❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
▼♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿
❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s
❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳
❆❧✐♥❡ ❋❡r♥❛♥❞❛ ❋❛q✉✐♥✐ ❍❡❧❡♥❛
❘❡❧❛tór✐♦ ♣❛r❛ ♦ ❊①❛♠❡ ●❡r❛❧ ❞❡ ◗✉❛❧✐✜❝❛çã♦ ❛♣r❡✲ s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s
❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
▼♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿
❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s
❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳
❆❧✐♥❡ ❋❡r♥❛♥❞❛ ❋❛q✉✐♥✐ ❍❡❧❡♥❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s
✺✶✵✳✵✼ ❍✹✼✹♠
❍❡❧❡♥❛✱ ❆❧✐♥❡ ❋❡r♥❛♥❞❛ ❋❛q✉✐♥✐
▼♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿ ❯♠❛ ❛❜♦r❞❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳✴ ❆❧✐♥❡ ❋❡r♥❛♥❞❛ ❋❛q✉✐♥✐ ❍❡❧❡♥❛ ✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✻✳
✼✷ ❢✳✱ ✐❧✳✱ ✜❣s✳✱ ❣rá❢s✳✱ t❛❜s✳
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳
❖r✐❡♥t❛❞♦r✿ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s
✶✳ ▼❛t❡♠át✐❝❛ ✲ ❊st✉❞♦ ❡ ❡♥s✐♥♦✳ ✷✳ ▼❛t❡♠át✐❝❛✳ ✸✳ ❋✉♥çã♦✳ ✹✳ ❋✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✳ ■✳ ❚ít✉❧♦
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❆❧✐♥❡ ❋❡r♥❛♥❞❛ ❋❛q✉✐♥✐ ❍❡❧❡♥❛
▼♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿ ❯♠❛ ❛❜♦r❞❛❣❡♠
♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛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✳ ❘✐❝❛r❞♦ ❞❡ ❙á ❚❡❧❡s ❖r✐❡♥t❛❞♦r
Pr♦❢✭❛✮✳ ❉r✭❛✮✳ ❈❛r✐♥❛ ❆❧✈❡s ■●❈❊ ✲ ❯◆❊❙P ❘✐♦ ❈❧❛r♦ ✭❙P✮
Pr♦❢✳ ❉r✳ ❏❛♠✐❧ ●♦♠❡s ❞❡ ❆❜r❡✉ ❏ú♥✐♦r ❯❋❙❈❛r ✲ ❙ã♦ ❈❛r❧♦s ✭❙P✮
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛♦s ♠❡✉s ♣❛✐s q✉❡✱ ❞❡♥tr♦ ❞❡ t♦❞❛ s✐♠♣❧✐❝✐❞❛❞❡ q✉❡ ❧❤❡s ❝❛❜❡✱ ♠❡ ♠♦str❛r❛♠ q✉❡ ❛ ❡❞✉❝❛çã♦ s❡r✐❛ ♦ ❛❣❡♥t❡ tr❛♥s❢♦r♠❛❞♦r ❞❡ ♠✐♥❤❛ ✈✐❞❛✳
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ❛♠❛❞♦ ♠❛r✐❞♦ q✉❡ ❡st❡✈❡ ❛♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ ❡st❡ ❝✉rs♦ ❞❡ ♠❡s✲ tr❛❞♦ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ❛ss✐♠ ♣❡r♠❛♥❡❝❡✉ ❛té ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❡ ♠✐♥❤❛ ❡❞✉❝❛çã♦ ❡ q✉❡ ❞❡✐①❛✲ r❛♠ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❛ ❢♦r♠❛çã♦ ❞❛ ♠✐♥❤❛ ❝✐❞❛❞❛♥✐❛✳
❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦♠♣❛♥❤❡✐r♦s ❞❡ Pr♦❢♠❛t✱ ♣❡❧❛s ❤♦r❛s ❞❡ ❡st✉❞♦✱ ♣❡❧❛s ❝♦♥✈❡rs❛s ❡ ♣♦r t♦❞❛s ❛s r✐s❛❞❛s✳
