ás✐❝♦
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❊❧✐r✐s ❈r✐st✐♥❛ ❘✐③③✐♦❧❧✐ ■●❈❊✴ ❯♥❡s♣ ✲❘✐♦ ❈❧❛r♦✴ ❙P
Pr♦❢✳ ❉r✳ ❊✈❡r❛❧❞♦ ❞❡ ▼❡❧❧♦ ❇♦♥♦tt♦ ■❈▼❈✴ ❯❙P ✲ ❙ã♦ ❝❛r❧♦s✴❙P
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r q✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s✱ ♣♦r ❡ss❛ ✈✐tór✐❛✳ ❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♣r♦✲ ❢❡ss♦r❛ ♦r✐❡♥t❛❞♦r❛ ❙✉③❡t❡ ▼❛r✐❛ ♣♦r ♠❡ ♦r✐❡♥t❛r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦ ❡ ♠❡ ❛♣♦✐❛r ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ à ❈❆P❊❙ q✉❡ ✜♥❛♥❝✐♦✉ ♦ Pr♦❣r❛♠❛ P❘❖❋▼❆❚ ❡ ♣r♦✲ ♣♦r❝✐♦♥♦✉ ❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ♣❛r❛ q✉❡ ♦ s♦♥❤♦ ❞❡ ❝✉rs❛r ♦ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛ ♥❛ ❯◆❊❙P ❢♦ss❡ ♣♦ssí✈❡❧✳
❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ ♣❛r❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞♦ Pr♦❣r❛♠❛ P❘❖❋▼❆❚✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♣❛r❛ ❛ ♣r♦❢❡ss♦r❛ ❙✉③✐♥❡✐✱ q✉❡ ♠✉✐t♦ s❡ ❞❡❞✐❝♦✉ ♣❛r❛ ❛ ❡①✲ ❡❝✉çã♦ ❞♦ Pr♦❣r❛♠❛ P❘❖❋▼❆❚ ♥❛ ❯◆❊❙P ✲ ❘✐♦ ❈❧❛r♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❡stá ✐♥s❡r✐❞♦ ♥♦ ❝♦♥t❡①t♦ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ❡ ❛♣r❡s❡♥t❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✳ P❛r❛ ❡①✐❜✐r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✱ ❡①♣❧♦r❛♠♦s ❝♦♥t❡ú❞♦s ♣r❡s❡♥t❡s ♥♦ ❈✉rrí❝✉❧♦ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❞❡ ▼❛t❡♠át✐❝❛✱ t❛✐s ❝♦♠♦ ❝♦♥❥✉♥t♦s✱ ❢✉♥çõ❡s✱ ✐♥t❡r✈❛❧♦s r❡❛✐s ❡ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s✳ ❯t✐❧✐③❛♠♦s ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❝❡ssí✈❡❧ ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ ❛♣r❡s❡♥t❛♠♦s ❛t✐✈✐❞❛❞❡s ❡♥✈♦❧✈❡♥❞♦ ♦✉tr♦s ❢r❛❝t❛✐s✱ ♦❜t✐❞♦s ❞❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ Pr♦❢❡ss✐♦♥❛❧ Pr♦❣r❛♠ ▼❛st❡r ♦❢ ▼❛t❤❡♠❛t✐❝s ✐♥ ◆❛t✐♦♥❛❧ ◆❡t✇♦r❦ ✲ P❘❖❋▼❆❚ ❛♥❞ ✐t ♣r❡s❡♥ts t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❈❛♥t♦r s❡t✳ ❚♦ st✉❞② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❈❛♥t♦r s❡t✱ ✇❡ ❡①♣❧♦r❡ ❝♦♥t❡♥ts ✇❤✐❝❤ ❛r❡ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ◆❛t✐♦♥❛❧ ❈✉rr✐❝✉❧✉♠ ❢♦r ❇❛s✐❝ ❊❞✉❝❛t✐♦♥ ✐♥ ▼❛t❤❡♠❛t✐❝s✱ s✉❝❤ ❛s s❡ts✱ ❢✉♥❝t✐♦♥s✱ r❡❛❧ ✐♥t❡r✈❛❧s ❛♥❞ ❣❡♦♠❡tr✐❝ ♣r♦❣r❡ss✐♦♥s✳ ❲❡ ✉s❡ ❛ ❛❝❝❡ss✐❜❧❡ ❧❛♥❣✉❛❣❡ t♦ ❍✐❣❤ ❙❝❤♦♦❧ st✉❞❡♥ts ❛♥❞ ✇❡ ♣r❡s❡♥t s♦♠❡ ❛❝t✐✈✐t✐❡s ✐♥✈♦❧✈✐♥❣ ♦t❤❡r ❢r❛❝t❛❧s✱ ♦❜t❛✐♥❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛② t♦ t❤❡ ❈❛♥t♦r s❡t✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✾
✶ ❙♦❜r❡ ❝♦♥❥✉♥t♦s ❡ ❢✉♥çõ❡s ✶✶
✶✳✶ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✶ ◆ú♠❡r♦s ♥❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸✳✷ ❈♦♥❥✉♥t♦s ✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸✳✸ ❈♦♥❥✉♥t♦s ■♥✜♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ✷✶
✷✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❉❡✜♥✐çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸ ❈♦♥str✉✐♥❞♦ ♦✉tr♦s ❢r❛❝t❛✐s ✸✺
✹ ❊♥❢♦q✉❡ ♣❡❞❛❣ó❣✐❝♦ ✸✽
■♥tr♦❞✉çã♦
●❡♦r❣ ❋❡r❞✐♥❛♥❞ ▲✉❞✇✐❣ P❤✐❧✐♣ ❈❛♥t♦r✱ ✜❧❤♦ ❞❡ ❡♠✐❣r❛♥t❡s ❞✐♥❛♠❛rq✉❡s❡s✱ ♥❛s❝❡✉ ❡♠ ❙✳ P❡t❡rs❜✉r❣♦✱ ❘úss✐❛✱ ❡♠ ✶✽✹✺✳ ❊♠ ✶✽✺✻ s✉❛ ❢❛♠í❧✐❛ tr❛♥s❢❡r✐✉✲s❡ ♣❛r❛ ❋r❛♥❦❢✉rt✱ ❆❧❡♠❛♥❤❛✳ ❖ ♣❛✐ ❞❡ ❈❛♥t♦r ❡r❛ ✉♠ ❥✉❞❡✉ ❝♦♥✈❡rt✐❞♦ ❞♦ ♣r♦t❡st❛♥t✐s♠♦ ❡ ❛ ♠ã❡ ♥❛s❝❡r❛ ♥❛ r❡❧✐❣✐ã♦ ❝❛tó❧✐❝❛✳ ●❡♦r❣ s❡ ✐♥t❡r❡ss♦✉ ♣r♦❢✉♥❞❛♠❡♥t❡ ♣❡❧♦s ❛r❣✉♠❡♥t♦s ❞❛ t❡♦❧♦❣✐❛ ♠❡❞✐❡✈❛❧ s♦❜r❡ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ♦ ✐♥✜♥✐t♦✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♥ã♦ q✉✐s s❡❣✉✐r ✉♠❛ ❝❛rr❡✐r❛ ♥❛ ❡♥❣❡♥❤❛r✐❛ ❝♦♠♦ ♦ s❡✉ ♣❛✐ s✉❣❡r✐❛✱ q✉✐s s❡ ❞❡❞✐❝❛r à ♠❛t❡♠át✐❝❛✱ à ✜❧♦s♦✜❛ ❡ à ❢ís✐❝❛✳ ❊st✉❞♦✉ ❡♠ ❩✉r✐q✉❡✱ ●ött✐♥❣❡♥ ❡ ❇❡r❧✐♠ ✲ ♦♥❞❡ r❡❝❡❜❡✉ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ ❑❛r❧ ❲❡✐❡rstr❛ss ❡ ♦❜t❡✈❡ ♦ ❞♦✉t♦r❛❞♦ ❡♠ ✶✽✻✼✱ ❝♦♠ ✉♠❛ t❡s❡ s♦❜r❡ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✳ ❉❡ ✶✽✻✾ ❛ ✶✾✵✺✱ ❞❡s❡♥✈♦❧✈❡✉ s✉❛ ❝❛rr❡✐r❛ ❞♦❝❡♥t❡ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❍❛❧❧❡✳ ❋❛❧❡❝❡✉ ❡♠ ✶✾✶✽✱ ♥♦ ❤♦s♣✐t❛❧ ❞❡ ❞♦❡♥ç❛s ♠❡♥t❛✐s ❞❡ ❍❛❧❧❡✳
❈❛♥t♦r ♣❡r❝❡❜❡✉ q✉❡ ♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ♥ã♦ sã♦ t♦❞♦s ✐❣✉❛✐s✳ ◆♦ ❝❛s♦ ✜♥✐t♦✱ ❞✐③❡♠♦s q✉❡ ♦s ❝♦♥❥✉♥t♦s ❞❡ ❡❧❡♠❡♥t♦s tê♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ✭❝❛r❞✐♥❛❧✮ s❡ ♣✉❞❡r❡♠ s❡r ♣♦st♦s ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✳ ❉❡ ♠♦❞♦ ✉♠ t❛♥t♦ s✐♠✐❧❛r✱ ❈❛♥t♦r s❡ ❞✐s♣ôs ❛ ❝♦♥str✉✐r ✉♠❛ ❤✐❡r❛rq✉✐❛ ❞♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ❝♦♥❢♦r♠❡ ❛ ▼ä❝❤t✐♥❣❦❡✐t ♦✉ ✧♣♦tê♥✲ ❝✐❛✧ ❞♦ ❝♦♥❥✉♥t♦✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s t❡♠ ❛ ♠❡s♠❛ ♣♦tê♥❝✐❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡❧❡s ♣♦❞❡♠ s❡r ♣♦st♦s ❡♠ ❝♦r✲ r❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✳ ❈❛♥t♦r ♠♦str♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢r❛çõ❡s r❛❝✐♦♥❛✐s é ❡♥✉♠❡rá✈❡❧ ♦✉ ❝♦♥tá✈❡❧✱ ♦✉ s❡❥❛✱ t❛♠❜é♠ ♣♦❞❡ s❡r ♣♦st♦ ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✲ ✉♥í✈♦❝❛ ❝♦♠ ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ t❡♠ ❛ ♠❡s♠❛ ♣♦tê♥❝✐❛✳ ❈♦♠ ❡ss❛s ❝♦♥st❛t❛çõ❡s✱ ❝♦♠❡ç❛r❛♠ ❛ ♣❡♥s❛r q✉❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s ♣♦ss✉í❛♠ ❛ ♠❡s♠❛ ♣♦tê♥❝✐❛✳ ❚♦❞❛✈✐❛✱ ❈❛♥t♦r ♣r♦✈♦✉ q✉❡ ✐ss♦ ❡st❛✈❛ ❧♦♥❣❡ ❞❡ s❡r ✈❡r❞❛❞❡✳ ❊❧❡ ❞❡♠♦♥str♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s t❡♠ ♣♦tê♥❝✐❛ ♠❛✐♦r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡ ♦ ❞❡t❡r♠✐♥♦✉ ♥ã♦✲❡♥✉♠❡rá✈❡❧✳ ❙❡✉s ✐♥❝rí✈❡✐s r❡s✉❧t❛❞♦s ❧❡✈❛r❛♠✲♦ ❛♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❞♦s ❈♦♥❥✉♥t♦s✳