❆❣r❛❞❡ç♦ à ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✱ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛✲ t❡♠át✐❝❛ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❛♣♦✐♦ ❛❝❛❞ê♠✐❝♦ ❡ à ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣♦r ✐♠♣❧❡♠❡♥t❛r ♦ Pr♦❢♠❛t ♥♦ ❇r❛s✐❧✳
❆❣r❛❞❡ç♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❘✐❝❛r❞♦✱ ♣♦r t♦❞❛ s✉❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
P♦r ✈❡③❡s s❡♥t✐♠♦s q✉❡ ❛q✉✐❧♦ q✉❡ ❢❛③❡♠♦s ♥ã♦ é s❡♥ã♦ ✉♠❛ ❣♦t❛ ❞❡ á❣✉❛ ♥♦ ♠❛r✳ ▼❛s ♦ ♠❛r s❡r✐❛ ♠❡♥♦r s❡ ❧❤❡ ❢❛❧t❛ss❡ ✉♠❛ ❣♦t❛✳
❘❡s✉♠♦
❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣❡♥s❛r ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❡s♣❡❝✐✲ ❛❧♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ t❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❛❧✉♥♦s✱ ♠♦t✐✈❛r❛♠ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦ q✉❡✱ ❛tr❛✈és ❞❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ♣r♦♣õ❡ ♦ ❡♥s✐♥♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳ ❈♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ♣♦ss✉❡♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s q✉❡ s❡ ❡st❡♥❞❡♠ ♣❡❧❛s ♠❛✐s ❞✐✈❡rs❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❞✐❛♥t❡ ❞❡ ❞❛❞♦s ❛❧❛r♠❛♥t❡s s♦❜r❡ ♦ ❝♦♥s✉♠♦ ❞❡ á❧❝♦♦❧ ♣♦r ❛❞♦❧❡s❝❡♥t❡s✱ ❡❧❛❜♦r❛♠♦s ✉♠❛ ♣r♦♣♦st❛ ❞❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ q✉❡ ♣❡r♠✐t❡ ❛ r❡✢❡①ã♦ s♦❜r❡ ♦ ❝♦♥s✉♠♦ ❞❡ á❧❝♦♦❧ ❡ ❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ❝♦♠ s✉♣♦rt❡ t❡ór✐❝♦ ❛♦ ♣r♦❢❡ss♦r✳
❆❜str❛❝t
❚❤❡ ♥❡❡❞ t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ❡s♣❡❝✐❛❧❧② ❢r♦♠ t♦♣✐❝s r❡❧❛t❡❞ t♦ ❞❛✐❧② ❧✐✈❡s ♦❢ st✉❞❡♥ts✱ ♠♦t✐✈❛t❡❞ t❤✐s ✇♦r❦ t❤❛t t❤r♦✉❣❤ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ ♣r♦♣♦s❡s t❤❡ t❡❛❝❤✐♥❣ ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s✳ ❚❛❦✐♥❣ ✐♥t♦ ❝♦♥s✐❞❡r❛✲ t✐♦♥ t❤❡ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s t❤❛t ❡①t❡♥❞ ❛❝r♦ss ♠❛♥② ❞✐✛❡r❡♥t ❛r❡❛s ♦❢ ❦♥♦✇❧❡❞❣❡ ❛♥❞ t❤❡ ❛❧❛r♠✐♥❣ st❛t✐st✐❝s ♦♥ t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛❧❝♦❤♦❧ ❜② t❡❡♥❛❣❡rs✱ ✐t ✇❛s ❡❧❛❜♦r❛t❡❞ ❛ ♣r♦♣♦s❛❧ ❢♦r ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ t❤❛t ❛❧❧♦✇s s♦♠❡ r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ ❝♦♥s✉♠♣t✐♦♥ ♦❢ ❛❧❝♦❤♦❧ ❛♥❞ ❝♦♥t❡①t✉❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ t❤❡♦r❡t✐❝❛❧ s✉♣♣♦rt t♦ t❤❡ t❡❛❝❤❡r✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❙✐t✉❛çã♦ ✶✳ ❋♦♥t❡✿ ❬✶❪ ✲ ♣❛❣ ✺✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ❙✐t✉❛çã♦ ✷✳ ❋♦♥t❡✿ ❬✶❪ ✲ ♣❛❣ ✶✽✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹ ❙✐t✉❛çã♦ ✸✳ ❋♦♥t❡✿ ❬✷❪ ✲ ♣❛❣ ✹✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✺ ❙✐t✉❛çã♦ ✹✳ ❋♦♥t❡✿ ❬✷❪ ✲ ♣❛❣ ✶✹✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✶ ❉✐❛❣r❛♠❛ ❞❡ ✢❡❝❤❛s ❡①❡♠♣❧✐✜❝❛♥❞♦ ❛ ❢✉♥çã♦f ❞❡X ❡♠ Y✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸✳✷ P♦♥t♦ P✭①✱②✮ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f(x) =c, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✹ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡IdR✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✺ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❛✜♠f(x) = ax+b, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸✳✻ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) =ax2+bx+c, ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✼ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = x2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✽ ●rá✜❝♦ ❞❛ ❢✉♥çã♦g(x) = 1
x2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✾ ●rá✜❝♦ ❞❛ ❢✉♥çã♦h(x) = x
2+ 4
x−2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✶✵ ■❧✉str❛çã♦ ❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦✳ ❋♦♥t❡✿ ❬✸❪✱ ♣❛❣ ✶✶✾✳ ✳ ✹✶ ✸✳✶✶ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✶✷ ●rá✜❝♦ ❞❛s ❢✉♥çõ❡s ✐♥✈❡rs❛sf(x) =x2 ❡ g(y) =√y ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ▲✐✈r♦s ❞✐❞át✐❝♦s ❛♥❛❧✐s❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✺✳✶ P♦r❝❡♥t❛❣❡♠ ❞❡ á❧❝♦♦❧ ♣♦r ❜❡❜✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❈♦♥❝❡♥tr❛çã♦ ❞❡ á❧❝♦♦❧ ♥♦ s❛♥❣✉❡ ✭❈❆❙✮ ❡ ♦s s✐♥t♦♠❛s ❝❧í♥✐❝♦s ❝♦rr❡s✲
♣♦♥❞❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✺✳✸ ❘✐s❝♦ ❞❡ ❛❝✐❞❡♥t❡s ❡ q✉❛♥t✐❞❛❞❡ ❞❡ á❧❝♦♦❧ ✐♥❣❡r✐❞♦✳ ❋♦♥t❡✿ ❚❛❜❡❧❛ ❛❞❛♣✲
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✾
✷ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ✷✶
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸ ❋✉♥çõ❡s ✸✶
✸✳✶ ❆ ❉❡✜♥✐çã♦ ❞❡ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❊❧❡♠❡♥t❛r❡s ❞❛s ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ●rá✜❝♦s ❞❡ ❋✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹ ❋✉♥çã♦ ❈♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✺ ❋✉♥çã♦ ✐♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❋✉♥çõ❡s ❊①♣♦♥❡♥❝✐❛✐s ❡ ▲♦❣❛rít♠✐❝❛s ✹✼