❈❛♥t♦r ♣❛ss♦✉ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡ s✉❛ ❝❛rr❡✐r❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❍❛❧❧❡✱ ♥✉♥❝❛ ❝♦♥✲ s❡❣✉✐♥❞♦ r❡❛❧✐③❛r ✉♠❛ ❞❡ s✉❛s ❣r❛♥❞❡s ❛s♣✐r❛çõ❡s q✉❡ ❡r❛ ❛ ❞❡ s❡r ♣r♦❢❡ss♦r ♥❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❞❡ ❇❡r❧✐♠✱ ❞❡✈✐❞♦ à ♣❡rs❡❣✉✐çã♦ ❞❡ ❑r♦♥❡❝❦❡r✱ q✉❡ ❞✉✈✐❞❛✈❛ ❞❛ t❡♦r✐❛ ❞❛ ✐♥✜♥✐❞❛❞❡ ❝♦♠♣❧❡t❛ ❞❡ ❈❛♥t♦r✳
❖ r❡❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ s✉❛s r❡❛❧✐③❛çõ❡s✱ ❞❡♣♦✐s ❞❡ ❛❧❣✉♠ t❡♠♣♦✱ ♠❡r❡❝❡r❛♠ ❛ ❡①❝❧❛✲ ♠❛çã♦ ❞❡ ❍✐❧❜❡rt✿ ✏◆✐♥❣✉é♠ ♥♦s ❡①♣❧✉s❛rá ❞♦ ♣❛r❛ís♦ q✉❡ ❈❛♥t♦r ❝r✐♦✉ ♣❛r❛ ♥ós✳✑
■♥tr♦❞✉çã♦ ✶✵
●❡♦r❣ ❈❛♥t♦r ✭✶✽✹✺ ✲ ✶✾✶✽✮
◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✱ ✉♠ ❡♥❣❡♥❤♦s♦ ❡①❡♠♣❧♦ ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❛ r❡t❛ q✉❡ é t❛♠❜é♠ ♥ã♦✲❡♥✉♠❡rá✈❡❧ ❡ é ❝♦♥s✐❞❡r❛❞♦ ♦ ♣r❡❝✉rs♦r ❞❛ ❣❡♦♠❡tr✐❛ ❢r❛❝t❛❧✱ r❛♠♦ ❞❛ ❣❡♦♠❡tr✐❛ ♥ã♦✲❡✉❝❧✐❞✐❛♥❛✱ q✉❡ ❞❡s❞❡ ✶✾✼✺ ✈❡♠ ♦❜t❡♥❞♦ ❛✈❛♥ç♦s s✐❣♥✐✜❝❛t✐✈♦s ❡♠ ❞✐✈❡rs♦s s❡t♦r❡s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❆tr❛✈és ❞❡st❡ ❝♦♥❥✉♥t♦✱ ❡①♣❧♦r❛r❡♠♦s ❝♦♥t❡ú❞♦s ❝♦♥t❡♠♣❧❛❞♦s ♥♦ ❈✉rrí❝✉❧♦ ◆❛❝✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ t❛✐s ❝♦♠♦ ❢✉♥çõ❡s✱ ❝♦♥❥✉♥t♦s✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ✐♥t❡r✈❛❧♦s r❡❛✐s✱ r❡❝♦rrê♥❝✐❛s✱ ♣r♦❣r❡ssõ❡s ❣❡♦♠étr✐❝❛s ❡ ♥♦çõ❡s ❞❡ ❧✐♠✐t❡✳ ❆♣♦st❛♠♦s ♥❛ ✐❞❡✐❛ ❞❡ q✉❡ ♦ ✉s♦ ❞❡ ♣r♦❝❡ss♦s r❡❝✉rs✐✈♦s ✐♥✜♥✐t♦s ♥❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ♣♦❞❡ s❡r ❞❡s❛✜❛❞♦r ❡ ❢❛s❝✐♥❛♥t❡ ♣❛r❛ ♦s ❛❧✉♥♦s✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ✉♠ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ♥ã♦ s❡ ❢❛❧❛ q✉❡ ♦s ✐♥✜♥✐t♦s ♥ã♦ sã♦ t♦❞♦s ✐❣✉❛✐s✱ ♠❛s r❡❝♦♠❡♥❞❛♠♦s q✉❡ ❡❧❡ ❥á t❡♥❤❛ ❛❝❡ss♦ ❛ ✈❡r❞❛❞❡ ♣r♦♣♦r❝✐♦♥❛❞❛ ♣♦r ❈❛♥t♦r ♥❡st❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦ ✲ ♠❡s♠♦ q✉❡ s✉♣❡r✜❝✐❛❧♠❡♥t❡✱ ✈✐st♦ q✉❡ ♦ ❝♦♥t❛t♦ ❝♦♠ ♦ ✐♥✜♥✐t♦ ♣♦❞❡ ❣❡r❛r ✐♥tr✐❣❛♥t❡s q✉❡stõ❡s✳
✶ ❙♦❜r❡ ❝♦♥❥✉♥t♦s ❡ ❢✉♥çõ❡s
◆❡st❡ ❝❛♣ít✉❧♦ ❞❡ ❝❛rát❡r ♣r❡❧✐♠✐♥❛r ❛♦ ❡st✉❞♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✱ ❛♣r❡s❡♥t❛r❡✲ ♠♦s ❛ ❧✐♥❣✉❛❣❡♠ ❜ás✐❝❛ ❞❡ ❝♦♥❥✉♥t♦s ❡ ❢✉♥çõ❡s ♥❛s ❞✉❛s ♣r✐♠❡✐r❛s s❡çõ❡s✱ ❧✐♥❣✉❛❣❡♠ ❡ss❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦s ❛❧✉♥♦s ❞♦s ❊♥s✐♥♦s ❋✉♥❞❛♠❡♥t❛❧ ❡ ▼é❞✐♦✳ ◆❛ t❡r❝❡✐r❛ s❡çã♦✱ ❞❛r❡♠♦s ✉♠❛ ❛t❡♥çã♦ ❡s♣❡❝✐❛❧ ❛♦s ❝♦♥❥✉♥t♦s ✜♥✐t♦s ❡ ✐♥✜♥✐t♦s✳
❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ sã♦ ❬✹❪ ❡ ❬✺❪✳
✶✳✶ ❈♦♥❥✉♥t♦s
❆ ♥♦çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ❝♦♥❥✉♥t♦ é ♣r❛t✐❝❛♠❡♥t❡ ❛ ♠❡s♠❛ q✉❡ s❡ ✉s❛ ♥❛ ❧✐♥❣✉❛❣❡♠ ❝♦rr❡♥t❡✿ é ♦ ♠❡s♠♦ q✉❡ ❛❣r✉♣❛♠❡♥t♦✱ ❝❧❛ss❡✱ ❝♦❧❡çã♦✱ s✐st❡♠❛✳ ❱❡❥❛♠♦s ❛❧❣✉♥s ❡①❡♠✲ ♣❧♦s✿
✶✮ ❈♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s❀
✷✮ ❈♦♥❥✉♥t♦ ❞♦s ❛❧❣❛r✐s♠♦s r♦♠❛♥♦s❀ ✸✮ ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣❛r❡s✳
❯♠ ❝♦♥❥✉♥t♦ é ❝♦♥st✐t✉í❞♦ ♣♦r ❡❧❡♠❡♥t♦s✱ ♣♦❞❡♥❞♦ ❛✐♥❞❛ ♥ã♦ ♣♦ss✉✐r ❡❧❡♠❡♥t♦ ❛❧✲ ❣✉♠✳
◗✉❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ♣♦ss✉✐ ❡❧❡♠❡♥t♦s✱ ❡st❡ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ❡ ❞❡♥♦✲ t❛❞♦ ♣♦r ∅✳
◆♦s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✱ ♦s ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s ♠❡♥❝✐♦♥❛❞♦s sã♦✿ ✶✮ ❛✱ ❡✱ ✐✱ ♦✱ ✉❀
✷✮ ■✱ ❱✱ ❳✱ ▲✱ ❈✱ ❉✱ ▼❀ ✸✮ 2,4,6,8,10,12, . . .✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
◗✉❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❡①❡♠♣❧♦✱ ❡❧❡ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ✉♥✐tár✐♦✳ ❆ r❡❧❛çã♦ ❡♥tr❡ ✉♠ ❡❧❡♠❡♥t♦ ❡ ✉♠ ❝♦♥❥✉♥t♦ ❝❤❛♠❛✲s❡ r❡❧❛çã♦ ❞❡ ♣❡rt✐♥ê♥❝✐❛✱ ✉♠ ❡❧❡♠❡♥t♦ ♣♦❞❡ ♣❡rt❡♥❝❡r ♦✉ ♥ã♦ ❛♦ ❝♦♥❥✉♥t♦ ❡ ❛♣❡♥❛s ✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s é ✈❡r❞❛❞❡✐r❛✳
❈♦♥❥✉♥t♦s ✶✷
P❛r❛ ✐♥❞✐❝❛r q✉❡x ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦A✱ ❡s❝r❡✈❡♠♦sx∈A✳ P❛r❛ ✐♥❞✐❝❛r q✉❡x ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ A✱ ❡s❝r❡✈❡♠♦sx6∈A✳
❉♦✐s r❡❝✉rs♦s ♣r✐♥❝✐♣❛✐s ♣❛r❛ ❞❡s❝r❡✈❡r ✉♠ ❝♦♥❥✉♥t♦ ❡ s❡✉s ❡❧❡♠❡♥t♦s sã♦ ✉t✐❧✐③❛✲ ❞♦s✿ ♦✉ ❡♥✉♠❡r❛♠♦s ✭❧✐st❛♠♦s✮ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ♦✉ ❞❛♠♦s ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦✳
• ◗✉❛♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ é ❞❛❞♦ ♣❡❧❛ ❡♥✉♠❡r❛çã♦ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✱ ✐♥❞✐❝❛♠♦s✲♦
❡s❝r❡✈❡♥❞♦ s❡✉s ❡❧❡♠❡♥t♦s ❡♥tr❡ ❝❤❛✈❡s✳ ❘❡t♦r♥❛♥❞♦ ❛♦s ❡①❡♠♣❧♦s ❛❝✐♠❛✱ t❡♠♦s✿ ✶✮ ❈♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s✿ {❛✱ ❡✱ ✐✱ ♦✱ ✉}❀
✷✮ ❈♦♥❥✉♥t♦ ❞♦s ❛❧❣❛r✐s♠♦s r♦♠❛♥♦s✿ {■✱ ❱✱ ❳✱ ▲✱ ❈✱ ❉✱ ▼}✳
❊st❛ ♥♦t❛çã♦ t❛♠❜é♠ é ❡♠♣r❡❣❛❞❛ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ é ✐♥✜♥✐t♦✿ ❡s❝r❡✈❡♠♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s q✉❡ ❡✈✐❞❡♥❝✐❛♠ ❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❡ ❡♠ s❡❣✉✐❞❛ ❝♦❧♦❝❛♠♦s r❡✲ t✐❝ê♥❝✐❛s✱ ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ❛❜❛✐①♦✿
✸✮ ❈♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♣r✐♠♦s ♣♦s✐t✐✈♦s✿ {2,3,5,7,11,13, . . .}✳
•P❛r❛ ❞❡s❝r❡✈❡r ✉♠ ❝♦♥❥✉♥t♦A ❛tr❛✈és ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝❛r❛❝t❡ríst✐❝❛P ❞❡ s❡✉s ❡❧❡♠❡♥t♦s x✱ ❡s❝r❡✈❡♠♦s
A={x|x ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P}
❡ ❧❡♠♦s✿ A é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s x t❛❧ q✉❡ x ❣♦③❛ ❞❛ ♣r♦♣r✐❡❞❛❞❡ P✳ ❱❡❥❛♠♦s ♦s ♣ró①✐♠♦s ❡①❡♠♣❧♦s✿
✶✮ {x|x é ❞✐✈✐s♦r ❞❡ ✺} é ✉♠❛ ❢♦r♠❛ ❞❡ ❞❡s❝r❡✈❡r ♦ ❝♦♥❥✉♥t♦ {−5,−1,1,5}✳
✷✮ {x|x é ✐♥t❡✐r♦ ❡ 0≤x≤200} é ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ✐♥❞✐❝❛r ♦ ❝♦♥❥✉♥t♦✿ {0,1,2,3,4,5, . . . ,198,199,200}✳
❆ r❡❧❛çã♦ ❡♥tr❡ ❝♦♥❥✉♥t♦s ❝❤❛♠❛✲s❡ r❡❧❛çã♦ ❞❡ ❝♦♥t✐♥ê♥❝✐❛✳ P❛r❛ ✐♥❞✐❝❛r ❡st❡ t✐♣♦ ❞❡ r❡❧❛çã♦✱ ✉t✐❧✐③❛♠♦s ♦s sí♠❜♦❧♦s ⊂ ✭❡stá ❝♦♥t✐❞♦✮ ❡6⊂ ✭♥ã♦ ❡stá ❝♦♥t✐❞♦✮✳
❊s❝r❡✈❡♠♦s A ⊂ B✱ ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦ A ❡stá ❝♦♥t✐❞♦ ♥✉♠ ❝♦♥❥✉♥t♦ B✳ P❛r❛ ✐♥❞✐❝❛r q✉❡ ✉♠ ❝♦♥❥✉♥t♦A♥ã♦ ❡stá ❝♦♥t✐❞♦ ♥✉♠ ❝♦♥❥✉♥t♦ B✱ ❡s❝r❡✈❡♠♦sA6⊂B✳
P❛r❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ ♦✉tr♦ é ♥❡❝❡ssár✐♦ q✉❡ t♦❞♦s ♦s s❡✉s ❡❧❡♠❡♥t♦s ♣❡rt❡♥ç❛♠ t❛♠❜é♠ ❛ ❡ss❡ ♦✉tr♦ ❝♦♥❥✉♥t♦✳ ◗✉❛♥❞♦ A ⊂ B✱ ❞✐③❡♠♦s q✉❡ A é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ B ♦✉ ✉♠❛ ♣❛rt❡ ❞❡ B✳ ❆ ♥♦t❛çã♦ B ⊃ A ♣❛r❛ ✐♥❞✐❝❛r q✉❡ A é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ B t❛♠❜é♠ é ✉t✐❧✐③❛❞❛ ❡ ❧ê✲s❡✿ B ❝♦♥té♠ A✳
P❛r❛ ❡①❡♠♣❧✐✜❝❛r ❛ r❡❧❛çã♦ ❞❡ ❝♦♥t✐♥ê♥❝✐❛✱ ✈❡❥❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❱ ❞❛s ✈♦❣❛✐s ❡stá ❝♦♥t✐❞♦ ❡♠ ❆ ♦ ❝♦♥❥✉♥t♦ ❛❧❢❛❜❡t♦ ✐♥❣❧ês❀ s✐♠❜♦❧✐❝❛♠❡♥t❡✿ V ⊂ A✳ ▲❡♠❜r❛♠♦s
❈♦♥❥✉♥t♦s ✶✸
❖♣❡r❛çõ❡s ❡♥tr❡ ❝♦♥❥✉♥t♦s
❆ r❡✉♥✐ã♦ ♦✉ ✉♥✐ã♦ ❞♦s ❝♦♥❥✉♥t♦sA❡B é ♦ ❝♦♥❥✉♥t♦A∪B✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❡A ♠❛✐s ♦s ❡❧❡♠❡♥t♦s ❞❡B✳ P♦rt❛♥t♦✱ ❛✜r♠❛r q✉❡ x∈A∪B s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ❛✜r♠❛çõ❡s s❡❣✉✐♥t❡s é ✈❡r❞❛❞❡✐r❛✿ x ∈ A ♦✉ x ∈ B✳ P♦❞❡♠♦s✱ ♣♦✐s✱
❡s❝r❡✈❡r✿
A∪B ={x|x∈A ♦✉ x∈B}. ■✮ ❈♦♥s✐❞❡r❡♠♦s ♦s ❝♦♥❥✉♥t♦s
A={x|x é ❞✐✈✐s♦r ❞❡ ✸}
❡
B ={x|x é ❞✐✈✐s♦r ❞❡ ✺}.
❊♥tã♦✱A∪B ={x|x é ❞✐✈✐s♦r ❞❡ ✸ ♦✉ x é ❞✐✈✐s♦r ❞❡ ✺}✱ ♦✉ s❡❥❛✱A∪B ={−5,−3,−1,1,3,5}✳
❆ ✐♥t❡rs❡çã♦ ❞♦s ❝♦♥❥✉♥t♦s A ❡ B é ♦ ❝♦♥❥✉♥t♦ A∩B✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❝♦♠✉♥s ❛ A ❡ B✳ ❆ss✐♠✱ ❛✜r♠❛r q✉❡x ∈A∩B s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ s❡ t❡♠✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ x∈A ❡ x∈B✳ P♦❞❡♠♦s✱ ♣♦rt❛♥t♦✱ ❡s❝r❡✈❡r✿
A∪B ={x|x∈A ❡ x∈B}. ❈♦♥s✐❞❡r❛♥❞♦ A ❡ B ❝♦♠♦ ❡♠ ✭■✮✱ t❡♠♦sA∩B ={−1,1}✳
P♦❞❡ ♦❝♦rr❡r q✉❡ ♥ã♦ ❡①✐st❛ ❡❧❡♠❡♥t♦ ❛❧❣✉♠ x t❛❧ q✉❡ x∈A ❡ x∈B✳ ◆❡st❡ ❝❛s♦✱ t❡♠✲s❡ A∩B =∅ ❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B sã♦ ❞✐t♦s ❞✐s❥✉♥t♦s✳
■■✮ ❙❡❥❛♠V ♦ ❝♦♥❥✉♥t♦ ❞❛s ✈♦❣❛✐s ❡C ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝♦♥s♦❛♥t❡s ❞♦ ❛❧❢❛❜❡t♦ ✐♥❣❧ês✳ ➱ ❝❧❛r♦ q✉❡ V ∩C =∅✳ ❖✉ s❡❥❛✱ V ❡C sã♦ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s✳
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦sA❡B é ♦ ❝♦♥❥✉♥t♦A−B✱ ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❡ A q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ ❛B✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ t❡♠♦s✿
A−B ={x|x∈A ❡ x6∈B}.
➱ ✐♠♣♦rt❛♥t❡ ♣❡r❝❡❜❡r q✉❡ ♥ã♦ s❡ ❡①✐❣❡ q✉❡B ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ A ♣❛r❛ ❢♦r♠❛r ❛ ❞✐❢❡r❡♥ç❛ A−B✳ ◗✉❛♥❞♦ A ❡ B sã♦ ❞✐s❥✉♥t♦s✱ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡ A ♣❡rt❡♥❝❡ ❛ B✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ A−B =A✳ ❊♠ q✉❛❧q✉❡r ❝❛s♦✱ t❡♠♦s✿ A−B =A−(A∩B)✳
◗✉❛♥❞♦ t❡♠♦sB ⊂ A✱ ❛ ❞✐❢❡r❡♥ç❛ A−B é ❞✐t❛ ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ B ❡♠ r❡❧❛çã♦ ❛ A ❡ ❡s❝r❡✈❡♠♦s✿
A−B =CAB.
◆♦t❡♠♦s q✉❡x∈ CAB s❡✱ ❡ s♦♠❡♥t❡ s❡✱x6∈B✳
■■■✮ ❙❡❥❛♠ A = {x ∈ Z;x ≥ −2} ❡ B = {x ∈ Z;x ≤ 1}✳ ❊♥tã♦ A−B = {x ∈
Z;x≥2} ❡B−A ={x∈Z;x≤ −3}✳
❆ ♥♦çã♦ ❞❡ ❞✐❢❡r❡♥ç❛ r❡❞✉③✲s❡ à ❞❡ ❝♦♠♣❧❡♠❡♥t❛r✱ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ❞❛❞♦s ❝♦♥✲ ❥✉♥t♦s A ❡ B✱ ❝♦♥t✐❞♦s ♥✉♠ ❝♦♥❥✉♥t♦ ❢✉♥❞❛♠❡♥t❛❧ E✱ r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ q✉❛❧ t♦♠❛♠♦s ❝♦♠♣❧❡♠❡♥t❛r❡s✱ t❡♠♦s✿
❋✉♥çõ❡s ✶✹
❉❡ ❢❛t♦✱ x∈A−B ⇔x∈A ❡ x6∈B ⇔x∈A ❡ x∈ CEB ⇔x∈A∩ CEB✳
❙❡❥❛♠ ❆ ❡ ❇ ❝♦♥❥✉♥t♦s ❡ ❞❛❞♦s ♦s ♦❜❥❡t♦s a, b✱ ♣❡rt❡♥❝❡♥t❡s ❛ ❆ ❡ ❇✱ r❡s♣❡❝t✐✈❛✲ ♠❡♥t❡✱ ♦ ♣❛r ♦r❞❡♥❛❞♦(a, b)✜❝❛ ❢♦r♠❛❞♦ q✉❛♥❞♦ s❡ ❡s❝♦❧❤❡ ✉♠ ❞❡ss❡s ♦❜❥❡t♦s ♣❛r❛ s❡r
❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛✱ ♥❡st❡ ❝❛s♦a✱ ❡ ♦ ♦❜❥❡t♦b♣❛r❛ s❡r ❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣❛r✳ ❉♦✐s ♣❛r❡s ♦r❞❡♥❛❞♦s (a, b)❡ (a′
, b′
) sã♦ ✐❣✉❛✐s q✉❛♥❞♦ s✉❛s ♣r✐♠❡✐r❛s ❝♦♦r❞❡♥❛❞❛s✱a ❡ a′✱ sã♦ ✐❣✉❛✐s ❡ s✉❛s s❡❣✉♥❞❛s ❝♦♦r❞❡♥❛❞❛s✱
b ❡ b′✱ t❛♠❜é♠✳ ❙❡♥❞♦ ❛ss✐♠ (a, b) = (a′
, b′
)⇔a=a′ ❡
b =b′
.
❖ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞♦s ❝♦♥❥✉♥t♦sA ❡ B é ♦ ❝♦♥❥✉♥t♦ A×B ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ t♦❞♦s ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (a, b) ❝✉❥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ ♣❡rt❡♥❝❡ ❛ A ❡ ❛ s❡❣✉♥❞❛ ❝♦♦r❞❡♥❛❞❛ ♣❡rt❡♥❝❡ ❛ B✳ ❉❡ss❡ ♠♦❞♦✿
A×B ={(a, b)|a∈A ❡ b∈B}. ■❱✮ ❙❡❥❛♠A={1,2,3} ❡B ={4,5}✳ ❊♥tã♦
A×B ={(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}.