✹✳✶ P♦tê♥❝✐❛s ❞❡ ❊①♣♦❡♥t❡ ❘❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷ ❆ ❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✸ ❆ ❋✉♥çã♦ ▲♦❣❛rít♠✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺ Pr♦♣♦st❛ ❞❡ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ✺✼ ✺✳✶ ❈♦♥s✉♠♦ ❞❡ á❧❝♦♦❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✾
✶ ■♥tr♦❞✉çã♦
❖ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ t❡♠ s✐❞♦ ✉♠ ♣r♦❝❡ss♦ ár❞✉♦ ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s✱ ❛ss✐♠ ❝♦♠♦ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ♥ã♦ s❡ ♠♦str❛ ❡✜❝✐❡♥t❡ ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s✳ ❖s í♥❞✐❝❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❡♠ ▼❛t❡♠át✐❝❛ ♥♦ ❇r❛s✐❧ t❡♠ s✐❞♦ ❜❛✐①íss✐♠♦s ❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ♥♦ ❞❡❝♦rr❡r ❞♦s ❛♥♦s ❞❡ ❡♥s✐♥♦ ❜ás✐❝♦ ❡ss❡s í♥❞✐❝❡s ✈ã♦ ❛❜❛✐①❛♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s✳
▼✉✐t❛s ✈❡③❡s ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ s❡ ❞á ❞❡ ❢♦r♠❛ ❝♦♠♣❧❡t❛♠❡♥t❡ t❡ór✐❝❛ ❡ ❞❡s❧✐❣❛❞❛ ❞❛ r❡❛❧✐❞❛❞❡ ❞♦ ❛❧✉♥♦ ♦ q✉❡ ❝♦♥tr✐❜✉✐ ❛✐♥❞❛ ♠❛✐s ♣❛r❛ ♦ ❞❡s✐♥t❡r❡ss❡ ❡ ❜❛✐①♦ ❛♣r❡♥❞✐③❛❞♦✳
❉❡♥tr♦ ❞❡st❡ ❝♦♥t❡①t♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛✲ rít♠✐❝❛s✱ ❛❧❣✉♠❛s ✈❡③❡s✱ é ♣r♦♣♦st♦ ❞❡ ❢♦r♠❛ ❞❡s❧✐❣❛❞❛ ❞❛ r❡❛❧✐❞❛❞❡ ❛♣❡s❛r ❞❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ ❛♣❧✐❝❛çõ❡s q✉❡ ♣♦ss✉❡♠✳ ❊♠ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s ❛ ❛❜♦r❞❛❣❡♠ ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s ♦❢❡r❡❝❡ ❛♦s ❛❧✉♥♦s ❝♦♥❤❡❝✐♠❡♥t♦s s✉♣❡r✜❝✐❛✐s s♦❜r❡ ❛s ❛♣❧✐❝❛çõ❡s ❞♦ t❡♠❛✱ ♣♦✐s tr❛③❡♠ ♣♦✉❝❛s s✐t✉❛çõ❡s r❡❛✐s ❡ ❞♦ ❝♦t✐❞✐❛♥♦✳
P♦r ❡ss❡s ❡ ♦✉tr♦s ♠♦t✐✈♦s✱ ♣r❡❝✐s❛♠♦s ❜✉s❝❛r ♠❡t♦❞♦❧♦❣✐❛s ❞❡ ❡♥s✐♥♦ q✉❡ ❡st✐♠✉✲ ❧❡♠ ♦s ❛❧✉♥♦s ❡ t❛♠❜é♠ ♠♦t✐✈❡♠ ♦s ♣r♦❢❡ss♦r❡s✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛✲ t❡♠át✐❝❛ s❡❥❛ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❝❛♣❛③ ❞❡ ❝♦♥tr✐❜✉✐r ❡❢❡t✐✈❛♠❡♥t❡ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ♠❛t❡♠át✐❝❛ ❡ ♣❡r♠✐t❡ q✉❡ ♦s ❛❧✉♥♦s ✉t✐❧✐③❡♠ ❡ ❛♣❧✐q✉❡♠
✏s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❛ s✐t✉❛çõ❡s ❞✐✈❡rs❛s✱ ✉t✐❧✐③❛♥❞♦✲♦s ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❞❛ ❝✐ê♥❝✐❛✱ ♥❛ ❛t✐✈✐❞❛❞❡ t❡❝♥♦❧ó❣✐❝❛ ❡ ♥❛s ❛t✐✈✐❞❛❞❡s ❝♦t✐❞✐❛✲ ♥❛s✑