✶✳✷ ❋✉♥çõ❡s
❯♠❛ ❢✉♥çã♦f :A →B ❝♦♥st❛ ❞❡ três ♣❛rt❡s✿ ✉♠ ❝♦♥❥✉♥t♦ A✱ ❝❤❛♠❛❞♦ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✱ ♦✉ ♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ❛ ❢✉♥çã♦ ❡stá ❞❡✜♥✐❞❛✱ ✉♠ ❝♦♥❥✉♥t♦ B✱ ❝❤❛♠❛❞♦ ❝♦♥✲ tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✱ ♦✉ ♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ❛ ❢✉♥çã♦ t♦♠❛ ✈❛❧♦r❡s✱ ❡ ✉♠❛ ❧❡✐ q✉❡ ♣❡r♠✐t❡ ❛ss♦❝✐❛r ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ x ∈ A✱ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ f(x) ∈ B✱ ❝❤❛♠❛❞♦ ♦ ✈❛❧♦r q✉❡ ❛ ❢✉♥çã♦ ❛ss✉♠❡ ❡♠ x ✭♦✉ ♥♦ ♣♦♥t♦ x✮✳ ❆ ❧❡✐ q✉❡ ♣❡r♠✐t❡ ♦❜t❡r ♦ ✈❛❧♦r f(x) ∈ B✱ q✉❛♥❞♦ é ❞❛❞♦ x∈A✱ é ❛r❜✐trár✐❛✱ ♣♦ré♠ s✉❥❡✐t❛ ❛ ❞✉❛s ❝♦♥❞✐çõ❡s✿
• ◆ã♦ ❞❡✈❡ ❤❛✈❡r ❡①❝❡çõ❡s✿ ❛ ✜♠ ❞❡ q✉❡ f t❡♥❤❛ ♦ ❝♦♥❥✉♥t♦ A ❝♦♠♦ ❞♦♠í♥✐♦✱ ❛ r❡❣r❛ ❞❡✈❡ ❢♦r♥❡❝❡r f(x) ♣❛r❛ t♦❞♦x∈A❀
• ◆ã♦ ❞❡✈❡ ❤❛✈❡r ❛♠❜✐❣✉✐❞❛❞❡s✿ ❛ ❝❛❞❛x∈A✱ ❛ r❡❣r❛ ❞❡✈❡ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ✉♠
ú♥✐❝♦ f(x)∈B✳
❯s❛♠♦s ❛ ♥♦t❛çã♦ x7→f(x) ♣❛r❛ ✐♥❞✐❝❛r q✉❡ f ❢❛③ ❝♦rr❡s♣♦♥❞❡r ❛ x♦ ✈❛❧♦r f(x)✳
◆ã♦ s❡ ♣♦❞❡ ❝♦♥❢✉♥❞✐rf ❝♦♠ f(x)✱ ♣♦✐s f é ❛ ❢✉♥çã♦✱ ❡♥q✉❛♥t♦ q✉❡ f(x) é ♦ ✈❛❧♦r
q✉❡ ❛ ❢✉♥çã♦ ❛ss✉♠❡ ♥✉♠ ♣♦♥t♦ x❞♦ s❡✉ ❞♦♠í♥✐♦✳
❉✉❛s ❢✉♥çõ❡sf :A→B ❡f′ :A′ →B′ sã♦ ✐❣✉❛✐s s❡✱ ❡ s♦♠❡♥t❡ s❡✱A=A′✱B =B′
❡ f(x) =f′(x) ♣❛r❛ t♦❞♦ x∈A✳ ■st♦ é✱ ❞✉❛s ❢✉♥çõ❡s sã♦ ✐❣✉❛✐s q✉❛♥❞♦ tê♠ ♦ ♠❡s♠♦
❞♦♠í♥✐♦✱ ♦ ♠❡s♠♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❡ ❛ ♠❡s♠❛ r❡❣r❛ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛✳
❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦f :A→B é ♦ s✉❜❝♦♥❥✉♥t♦G(f)❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦A×B ❢♦r♠❛❞♦ ♣❡❧♦s ♣❛r❡s ♦r❞❡♥❛❞♦s (x, f(x))✱ ♦♥❞❡ x∈A é ❛r❜✐trár✐♦✳ ❖✉ s❡❥❛✱
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✶✺
❯♠❛ ❢✉♥çã♦f :A→B é ❞✐t❛ ✐♥❥❡t✐✈❛ ✭♦✉ ❜✐✉♥í✈♦❝❛✮ q✉❛♥❞♦✱ ❞❛❞♦s x, y q✉❛✐sq✉❡r ❡♠ A✱ f(x) =f(y) ✐♠♣❧✐❝❛ x= y✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ q✉❛♥❞♦x 6=y✱ ❡♠A✱ ✐♠♣❧✐❝❛
f(x)6=f(y)✱ ❡♠ B✳
❯♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ ❢✉♥çã♦ ✐♥❥❡t✐✈❛ é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ f :Z →Z ❞❛❞❛ ♣♦r f(x) =x ♣❛r❛ t♦❞♦x∈Z✳ ❈❧❛r❛♠❡♥t❡ f(x) = f(y)⇔x=y✳
❯♠❛ ❢✉♥çã♦ f : A→ B é ❞✐t❛ s♦❜r❡❥❡t✐✈❛ ✭♦✉ s♦❜r❡ B✮ q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ y ∈B✱
❡①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ x∈A t❛❧ q✉❡ f(x) = y✳
❈♦♥s✐❞❡r❛♥❞♦Q♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ♦✉ s❡❥❛✱Q=
p
q|p∈Z, q∈Z ❡q 6= 0
✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r✱ ❢❛❝✐❧♠❡♥t❡✱ q✉❡ ❛ ❢✉♥çã♦f :Z×Z∗ →Q❞❛❞❛ ♣♦r f(p, q) =
p q✱ ♣❛r❛ t♦❞♦ (p, q)∈Z×Z∗✱ é s♦❜r❡❥❡t✐✈❛✱ ♦♥❞❡ Z∗ =Z− {0}✳
❆❣♦r❛✱ s❡❥❛ f : Z → Z ❞❡✜♥✐❞❛ ♣❡❧❛ ❧❡✐ f(x) = x2 ♣❛r❛ t♦❞♦ x ∈ Z✳ ❖❜s❡r✈❡♠♦s
q✉❡ f ♥ã♦ é ✐♥❥❡t✐✈❛✱ ♣♦✐s f(−2) = f(2)✱ ❡♠❜♦r❛ −2 6= 2✳ ❚❛♠♣♦✉❝♦ f é s♦❜r❡❥❡t✐✈❛✱ ❛✜♥❛❧ ♥ã♦ ❡①✐st❡x∈Zt❛❧ q✉❡x2 =−2✱ ♣♦r ❡①❡♠♣❧♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s
g :Z →Z✱ ❞❡✜♥✐❞❛ ♣♦r g(x) = 2x+ 1✱ ❡♥tã♦g é ✐♥❥❡t✐✈❛✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ g(x) =g(y)
❡♥tã♦ 2x+ 1 = 2y+ 1✱ ♦✉ s❡❥❛✱2x= 2y✱ ❞♦♥❞❡x=y✳ P♦ré♠✱ g ♥ã♦ é s♦❜r❡❥❡t✐✈❛✱ ♣♦✐s ♥ã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ x t❛❧ q✉❡ 2x+ 1 = 0✱ ♣♦r ❡①❡♠♣❧♦✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s
P = {2n|n ∈ Z+∗} ✭P é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♣❛r❡s✮✱ ♦♥❞❡ Z +
∗ =
{z ∈Z|z >0}✱ ❡ ❞❡✜♥✐♠♦s h:Z+∗ →P ♣♦♥❞♦ h(2n) = 2n ❡ h(2n−1) = 2n ♣❛r❛ t♦❞♦
n ∈Z+∗✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ h é s♦❜r❡❥❡t✐✈❛✱ t♦❞❛✈✐❛h ♥ã♦ é ✐♥❥❡t✐✈❛✳
❯♠❛ ❢✉♥çã♦ f : A → B é ❞✐t❛ ❜✐❥❡t✐✈❛ ✭✉♠❛ ❜✐❥❡çã♦✱ ♦✉ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛✮ q✉❛♥❞♦ é ✐♥❥❡t✐✈❛ ❡ s♦❜r❡❥❡t✐✈❛ ❝♦♥❝♦♠✐t❛♥t❡♠❡♥t❡✳
❆ ♠❛✐s s✐♠♣❧❡s ❞❛s ❜✐❥❡çõ❡s é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ s✉♣r❛♠❡♥❝✐♦♥❛❞❛✳
❉❛❞♦s ✉♠❛ ❢✉♥çã♦f :A→B ❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ X ⊂A✱ ❝❤❛♠❛✲s❡ ❛ ✐♠❛❣❡♠ ❞❡X ♣❡❧❛ ❢✉♥çã♦ f ❛♦ ❝♦♥❥✉♥t♦ f(X) ❢♦r♠❛❞♦ ♣❡❧♦s ✈❛❧♦r❡s f(x) q✉❡ f ❛ss✉♠❡ ♥♦s ♣♦♥t♦s x∈X✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✿
f(X) ={f(x)|x∈X}={y∈B|y =f(x), x∈X}.
❊✈✐❞❡♥t❡♠❡♥t❡✱f(X)é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡B✳ P❛r❛ q✉❡f :A→B s❡❥❛ s♦❜r❡❥❡t✐✈❛ é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ ✶ q✉❡ f(A) =B✳
✶✳✸ ❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s
❖ ✐♥✜♥✐t♦ r❡✈❡❧❛✲s❡ ❛❧❣♦ ✐♥❡①♣❧✐❝á✈❡❧ ♣❛r❛ ♠✉✐t♦s✳ ■♠❛❣✐♥❛r q✉❡ ❡①✐st❡♠ ❞✐❢❡r❡♥t❡s ✐♥✜♥✐t♦s ♣♦❞❡ s❡r ✉♠❛ q✉❡stã♦ ✐♥tr✐❣❛♥t❡ ♣❛r❛ ✉♠ ❛❞♦❧❡s❝❡♥t❡✳ P♦r ✐ss♦✱ s✉❣❡r✐♠♦s
✶✏é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡✑ ♦✉ ✏s❡ ❡ s♦♠❡♥t❡ s❡✑ ❡①♣r❡ss❛♠ ❛ ♠❡s♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ❛♣❛r❡❝❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ♥♦ t❡①t♦✳ ❖ q✉❡ ❛ ♦r❛çã♦ ❞❡❝❧❛r❛t✐✈❛ ❛❝✐♠❛ q✉❡r ❞✐③❡r é✿ s❡ f :A →B é
s♦❜r❡❥❡t✐✈❛ ❡♥tã♦f(A) =B ❡ s❡ f(A) =B ❡♥tã♦ f :A→B é s♦❜r❡❥❡t✐✈❛✳ ❙❡ ♦ ❧❡✐t♦r✲❛❧✉♥♦ ♥ã♦ t✐✈❡r
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✶✻
q✉❡ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♥t❡ ❛♦ ❛❧✉♥♦ q✉❡ ♥ã♦ ❤á s♦♠❡♥t❡ ✉♠ t✐♣♦ ❞❡ ✐♥✜♥✐t♦ ✲ ❞❡s❝♦❜❡rt❛ ❡ss❛ q✉❡ ❢♦✐ ❢❡✐t❛ ♣♦r ●❡♦r❣ ❈❛♥t♦r ❡♠ ✶✽✼✹ ✲ q✉❡ é ♣♦ssí✈❡❧ ❞✐st✐♥❣✉í✲❧♦s q✉❛♥t♦ ❛♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s✱ ♣♦r ♠❛✐s ❡str❛♥❤♦ q✉❡ ✐ss♦ ♣♦ss❛ ♣❛r❡❝❡r✱ ❡ ❛♣r❡s❡♥t❡ ❛s ♥♦çõ❡s ❞❡ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s ❡ ♥ã♦✲❡♥✉♠❡rá✈❡✐s✱ ♠❡s♠♦ q✉❡ ❝♦♠ ♣♦✉❝♦ r✐❣♦r ♠❛t❡♠át✐❝♦✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛ ♦ ♥í✈❡❧ ❜ás✐❝♦ ❞❡ ❡♥s✐♥♦✳ ❆ ♣❛rt✐r ❞♦s ❝♦♥❝❡✐t♦s ❣❡r❛✐s ❡ ❞♦s ❢❛t♦s ❜ás✐❝♦s ❛ r❡s♣❡✐t♦ ❞❡ ❝♦♥❥✉♥t♦s ❡ ❢✉♥çõ❡s✱ ✈✐st♦s ♥❛s s❡çõ❡s ❛♥t❡r✐♦r❡s✱ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♣♦❞❡✱ tr❛♥q✉✐❧❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛r ❛s ♥♦çõ❡s ❞❡ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❡ ✐♥✜♥✐t♦ ❛♦s s❡✉s ❛❧✉♥♦s✳
◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♥♦çõ❡s ❞❡ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❡♥✉✲ ♠❡rá✈❡❧ ❡ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ♥ã♦✲❡♥✉♠❡rá✈❡❧✳ ❙❡r❡♠♦s ❝♦♥❝✐s♦s ❡ ♥ã♦ ♥♦s ❛t❡r❡♠♦s à ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❢❛t♦s r❡❧❡✈❛♥t❡s✳ ❆❣✐♥❞♦ ❛ss✐♠✱ t❡♠♦s ❛ ♣r❡t❡♥sã♦ ❞❡ q✉❡ ✉♠ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♥s✐❣❛ ❡♥t❡♥❞❡r ❡st❛ s❡çã♦✱ ❜❡♠ ❝♦♠♦ t♦❞♦ ♦ tr❛❜❛❧❤♦ ♣r❡s❡♥t❡✳ ❈♦♥t✉❞♦✱ q✉❛♥❞♦ ♥ã♦ ❞❡♠♦♥str❛r♠♦s ✉♠ ❢❛t♦ ✐♠♣♦rt❛♥t❡✱ ✐♥❞✐❝❛r❡♠♦s ✉♠❛ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦s ❧❡✐t♦r❡s ✐♥t❡r❡ss❛❞♦s✳
❆ ♥♦çã♦ ❞❡ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧ ❡stá ❡str✐t❛♠❡♥t❡ ❧✐❣❛❞❛ ❛♦ ❝♦♥❥✉♥t♦N❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ P♦r ✐ss♦✱ ✐♥✐❝✐❛r❡♠♦s ❡st❛ s❡çã♦ ❢❛③❡♥❞♦ ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❛ ♣❛rt✐r ❞♦s ❛①✐♦♠❛s ❞❡ P❡❛♥♦✳
✶✳✸✳✶ ◆ú♠❡r♦s ♥❛t✉r❛✐s
❆ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦❞❡ s❡r ❞❡❞✉③✐❞❛ ❞♦s três ❛①✐♦♠❛s✷ ❛❜❛✐①♦✱ ❝♦✲ ♥❤❡❝✐❞♦s ❝♦♠♦ ❛①✐♦♠❛s ❞❡ P❡❛♥♦ ✸✳
❙ã♦ ❞❛❞♦s✱ ❝♦♠♦ ♦❜❥❡t♦s ♥ã♦✲❞❡✜♥✐❞♦s✱ ✉♠ ❝♦♥❥✉♥t♦N✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❝❤❛♠❛✲ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ ❡ ✉♠❛ ❢✉♥çã♦ s : N → N✳ P❛r❛ ❝❛❞❛ n ∈ N✱ ♦ ♥ú♠❡r♦ s(n)✱
✈❛❧♦r q✉❡ ❛ ❢✉♥çã♦ ❛ss✉♠❡ ♥♦ ♣♦♥t♦ n✱ é ❝❤❛♠❛❞♦ ♦ s✉❝❡ss♦r ❞❡ n✳ ❆ ❢✉♥çã♦ s s❛t✐s❢❛③ ❛♦s s❡❣✉✐♥t❡s ❛①✐♦♠❛s✿
✭P✶✮ s :N→N é ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t✐✈❛✱ ♦✉ s❡❥❛✱m, n∈N✱ s(m) =s(n)⇒m=n✳ ❊♠
♣❛❧❛✈r❛s✱ ❞♦✐s ♥ú♠❡r♦s q✉❡ tê♠ ♦ ♠❡s♠♦ s✉❝❡ss♦r sã♦ ✐❣✉❛✐s✳
✭P✷✮ ❊①✐st❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ ♥❛t✉r❛❧✱ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ 1✱ ❞❡♥♦♠✐♥❛❞♦ ✉♠✱
q✉❡ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥❡♥❤✉♠ ♦✉tr♦ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❊♠ ♦✉tr♦s t❡r♠♦s✿ N−s(N) = {1}.
✭P✸✮ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦✿ ❙❡ X ⊂ N é ✉♠ s✉❜❝♦♥❥✉♥t♦ t❛❧ q✉❡ 1 ∈X ❡✱ ♣❛r❛ t♦❞♦ n ∈X t❡♠✲s❡ t❛♠❜é♠ s(n)∈X✱ ❡♥tã♦ X =N✳
✷◆❛ ❧ó❣✐❝❛ tr❛❞✐❝✐♦♥❛❧✱ ✉♠ ❛①✐♦♠❛ ♦✉ ♣♦st✉❧❛❞♦ é ✉♠❛ s❡♥t❡♥ç❛ ♦✉ ♣r♦♣♦s✐çã♦ q✉❡ ♥ã♦ é ♣r♦✈❛❞❛ ♦✉ ❞❡♠♦♥str❛❞❛ ❡ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ó❜✈✐❛ ♦✉ ❝♦♠♦ ✉♠ ❝♦♥s❡♥s♦ ✐♥✐❝✐❛❧ ♥❡❝❡ssár✐♦ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ♦✉ ❛❝❡✐t❛çã♦ ❞❡ ✉♠❛ t❡♦r✐❛✳
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✶✼
❖ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ♣♦❞❡ t❛♠❜é♠ s❡r ❡♥✉♥❝✐❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❙❡❥❛P ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ r❡❧❛t✐✈❛ ❛ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❙❡1❣♦③❛r ❞❛ ♣r♦♣r✐❡❞❛❞❡P
❡ s❡✱ ❞♦ ❢❛t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❣♦③❛r ❞❡ P ♣✉❞❡r✲s❡ ❝♦♥❝❧✉✐r q✉❡ n+ 1 ❣♦③❛ ❞❛
♣r♦♣r✐❡❞❛❞❡ P✱ ❡♥tã♦ t♦❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❣♦③❛♠ ❞❡ss❛ ♣r♦♣r✐❡❞❛❞❡✳
❯♠❛ ❞❡♠♦♥str❛çã♦ ♥❛ q✉❛❧ ♦ ❛①✐♦♠❛(P3)é ✉t✐❧✐③❛❞♦✱ ❝❤❛♠❛✲s❡ ❞❡♠♦♥str❛çã♦ ♣♦r
✐♥❞✉çã♦✳
P❛r❛ ✐❧✉str❛r ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦✱ ♠♦str❛r❡♠♦s q✉❡s(n)6=n♣❛r❛ t♦❞♦ n ∈N✳ ❉❡ ❢❛t♦✱ s❡❥❛X ={n ∈N|s(n)6=n}✳ ❈♦♠♦ ✶ ♥ã♦ é s✉❝❡ss♦r ❞❡ ♥ú♠❡r♦ ❛❧❣✉♠✲
❡♠ ♣❛rt✐❝✉❧❛r 1 6= s(1) ✲ t❡♠✲s❡ 1 6∈ X✳ ❆❧é♠ ❞✐ss♦✱ s❡ n ∈ X ❡♥tã♦ s(n) 6= n✳ P❡❧❛
✐♥❥❡t✐✈✐❞❛❞❡ ❞❡ s✭❛①✐♦♠❛(P1)✮✱ t❡♠✲s❡s(s(n))6=s(n)✳ ■st♦ ♥♦s ♠♦str❛ q✉❡s(n)∈X✳
❆ss✐♠✱ n∈X ⇒s(n)∈X✳ ❈♦♠♦1∈X✱ s❡❣✉❡ ❞♦ ❆①✐♦♠❛ (P3) q✉❡ X =N✱ ♦✉ s❡❥❛✱ n 6=s(n) ♣❛r❛ t♦❞♦n ∈N✳
❆❞♠✐t❛♠♦s q✉❡✱ ❞❛❞❛ f : X → X ✉♠❛ ❢✉♥çã♦ ❝✉❥♦ ❞♦♠í♥✐♦ ❡ ❝♦♥tr❛❞♦♠í♥✐♦ sã♦ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ X✱ ♣♦❞❡♠♦s ❛ss♦❝✐❛r✱ ❞❡ ♠♦❞♦ ú♥✐❝♦✱ ❛ ❝❛❞❛ n ∈ N ✉♠❛ ❢✉♥çã♦ fn : X → X✱ ❝❤❛♠❛❞❛ ❛ n✲és✐♠❛ ✐t❡r❛❞❛ ❞❡ f✱ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ f1 = f ❡
fs(n) =f◦fn✳
❯s❛♥❞♦ ❛s ✐t❡r❛❞❛s ❞❛ ❢✉♥çã♦s:N→N✱ ❞❡✜♥✐r❡♠♦s ❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❉❛❞♦s m, n∈N✱ s✉❛ s♦♠❛ m+n ∈N é ❞❡✜♥✐❞❛ ♣♦r✿
m+n=sn(m).
❖✉ s❡❥❛✱ s♦♠❛r m ❝♦♠ 1 é t♦♠❛r ♦ s✉❝❡ss♦r ❞❡ m✱ ❡♥q✉❛♥t♦ q✉❡ s♦♠❛r m ❝♦♠ n é ♣❛rt✐r ❞❡ m ❡ ✐t❡r❛r n ✈❡③❡s ❛ ♦♣❡r❛çã♦ ❞❡ t♦♠❛r ♦ s✉❝❡ss♦r✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠♦s✱ ♣♦r ❞❡✜♥✐çã♦✿
m+ 1 = s(m)
m+s(n) = s(m+n).