❝♦♥❢♦r♠❡ ✐♥❞✐❝❛♠ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭P❈✲ ◆❊▼✮ ❬✺❪✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ ❞❡ ❞✐ss❡rt❛çã♦ é ❡❧❛❜♦r❛r ✉♠❛ ♣r♦♣♦st❛ ❞❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝❛♣❛③ ❞❡ ✈✐♥❝✉❧❛r ✉♠ t❡♠❛ ♠✉✐t♦ ❝♦♠✉♠ ♥♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❥♦✈❡♥s ❛♦ ❡♥✲ s✐♥♦ ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s✳ P❛r❛ ✐ss♦✱ ♥♦ ❝❛♣ít✉❧♦ ✐♥✐❝✐❛❧ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❛♥❛❧✐s❛♥❞♦✲❛ ❝♦♠♦ ♣r♦♣♦st❛ ❞❡ ❡♥s✐♥♦ ❛❧é♠ ❞❡ ❜✉s❝❛r ❝♦♠♦ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ✈❡♠ s❡♥❞♦ ❛❜♦r❞❛❞❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ◆♦s ♣ró①✐♠♦s ❞♦✐s ❝❛♣ít✉❧♦s ❢♦r♠❛r❡♠♦s ❛ ❜❛s❡ ♠❛t❡♠át✐❝❛ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ tr❛③❡♥❞♦ ❞❡✜♥✐çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❢✉♥çõ❡s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❛❧é♠ ❞❡ tr❛t❛r s♦❜r❡ ♦s ✐♠♣♦rt❛♥t❡s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡ ✐♥✈❡rs❛s✳ ❈♦♠ ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s s❡r❡♠♦s ❝❛♣❛③❡s ❞❡ tr❛❜❛❧❤❛r ❝♦♠ ❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❡ ❧♦❣❛rít♠✐❝❛s
✷✵ ■♥tr♦❞✉çã♦
✷ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
◆❡st❡ ❝❛♣ít✉❧♦ ♣r❡t❡♥❞❡♠♦s ♦❢❡r❡❝❡r ❛♦ ❧❡✐t♦r ✉♠❛ ❜r❡✈❡ r❡✢❡①ã♦ ❛ r❡s♣❡✐t♦ ❞♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦s t❡♠♣♦s ❛t✉❛✐s ❡ ❞✐s❝✉t✐r ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡♥s✐♥♦ ♥❡st❡ ❝♦♥t❡①t♦✳ ◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡ é ❢❡✐t❛ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✱ ❛♣r❡s❡♥t❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❛❞❛ ♣♦r ❛❧❣✉♥s ❛✉t♦r❡s ❞❡ ❧✐✈r♦s s♦❜r❡ ♦ t❡♠❛ ❡ ♥❛ s❡❣✉♥❞❛ ♣❛rt❡✱ ✉♠❛ ❜r❡✈❡ ❛♥á❧✐s❡ s♦❜r❡ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ❛♣r♦❢✉♥❞❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ❡ ❛♣r❡s❡♥t❛♥❞♦✲❛ ❝♦♠♦ ♣r♦♣♦st❛ ❞❡ ❡♥s✐♥♦✳ ◆♦ ❡♥❝❡rr❛♠❡♥t♦ ❞♦ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❞❡ ✷ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣❛r❛ ✈❡r✐✜❝❛r ❝♦♠♦ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ✈❡♠ s❡♥❞♦ ❛❜♦r❞❛❞❛s ♥❡st❛s ♦❜r❛s✳
✷✳✶ ■♥tr♦❞✉çã♦
P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦ s✉r❣✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ s❡ ❞❡✉ ♥♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ♦ ❤♦♠❡♠ s❡♥t✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝♦♥t❛r ❡ ❡♥✉♠❡r❛r ♦❜❥❡t♦s✱ s❡❣✉♥❞♦ ❇♦②❡r ❬✻❪
✏❛ ♠❛t❡♠át✐❝❛ ♦r✐❣✐♥❛❧♠❡♥t❡ s✉r❣✐✉ ❝♦♠♦ ♣❛rt❡ ❞❛ ✈✐❞❛ ❞✐ár✐❛ ❞♦ ❤♦✲ ♠❡♠✱ ❡ s❡ ❤á ✈❛❧✐❞❛❞❡ ♥♦ ♣r✐♥❝í♣✐♦ ❜✐♦❧ó❣✐❝♦ ❞❛ ✏s♦❜r❡✈✐✈ê♥❝✐❛ ❞♦ ♠❛✐s ❛♣t♦✑ ❛ ♣❡rs✐stê♥❝✐❛ ❞❛ r❛ç❛ ❤✉♠❛♥❛ ♣r♦✈❛✈❡❧♠❡♥t❡ t❡♠ r❡❧❛çã♦ ❝♦♠ ♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ♥♦ ❤♦♠❡♠ ❞❡ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✑✳