P♦❞❡♠♦s ❞✐s♣❡♥s❛r ❛ ♥♦t❛çã♦s(n)♣❛r❛ r❡♣r❡s❡♥t❛r ♦ s✉❝❡ss♦r ❞❡n❡ ✉s❛r ❛ ♥♦t❛çã♦ ❞❡✜♥✐t✐✈❛ n+ 1 ♣❛r❛ ✐♥❞✐❝❛r ❡st❡ s✉❝❡ss♦r✳
❆s ♣r♦♣r✐❡❞❛❞❡s ❢♦r♠❛✐s ❞❛ ❛❞✐çã♦ ❡stã♦ ❧✐st❛❞❛s ❛❜❛✐①♦✿
• ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (m+n) +p=m+ (n+p)✱ ♣❛r❛m, n, p ∈N❀
• ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ m+n=n+m✱ ♣❛r❛m, n∈N❀
• ▲❡✐ ❞♦ ❈♦rt❡✿ m+n =m+p⇒n=p✱ ♣❛r❛m, n, p ∈N❀
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✶✽
❆s ❞❡♠♦♥str❛çõ❡s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛❝✐♠❛ sã♦ ❢❡✐t❛s ♣♦r ✐♥❞✉çã♦✳
❆ r❡❧❛çã♦ ❞❡ ♦r❞❡♠ ❡♥tr❡ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ❞❡✜♥✐❞❛ ❡♠ t❡r♠♦s ❞❛ ❛❞✐çã♦✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐sm, n✱ ❞✐③❡♠♦s q✉❡ m é ♠❡♥♦r ❞♦ q✉❡ n ❡ ❡s❝r❡✈❡♠♦s
m < n,
♣❛r❛ s✐❣♥✐✜❝❛r q✉❡ ❡①✐st❡ p ∈ N t❛❧ q✉❡ n = m+p✳ ◆❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s
q✉❡ n é ♠❛✐♦r ❞♦ q✉❡ m✳ ❆ ♥♦t❛çã♦ m≤n ❡①♣r❡ss❛ q✉❡m é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ n✳
❆ r❡❧❛çã♦ ❞❡ ♦r❞❡♠< ❣♦③❛ ❞❡ três ♣r♦♣r✐❡❞❛❞❡s✱ sã♦ ❡❧❛s✿
• ❚r❛♥s✐t✐✈✐❞❛❞❡✿ s❡ m < n ❡ n < p❡♥tã♦ m < p✱ ♣❛r❛ m, n, p∈N❀
• ❚r✐❝♦t♦♠✐❛✿ ❞❛❞♦s m, n∈N✱ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❞❛s três ❛❧t❡r♥❛t✐✈❛s ♣♦❞❡ ♦❝♦rr❡r✿ ♦✉ m=n✱ ♦✉ m < n ♦✉ n < m✳
• ▼♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛ ❛❞✐çã♦✿ s❡ m, n ∈ N sã♦ t❛✐s q✉❡ m < n ❡♥tã♦✱ ♣❛r❛ t♦❞♦ p∈N✱ t❡♠✲s❡m+p < n+p✳
■♥tr♦❞✉③✐r❡♠♦s ❛❣♦r❛ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ P❛r❛ ❝❛❞❛m ∈N✱ s❡❥❛ fm : N → N ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r fm(p) = p+m✱ ♦✉ s❡❥❛✱ fm é ❛ ❢✉♥çã♦ ✏s♦♠❛r ♣✑✳
❯t✐❧✐③❛r❡♠♦s ❡st❛ ❢✉♥çã♦ ♣❛r❛ ❞❡✜♥✐r ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
❖ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ m.1 = m ❡ m.(n+ 1) = (fm)n(m)✳ ❱❛♠♦s ❡♥t❡♥❞❡r ♠❡❧❤♦r ❡st❛ ❞❡✜♥✐çã♦✳ P♦✐s ❜❡♠✱ ♠✉❧t✐♣❧✐❝❛r
✉♠ ♥ú♠❡r♦m♣♦r1♥ã♦ ♦ ❛❧t❡r❛✳ ▼✉❧t✐♣❧✐❝❛rm♣♦r ✉♠ ♥ú♠❡r♦ ♠❛✐♦r ❞♦ q✉❡1✱ ♦✉ s❡❥❛✱
♣♦r ✉♠ ♥ú♠❡r♦ ❞❛ ❢♦r♠❛n+1✱ é ✐t❡r❛rn✲✈❡③❡s ❛ ♦♣❡r❛çã♦ ❞❡ s♦♠❛rm✱ ❝♦♠❡ç❛♥❞♦ ❝♦♠ m✳ ❉❡ss❡ ♠♦❞♦✱ ♣♦r ❡①❡♠♣❧♦✱m.2 = fm(m) =m+m✱m.3 = (fm)2(m) =m+m+m✳
❘❡❝♦r❞❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ (fm)n✱ ✈❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ m.(n + 1) ❡stá ❞❡✜♥✐❞♦
✐♥❞✉t✐✈❛♠❡♥t❡ ♣❡❧❛s ♣r♦♣r✐❡❞❛❞❡s ❛ s❡❣✉✐r✿ m.1 = m
m.(n+ 1) = m.n+m.
❆s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stã♦ ❧✐st❛❞❛s ❛❜❛✐①♦✿
• ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (m.n).p=m.(n.p)✱ ♣❛r❛m, n, p∈N❀
• ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ m.n=n.m✱ ♣❛r❛ m, n∈N❀
• ▲❡✐ ❞♦ ❈♦rt❡✿ m.p=n.p⇒m=n✱ ♣❛r❛ m, n, p∈N❀
• ❉✐str✐❜✉t✐✈✐❞❛❞❡✿ m.(n+p) =m.n+m.p✱ ♣❛r❛m, n, p ∈N❀
• ▼♦♥♦t♦♥✐❝✐❞❛❞❡✿ m < n ⇒m.p < n.p✱ ♣❛r❛ m, n, p∈N✳
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✶✾
✶✳✸✳✷ ❈♦♥❥✉♥t♦s ✜♥✐t♦s
❉❡♥♦t❛r❡♠♦s ♣♦r In ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ ✶ ❛té n✱ ♦✉ s❡❥❛✱
In={1,2,3, . . . , n}✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❞❛❞♦ n∈N✱
In ={p∈N|1≤p≤n}.
❯♠ ❝♦♥❥✉♥t♦ X é ❞✐t♦ ✜♥✐t♦ q✉❛♥❞♦ é ✈❛③✐♦ ♦✉ q✉❛♥❞♦ ❡①✐st❡✱ ♣❛r❛ ❛❧❣✉♠ n ∈N✱ ✉♠❛ ❜✐❥❡çã♦
ψ :In→X.
◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ X t❡♠ ③❡r♦ ❡❧❡♠❡♥t♦s✳ ◆♦ s❡❣✉♥❞♦ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ n∈N é ♦ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ X✱ ✐st♦ é✱ q✉❡ X ♣♦ss✉✐n ❡❧❡♠❡♥t♦s✳
❖s s❡❣✉✐♥t❡s ❢❛t♦s ❞❡❝♦rr❡♠ ❞❛s ❞❡✜♥✐çõ❡s✿
• ❝❛❞❛ ❝♦♥❥✉♥t♦ In é ✜♥✐t♦ ❡ ♣♦ss✉✐ n ❡❧❡♠❡♥t♦s❀
• s❡ f : X → Y é ✉♠❛ ❜✐❥❡çã♦✱ ✉♠ ❞❡ss❡s ❝♦♥❥✉♥t♦s é ✜♥✐t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ♦✉tr♦ t❛♠❜é♠ é✳
■♥t✉✐t✐✈❛♠❡♥t❡✱ ✉♠❛ ❜✐❥❡çã♦ ψ : In → X é ✉♠❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ X✳
P♦♥❞♦ ψ(1) = x1, ψ(2) = x2, . . . , ψ(n) = xn✱ t❡♠♦s X = {x1, x2, . . . , x3}✳ ❊st❛ é ❛
r❡♣r❡s❡♥t❛çã♦ ♦r❞✐♥ár✐❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳
◆♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦s ❛❧✉♥♦s r❡✈ê❡♠ q✉❡ ❝♦♥❥✉♥t♦s ✜♥✐t♦s sã♦ ❝♦♥✲ ❥✉♥t♦s q✉❡ ❝♦♥tê♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❤á ❞❡ s❡ ❝♦♥✈✐r q✉❡ ❛ ❞❡✜♥✐çã♦ ❛♣r❡s❡♥t❛❞❛ ❛❝✐♠❛ ♥ã♦ ♦❢❡r❡❝❡ r✐s❝♦ ❛❧❣✉♠ ❞❡ ♥ã♦ ❡♥t❡♥❞✐♠❡♥t♦ ♣♦r ♣❛rt❡ ❞♦s ♠❡s♠♦s✱ ✉♠❛ ✈❡③ q✉❡ ❡❧❡s tê♠ ❛❝❡ss♦ às ♥♦çõ❡s ❜ás✐❝❛s ❞❡ ❢✉♥çõ❡s ✲ ❡ ✐ss♦ ✐♥✲ ❝❧✉✐ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❜✐❥❡çã♦ ✲ ♥♦ ♥♦♥♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✳ P❛r❛ q✉❡ ♥ã♦ r❡st❡ ❞ú✈✐❞❛s✱ ♣♦❞❡♠♦s ✐❧✉str❛r ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ❝♦♠ ♦ ❝♦♥❥✉♥t♦ V ❞❛s ✈♦❣❛✐s✳ ❱❡❥❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ ψ ❞❡ I5 s♦❜r❡ V✳ ❈♦♠ ❡❢❡✐t♦✱ ❜❛st❛ ❞❡✜♥✐r ψ : I5 → V ♣♦♥❞♦
ψ(1) =a, ψ(2) =e, ψ(3) =i, ψ(4) =o, ψ(5) =u✳ ▲♦❣♦ V é ✜♥✐t♦ ❡ t❡♠ ✺ ❡❧❡♠❡♥t♦s✳ ❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳ ❖ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✱ ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✺❪✱ ♥♦s ❣❛r❛♥t❡ ❡st❡ ❢❛t♦✳
✏❙❡ ❡①✐st❡♠ ❞✉❛s ❜✐❥❡çõ❡s ψ :In →X ❡ φ :Im →X✱ ❞❡✈❡✲s❡ t❡r m =n✳✑
✶✳✸✳✸ ❈♦♥❥✉♥t♦s ■♥✜♥✐t♦s
✏❚ã♦ ❝♦rr❡t♦ ❡ tã♦ ❜♦♥✐t♦✱ ♦ ✐♥✜♥✐t♦ é r❡❛❧♠❡♥t❡ ✉♠ ❞♦s ❞❡✉s❡s ♠❛✐s ❧✐♥❞♦s✳✑ ✭▲❡❣✐ã♦ ❯r❜❛♥❛✮ ❯♠ ❝♦♥❥✉♥t♦ é ❞✐t♦ ✐♥✜♥✐t♦ q✉❛♥❞♦ ♥ã♦ é ✜♥✐t♦✳ P♦rt❛♥t♦✱X é ✐♥✜♥✐t♦ q✉❛♥❞♦ ♥ã♦ é ✈❛③✐♦ ♥❡♠ ❡①✐st❡✱ s❡❥❛ q✉❛❧ ❢♦r n∈N✱ ✉♠❛ ❜✐❥❡çã♦ f :In→X✳
❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s é ✐♥✜♥✐t♦✳ ❉❡ ❢❛t♦✱ ❞❛❞❛ q✉❛❧q✉❡r ❢✉♥çã♦φ:In→
N✱ ❝♦♠ n >1✱ t♦♠❡
❈♦♥❥✉♥t♦s ❋✐♥✐t♦s ❡ ■♥✜♥✐t♦s ✷✵
❊♥tã♦✱ p ∈ N ❡ p > φ(j) ♣❛r❛ t♦❞♦ j = 1,· · · , n✳ ❆ss✐♠ p 6∈ φ(In) ❡✱ ♣♦rt❛♥t♦✱ φ
♥ã♦ é s♦❜r❡❥❡t✐✈❛✳
❊♠ ✶✽✼✷✱ ♦ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦ ❘✐❝❤❛r❞ ❉❡❞❡❦✐♥❞ ❝❛r❛❝t❡r✐③♦✉ ♦s ❝♦♥❥✉♥t♦s ✐♥✜♥✐✲ t♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
✏❯♠ ❝♦♥❥✉♥t♦ X é ✐♥✜♥✐t♦ s❡✱ s❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ f :X →Y✱ ❞❡X s♦❜r❡ ✉♠❛ ♣❛rt❡ ♣ró♣r✐❛ Y ⊂X✳✑
❖❜s❡r✈❛♠♦s q✉❡Y é ✉♠❛ ♣❛rt❡ ♣ró♣r✐❛ ❞❡X s❡Y ⊂X ❡X−Y 6=∅✳ P❛r❛ ❛♥❛❧✐s❛r
❛ ❞❡♠♦♥str❛çã♦ ❞♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛✱ ✐♥❞✐❝❛♠♦s q✉❡ ♦ ❧❡✐t♦r ❝♦♥s✉❧t❡ ❛ r❡❢❡rê♥❝✐❛ ❬✺❪✳ ❉♦✐s ❛♥♦s ❞❡♣♦✐s✱ ●❡♦r❣ ❈❛♥t♦r ♠♦str♦✉ q✉❡ ♦s ✐♥✜♥✐t♦s ♥ã♦ sã♦ t♦❞♦s ✐❣✉❛✐s ❡ ♦s ❝❧❛ss✐✜❝♦✉ ❡♠ ✐♥✜♥✐t♦s ❡♥✉♠❡rá✈❡✐s ❡ ✐♥✜♥✐t♦s ♥ã♦✲❡♥✉♠❡rá✈❡✐s✳ ❱❡r❡♠♦s✱ ❛❣♦r❛✱ ❡ss❛s ❞✉❛s ♥♦çõ❡s ❞❡ ✐♥✜♥✐t♦s ❞✐❛❣♥♦st✐❝❛❞❛s ♣♦r ❈❛♥t♦r✳
❯♠ ❝♦♥❥✉♥t♦ X é ❞✐t♦ ❡♥✉♠❡rá✈❡❧ q✉❛♥❞♦ é ✜♥✐t♦ ♦✉ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ f :N→X✳ ◆❡st❡ ❝❛s♦✱ f ❝❤❛♠❛✲s❡ ✉♠❛ ❡♥✉♠❡r❛çã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡X✳ ❊s❝r❡✈❡♥❞♦ f(1) =x1✱ f(2) =x2, . . . , f(n) =xn, . . .✱ t❡♠✲s❡X ={x1, x2, x3, . . . , xn, . . .}✳
❖ ❝♦♥❥✉♥t♦Z ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s é ✐♥✜♥✐t♦ ❡♥✉♠❡rá✈❡❧✳
❈♦♠ ❡❢❡✐t♦✱ ❛ ❢✉♥çã♦ϕ :N→Z❞❡✜♥✐❞❛ ♣♦rϕ(1) = 0✱ϕ(2n) = n❡ϕ(2n+1) =−n✱
♣❛r❛ n ∈N✱ é ✉♠❛ ❜✐❥❡çã♦ ❞❡N s♦❜r❡ Z✳
❈❛♥t♦r ♠♦str♦✉ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s Q é ❡♥✉♠❡rá✈❡❧✳ ▼♦str♦✉ t❛♠❜é♠ q✉❡ ❛ r❡✉♥✐ã♦ ❞❡ ❝♦♥❥✉♥t♦s ❡♥✉♠❡rá✈❡✐s é ✉♠ ❝♦♥❥✉♥t♦ ❡♥✉♠❡rá✈❡❧✳ P❛r❛ ✈❡r ❛s ♣r♦✈❛s ❞❡st❡s r❡s✉❧t❛❞♦s✱ q✉❡ ❡①✐❣❡♠ ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ ❡st✉❞♦✱ t❛♠❜é♠ r❡❝♦♠❡♥❞❛♠♦s ❛ r❡❢❡rê♥❝✐❛ ❬✺❪✳
❯♠ ❝♦♥❥✉♥t♦X é ❞✐t♦ ♥ã♦✲❡♥✉♠❡rá✈❡❧ q✉❛♥❞♦ ♥ã♦ é ❡♥✉♠❡rá✈❡❧✱ ❝♦♠♦ ❛ ♣ró♣r✐❛ ♥♦♠❡♥❝❧❛t✉r❛ ❥á ❞✐③✳ ❖✉ s❡❥❛✱ q✉❛♥❞♦ é ✐♥✜♥✐t♦ ❡ ♥ã♦ ❡①✐st❡ ✉♠❛ ❜✐❥❡çã♦ f :N →X✳
✷ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✱ t❛♠❜é♠ ❞❡♥♦♠✐♥❛❞♦ ❝♦♥✲ ❥✉♥t♦ ❞❡ ❈❛♥t♦r ❞♦s t❡rç♦s ♠é❞✐♦s✱ ❝♦♠ ❛ ❝❧áss✐❝❛ ❝♦♥str✉çã♦ ❞❡ r❡t✐r❛❞❛s ❞❡ t❡rç♦s ♠é❞✐♦s ❛❜❡rt♦s✱ ✐♥✐❝✐❛❞❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [0,1]✳
❊♠ ✈✐rt✉❞❡ ❞♦s ♣ré✲r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❝♦♥str✉çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r ❡ ❞❛s s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❞❡❞✐❝❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ♣❛r❛ ❡st❡ ❝❛♣ít✉❧♦ sã♦ ❬✶❪✱ ❬✺❪ ❡ ❬✻❪✳
✷✳✶ Pr❡❧✐♠✐♥❛r❡s
◆❡st❛ s❡çã♦✱ ❡①♣♦r❡♠♦s✱ ❞❡ ❢♦r♠❛ s✉❝✐♥t❛✱ ❝♦♥❝❡✐t♦s ❡ ❢❛t♦s ❞❡ ❆♥á❧✐s❡ ❘❡❛❧✱ ✉t✐✲ ❧✐③❛♥❞♦ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ❛❝❡ssí✈❡❧ ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❡①✐❜✐r✱ s✉♣❡r✜❝✐❛❧♠❡♥t❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❛♣r❡✲ s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦r♣♦ ❛♦ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♥♦ q✉❡ s❡❣✉❡✳
❯♠ ❝♦r♣♦ é ✉♠ ❝♦♥❥✉♥t♦ K ♠✉♥✐❞♦ ❞❡ ❞✉❛s ♦♣❡r❛çõ❡s✿ ❆❞✐çã♦✿ + :K×K → K
(x, y) 7→ x+y,
❡
▼✉❧t✐♣❧✐❝❛çã♦✿ .:K×K → K
(x, y) 7→ x.y.
❖s ❛①✐♦♠❛s ❞❡ ❝♦r♣♦ sã♦ ♦s s❡❣✉✐♥t❡s✿
❆✳ ❆①✐♦♠❛s ❞❛ ❛❞✐çã♦✿
A1. ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (x+y) +z =x+ (y+z)✱ ♣❛r❛x, y, z ∈K❀ A2. ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ x+y=y+x✱ ♣❛r❛ x, y ∈K❀
A3. ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❡①✐st❡0∈Kt❛❧ q✉❡x+ 0 =x✱ s❡❥❛ q✉❛❧ ❢♦rx∈K✳ ❖ ❡❧❡♠❡♥t♦
0 ❝❤❛♠❛✲s❡ ③❡r♦✳
Pr❡❧✐♠✐♥❛r❡s ✷✷
A4. ❙✐♠étr✐❝♦✿ t♦❞♦ ❡❧❡♠❡♥t♦ x ∈ K ♣♦ss✉✐ ✉♠ s✐♠étr✐❝♦ (−x) ∈ K✱ t❛❧ q✉❡ x + (−x) = 0✳
❆ s♦♠❛x+ (−y) s❡rá ✐♥❞✐❝❛❞❛ ❝♦♠ ❛ ♥♦t❛çã♦ x−y ❡ ❝❤❛♠❛❞❛ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ x ❡ y✳ ❆ ♦♣❡r❛çã♦(x, y)7→x−y ❝❤❛♠❛✲s❡ s✉❜tr❛çã♦✳
▼✳ ❆①✐♦♠❛s ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✿
M1. ❆ss♦❝✐❛t✐✈✐❞❛❞❡✿ (x.y).z=x.(y.z)✱ ♣❛r❛x, y, z ∈K❀ M2. ❈♦♠✉t❛t✐✈✐❞❛❞❡✿ x.y =y.x✱ ♣❛r❛x, y ∈K❀
M3. ❊❧❡♠❡♥t♦ ♥❡✉tr♦✿ ❡①✐st❡ 1∈K t❛❧ q✉❡ x.1 =x✱ s❡❥❛ q✉❛❧ ❢♦r x∈K✳ ❖ ❡❧❡♠❡♥t♦
1 ❝❤❛♠❛✲s❡ ✉♠✳
M4. ■♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿ t♦❞♦ ❡❧❡♠❡♥t♦ x ∈ K t❛❧ q✉❡ x 6= 0 ♣♦ss✉✐ ✉♠ ✐♥✈❡rs♦
x−1✱ t❛❧ q✉❡ x.x−1 = 1✳
❉❛❞♦s x, y ∈ K✱ ❝♦♠ y 6= 0✱ ❡s❝r❡✈❡✲s❡ t❛♠❜é♠ x
y ❡♠ ✈❡③ ❞❡ x.y
−1✳ ❆ ♦♣❡r❛çã♦ (x, y)7→ x
y ❞❡✜♥✐❞❛ ♣❛r❛ xq✉❛❧q✉❡r ❡ y6= 0 ❡♠ K ❝❤❛♠❛✲s❡ ❞✐✈✐sã♦✳
P♦r ✜♠✱ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♥✉♠ ❝♦r♣♦K❛❝❤❛♠✲s❡ r❡❧❛❝✐♦♥❛❞❛s ♣♦r ✉♠ ❛①✐♦♠❛✱ ❝♦♠ ♦ q✉❛❧ ✜❝❛ ❝♦♠♣❧❡t❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦r♣♦✳
D1. ❆①✐♦♠❛ ❞❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡✿ ❞❛❞♦s x, y, z q✉❛✐sq✉❡r ❡♠ K✱ t❡♠✲s❡ x.(y +z) =
x.y+x.z✳
❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s Q é ✉♠ ❝♦r♣♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s p q +
p′
q′ =
p.q′ +p′
.q q.q′ ❡
p q.
p′
q′ =
p.p′
q.q′✳
❯♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ é ✉♠ ❝♦r♣♦K♥♦ q✉❛❧ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ P ⊂K✱ ❝❤❛♠❛❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ♣♦s✐t✐✈♦s ❞❡ K✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
P1. ❆ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❡ ❡❧❡♠❡♥t♦s ♣♦s✐t✐✈♦s sã♦ ❡❧❡♠❡♥t♦s ♣♦s✐t✐✈♦s✳ ❖✉ s❡❥❛✱ x, y ∈P ⇒x+y ∈P ❡ x.y ∈P❀
P2. ❉❛❞♦ x∈K✱ ♦❝♦rr❡ s♦♠❡♥t❡ ✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s s❡❣✉✐♥t❡s✿ ♦✉ x= 0✱ ♦✉ x∈P✱ ♦✉ −x∈P✳ ❆ss✐♠✱ s❡♥❞♦ −P ={x∈K| −x∈P}✱ t❡♠♦s
K=P ∪(−P)∪ {0},
Pr❡❧✐♠✐♥❛r❡s ✷✸
❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐sQé ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ♥♦ q✉❛❧P =
p
q|p, q ∈N
✳
◆✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ K✱ ❡s❝r❡✈❡✲s❡ x < y ❡ ❞✐③✲s❡ q✉❡ x é ♠❡♥♦r ❞♦ q✉❡ y q✉❛♥❞♦ y−x ∈P✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡✲s❡ t❛♠❜é♠ y > x ❡ ❞✐③✲s❡ q✉❡ y é ♠❛✐♦r ❞♦ q✉❡ x✳ ❆ ♥♦t❛çã♦x≤y é ✉s❛❞❛ ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ✉♠❛ ❞❛s ❛❧t❡r♥❛t✐✈❛s ♦❝♦rr❡✿ ♦✉y−x∈P✱ ♦✉ y−x= 0✳ ◆❡st❡ ❝❛s♦✱ ❞✐③✲s❡ q✉❡ x é ♠❡♥♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ y ♦✉ q✉❡ y é ♠❛✐♦r ❞♦ q✉❡ ♦✉ ✐❣✉❛❧ ❛ x✳
◆✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦✱ ❡①✐st❡ ❛ ✐♠♣♦rt❛♥t❡ ♥♦çã♦ ❞❡ ✐♥t❡r✈❛❧♦✳
• ■♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s✿ ❉❛❞♦s a, b∈ K✱ a < b✱ ❞❡✜♥✐♠♦s ♦s ✐♥t❡r✈❛❧♦s ❧✐♠✐t❛❞♦s ❞❡ ❡①tr❡♠♦s a ❡ b ❝♦♠♦ s❡♥❞♦ ♦s ❝♦♥❥✉♥t♦s✿
◦ ■♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦✿ [a, b] ={x∈K|a≤x≤b}❀
◦ ■♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ à ❡sq✉❡r❞❛✿ [a, b) = {x∈K|a≤x < b}❀ ◦ ■♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ à ❞✐r❡✐t❛✿ (a, b] ={x∈K|a < x≤b}❀ ◦ ■♥t❡r✈❛❧♦ ❛❜❡rt♦✿ (a, b) = {x∈K|a < x < b}✳
❉✐③❡♠♦s q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ t♦❞♦s ♦s ✐♥t❡r✈❛❧♦s ❛❝✐♠❛ é b−a✳
•■♥t❡r✈❛❧♦s ✐❧✐♠✐t❛❞♦s✿ ❉❛❞♦a∈K✱ ❞❡✜♥✐♠♦s ♦s ✐♥t❡r✈❛❧♦s ✐❧✐♠✐t❛❞♦s ❞❡ ♦r✐❣❡♠ a ❝♦♠♦ s❡♥❞♦ ♦s ❝♦♥❥✉♥t♦s✿
◦ ❙❡♠✐rr❡t❛ ❡sq✉❡r❞❛ ❢❡❝❤❛❞❛ ❞❡ ♦r✐❣❡♠ a✿ (−∞, a] ={x∈K|x≤a}❀ ◦ ❙❡♠✐rr❡t❛ ❡sq✉❡r❞❛ ❛❜❡rt❛ ❞❡ ♦r✐❣❡♠ a✿ (−∞, a) ={x∈K|x < a}❀ ◦ ❙❡♠✐rr❡t❛ ❞✐r❡✐t❛ ❢❡❝❤❛❞❛ ❞❡ ♦r✐❣❡♠ a✿ [a,+∞) = {x∈K|x≥a}❀ ◦ ❙❡♠✐rr❡t❛ ❞✐r❡✐t❛ ❛❜❡rt❛ ❞❡ ♦r✐❣❡♠ a✿ (a,+∞) ={x∈K|x > a}✳
◆✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦K✱ ❞❡✜♥✐♠♦s ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦x✱ ❝♦♠♦ s❡♥❞♦ x✱ s❡ x ≥ 0 ❡ −x s❡ x <0✳ ❯s❛♠♦s ♦ sí♠❜♦❧♦ |x| ♣❛r❛ ✐♥❞✐❝❛r ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ x✳
P♦rt❛♥t♦✱ ❞❛❞♦ x∈K✱ t❡♠✲s❡
|x|=
(
x s❡x≥0
−x s❡x <0.
❱❡❥❛♠♦s✱ ♣♦✐s✱ q✉❡ |x| é ♦ ♠❛✐♦r ❞♦s ❡❧❡♠❡♥t♦s x ❡ −x✳ P♦❞❡rí❛♠♦s ❡♥tã♦ ❞❡✜♥✐r |x|= max{−x, x}.
Pr❡❧✐♠✐♥❛r❡s ✷✹
❯♠ s✉❜❝♦♥❥✉♥t♦X ❞❡ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦Ké ❞✐t♦ ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st❡ b∈K t❛❧ q✉❡b ≥x♣❛r❛ t♦❞♦x∈X✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠✲s❡ X ⊂(−∞, b]✳
❈❛❞❛ b ∈K ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❝❤❛♠❛✲s❡ ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ❞❡ X✳
❯♠ s✉❜❝♦♥❥✉♥t♦X ❞❡ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦Ké ❞✐t♦ ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ q✉❛♥❞♦ ❡①✐st❡ a∈Kt❛❧ q✉❡a ≤x♣❛r❛ t♦❞♦x∈X✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ t❡♠✲s❡X ⊂[a,+∞)✳
❈❛❞❛ a∈K ❝♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❝❤❛♠❛✲s❡ ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ❞❡ X✳
❯♠ s✉❜❝♦♥❥✉♥t♦ X ❞❡ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ K é ❞✐t♦ ❧✐♠✐t❛❞♦ q✉❛♥❞♦ é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r♠❡♥t❡✱ ✐st♦ é✱ q✉❛♥❞♦ ❡①✐st❡♠ a, b∈K t❛✐s q✉❡ X ⊂[a, b]✳
❯♠ ❡❧❡♠❡♥t♦ b ∈ K é ❞✐t♦ s✉♣r❡♠♦ ❞♦ s✉❜❝♦♥❥✉♥t♦ X q✉❛♥❞♦ b é ❛ ♠❡♥♦r ❞❛s ❝♦t❛s s✉♣❡r✐♦r❡s ❞❡ X ❡♠ K✳
❯♠ ❡❧❡♠❡♥t♦a∈Ké ❞✐t♦ í♥✜♠♦ ❞♦ s✉❜❝♦♥❥✉♥t♦X q✉❛♥❞♦ aé ❛ ♠❛✐♦r ❞❛s ❝♦t❛s ✐♥❢❡r✐♦r❡s ❞❡ X ❡♠ K✳
❯♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦K é ❞✐t♦ ❝♦♠♣❧❡t♦ q✉❛♥❞♦ t♦❞♦ s✉❜❝♦♥❥✉♥t♦X ⊂K ♥ã♦✲✈❛③✐♦ ❡ ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ ♣♦ss✉✐ s✉♣r❡♠♦ ❡♠ K✳
❆❞♦t❛r❡♠♦s✱ ♥❡st❡ ♠♦♠❡♥t♦✱ ✉♠ ❛①✐♦♠❛ ❢✉♥❞❛♠❡♥t❛❧✳
❆①✐♦♠❛✳ ❊①✐st❡ ✉♠ ❝♦r♣♦ ♦r❞❡♥❛❞♦ ❝♦♠♣❧❡t♦ ❝❤❛♠❛❞♦ ♦ ❝♦r♣♦ ❞♦ ♥ú♠❡r♦s r❡❛✐s✱ ❞❡♥♦t❛❞♦ ♣♦r R✳
◆♦ q✉❡ s❡❣✉❡✱ ❧✐st❛r❡♠♦s✱ ❧✐t❡r❛❧♠❡♥t❡✱ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ✐♠♣r❡s❝✐♥❞í✈❡✐s ♣❛r❛ ❛ ❡①✐❜✐çã♦ ❞❛s s✉r♣r❡❡♥❞❡♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ❈❛♥t♦r✳ ❈❛❜❡ r❡ss❛❧t❛r q✉❡ ♥ã♦ ♣r❡t❡♥❞❡♠♦s s❡r r✐❣♦r♦s♦s ♥♦ tr❛t❛♠❡♥t♦ ❞❡ t❛✐s ♣r❡❧✐♠✐♥❛r❡s✱ ✈✐st♦ q✉❡ ♦ t❡①t♦ ❡stá ❞✐r❡❝✐♦♥❛❞♦ ❛♦s ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❖ ❧❡✐t♦r✲♣r♦❢❡ss♦r ♣♦❞❡ ❝♦♥s✉❧t❛r ❬✺❪ ♣❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣r♦❢✉♥❞❛❞♦✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ é ✉♠❛ ❧✐st❛ ♦r❞❡♥❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s❀ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ sã♦ ✐♥❞❡①❛❞♦s ♣♦r ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s é ✉♠❛ ❢✉♥çã♦ a : N → R q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ a(n)✳ ❖ ✈❛❧♦ra(n)✱ ♣❛r❛ t♦❞♦ n∈N✱ é r❡♣r❡s❡♥t❛❞♦ ♣♦ran ❡ ❞❡♥♦♠✐♥❛❞♦
♥✲és✐♠♦ t❡r♠♦ ❞❛ s❡q✉ê♥❝✐❛✳
❊s❝r❡✈❡r❡♠♦s (a1, a2, . . . , an, . . .) ✱ ♦✉ (an)n∈N✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ (an)✱ ♣❛r❛ ✐♥❞✐❝❛r
❛ s❡q✉ê♥❝✐❛ a✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❙❡ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ q✉❛❧q✉❡r ǫ✱ ❡①✐st✐r ✉♠ í♥❞✐❝❡ ❞❛ s❡q✉ê♥❝✐❛ (an) ❛ ♣❛rt✐r ❞♦ q✉❛❧✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s t❡r♠♦s ❞❡ (an) é
♠❡♥♦r ❞♦ q✉❡ ǫ✱ ❛ s❡q✉ê♥❝✐❛ (an) s❡rá ❞✐t❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳
❖ q✉❡ q✉❡r❡♠♦s ❞✐③❡r ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ q✉❛✐sq✉❡r ❞♦✐s t❡r♠♦s ❞❛ s❡q✉ê♥✲ ❝✐❛ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛❄ ❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦ d : R×R → R+ ❞❡✜♥✐❞❛ ♣♦r d(x, y) = |x−y|✱ ♣❛r❛ (x, y) ∈ R✱ ♦♥❞❡ R+ = {x ∈
R|x≥0}✳