❆♦ s❡ ❞❡♣❛r❛r ❝♦♠ ✉♠❛ s✐t✉❛çã♦ q✉❡ ♦ ❝♦❧♦❝♦✉ ❞✐❛♥t❡ ❞❡ ✉♠❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❡♠ t♦r♥♦ ❞❡ ✈❛❧♦r❡s ♥✉♠ér✐❝♦s✱ ✐♥❝♦♥s❝✐❡♥t❡♠❡♥t❡✱ ♦ ❤♦♠❡♠ tr❛♥s❢♦r♠♦✉ ❡ss❛ s✐t✉❛çã♦ r❡❛❧ ❡♠ ✉♠ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ♣❛r❛ ♣♦❞❡r t♦♠❛r ❛ ♠❡❧❤♦r ❞❡❝✐sã♦✱ ♦✉ s❡❥❛✱ ♥❡st❡ ♠♦♠❡♥t♦✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ s✉r❣✐♠❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛ t✐✈❡♠♦s t❛♠❜é♠ ♦ s✉r❣✐♠❡♥t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳
▼❛s ♦ q✉❡ é ❛ ▼♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛❄ ❱❡❥❛♠♦s ❝♦♠♦ ❛❧❣✉♥s ❛✉t♦r❡s ❛ ❞❡✜♥❡♠✿ P❛r❛ ❇❛ss❛♥❡③✐ ❬✹❪✿
✏❆ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♥s✐st❡ ♥❛ ❛rt❡ ❞❡ tr❛♥s❢♦r♠❛r ♣r♦❜❧❡♠❛s ❞❛ r❡❛❧✐❞❛❞❡ ❡♠ ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s ❡ r❡s♦❧✈ê✲❧♦s ✐♥t❡r♣r❡t❛♥❞♦ s✉❛s s♦❧✉çõ❡s ♥❛ ❧✐♥❣✉❛❣❡♠ ❞♦ ♠✉♥❞♦ r❡❛❧✑✳
✷✷ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
P❛r❛ ❆❧♠❡✐❞❛ ❬✼❪✿
✏❆ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ✈✐s❛ ♣r♦♣♦r s♦❧✉çõ❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ♣♦r ♠❡✐♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s✳ ❖ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦✱ ♥❡ss❡ ❝❛s♦✱ é ♦ q✉❡ ❞á ❢♦r♠❛ à s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ é ❛ ❛t✐✈✐❞❛❞❡ ❞❡ ❜✉s❝❛ ❞❡ss❛ s♦❧✉çã♦✑✳
P❛r❛ ❇✐❡♠❜❡♥❣✉t ❬✽❪✿
✏➱ ♦ ♣r♦❝❡ss♦ q✉❡ ❡♥✈♦❧✈❡ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦✑ ❡ ❝♦♠♣❧❡♠❡♥t❛
✏✉♠ ❝♦♥❥✉♥t♦ ❞❡ sí♠❜♦❧♦s ❡ r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ ♣r♦❝✉r❛ tr❛❞✉③✐r✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ✉♠ ❢❡♥ô♠❡♥♦ ❡♠ q✉❡stã♦ ♦✉ ♣r♦❜❧❡♠❛ ❞❡ s✐t✉❛çã♦ r❡❛❧✱ ❞❡♥♦♠✐♥❛✲s❡ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦✑✳
❙❡♥❞♦ ❛ss✐♠✱ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ tr❛❞✉③ ❛s s✐t✉❛çõ❡s ❡ ✐♥❢♦r♠❛çõ❡s ❞♦ ❝♦t✐✲ ❞✐❛♥♦ ♣❛r❛ ♦ ❝♦♥t❡①t♦ ♠❛t❡♠át✐❝♦✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ❛♥á❧✐s❡ ❞❛s ♣♦ssí✈❡✐s s♦❧✉çõ❡s ❡ ❛ ❜✉s❝❛ ❞❛ s♦❧✉çã♦ ❛❧♠❡❥❛❞❛✳
✷✳✷ ❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛
❆t✉❛❧♠❡♥t❡✱ t❡♠♦s ❛ ❝♦♥s❝✐ê♥❝✐❛ ❞❡ q✉❡ ♦ ♣❛♣❡❧ ❞❛ ❡❞✉❝❛çã♦ é ❢♦r♠❛r ❜♦♥s ❝✐❞❛❞ã♦s ♣❛r❛ ♦ ♠✉♥❞♦✱ ❝❛♣❛③❡s ❞❡ r❡✢❡t✐r s♦❜r❡ s✉❛ ❢✉♥çã♦ ♥❛ s♦❝✐❡❞❛❞❡ ❡ s❡ ♣♦s✐❝✐♦♥❛r ❝r✐t✐❝❛✲ ♠❡♥t❡✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ❡♠ ♥♦ss❛ ♦♣✐♥✐ã♦✱ ❛ ♠❛t❡♠át✐❝❛ é ✉♠❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣❡♥s❛♠❡♥t♦ ❝rít✐❝♦✱ ❞❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s ❡ ❞♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ♥ã♦ ♣♦❞❡♥❞♦✱ ❞❡ ❢♦r♠❛ ❛❧❣✉♠❛✱ t❡r s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ♥❡❣❛❞♦ ❛♦s ❝✐❞❛❞ã♦s✳ ❇❛ss❛♥❡③✐ ❬✹❪ ❞✐③✿
✏♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠ ♥♦✈♦ ♠♦❞❡❧♦ ❞❡ ❡❞✉❝❛çã♦ ♠❡♥♦s ❛❧✐❡✲ ♥❛❞♦ ❡ ♠❛✐s ❝♦♠♣r♦♠❡t✐❞♦ ❝♦♠ ❛ r❡❛❧✐❞❛❞❡ ❞♦s ✐♥❞✐✈í❞✉♦s✱ ❞❡✈❡♠♦s ❧❛♥ç❛r ♠ã♦ ❞❡ ✐♥str✉♠❡♥t♦s ♠❛t❡♠át✐❝♦s ✐♥t❡rr❡❧❛❝✐♦♥❛❞♦s ❛ ♦✉tr❛s ár❡❛s ❞♦ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ❤✉♠❛♥♦✑✳
❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❬✾❪ ❛♣♦♥t❛♠ ❛❧❣✉♥s ♦❜❥❡t✐✈♦s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ q✉❡ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ♠♦❞❡❧❛❣❡♠ s❡r✐❛ ❜❛st❛♥t❡ ♣❡rt✐♥❡♥t❡✱ ❛ s❛❜❡r✿
✶✳ ■❞❡♥t✐✜❝❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ❝♦♠♦ ♠❡✐♦s ♣❛r❛ ❝♦♠♣r❡❡♥❞❡r ❡ tr❛♥s✲ ❢♦r♠❛r ♦ ♠✉♥❞♦ à s✉❛ ✈♦❧t❛❀
❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✷✸
✸✳ ❊st❛❜❡❧❡❝❡r ❝♦♥❡①õ❡s ❡♥tr❡ t❡♠❛s ♠❛t❡♠át✐❝♦s ❞❡ ❞✐❢❡r❡♥t❡s ❝❛♠♣♦s ❡ ❡♥tr❡ ❡ss❡s t❡♠❛s ❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ♦✉tr❛s ár❡❛s ❝✉rr✐❝✉❧❛r❡s❀ ❡♥tr❡ ♦✉tr♦s✳
P♦r ♠✉✐t♦ t❡♠♣♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❢♦✐ tr❛♥s♠✐t✐❞♦ ❞❡ ❢♦r♠❛ ❛❜str❛t❛ ❡ ❞❡s✈✐♥❝✉❧❛❞♦ ❞❛ r❡❛❧✐❞❛❞❡✱ s❡♥❞♦ ✈♦❧t❛❞♦ ❛♣❡♥❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❛❧❣♦r✐t♠♦s ❝♦♠ ♣r♦❝❡❞✐♠❡♥t♦s ♠❡❝â♥✐❝♦s ❡ r❡♣❡t✐t✐✈♦s✱ s❡♠ ❧❡✈❛r ❡♠ ❝♦♥t❛ s✉❛ ♦r✐❣❡♠ ❡ tr❛❥❡tór✐❛ ❤✐stór✐❝❛✳ ❈r✐♦✉✲s❡✱ ❡♥tã♦✱ ✉♠ ✐♠❛❣✐♥ár✐♦ ♣♦♣✉❧❛r s♦❜r❡ ❛ ♠❛t❡♠át✐❝❛✱ q✉❡ ❛ ♠✐st✐✜❝❛ ❝♦♠♦ ✉♠❛ ❝✐ê♥❝✐❛ ❢♦r♠✉❧❛❞❛ ❡ ❜❛s❡❛❞❛ ❡♠ s✉❛s ♣ró♣r✐❛s r❡❣r❛s ❡ ❧❡✐s✳
❆ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ é ✉♠❛ ❢♦r♠❛ ❞❡ ✐♥t❡❣r❛r ♥♦✈❛♠❡♥t❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❝♦♠ ❛ r❡❛❧✐❞❛❞❡ ❡ ♣♦❞❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡♠ t♦❞♦s ♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✱ ♣♦✐s ♣♦ss✐❜✐❧✐t❛ ❛ ✐♥t❡❣r❛çã♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❝♦♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❛❧é♠ ❞❡ ♣❡r♠✐t✐r ❛ ♣rát✐❝❛ ❞❛ ♠✉❧t✐❞✐s❝✐♣❧✐♥❛r✐❞❛❞❡✳
P❛r❛ ❡♥t❡♥❞❡r ♠❡❧❤♦r ❝♦♠♦ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❢✉♥❝✐♦♥❛✱ ✈❛♠♦s ❛♥❛❧✐s❛r ❛ ✜❣✉r❛ ✷✳✶✳
❋✐❣✉r❛ ✷✳✶✿ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
✷✹ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
◆♦ ❞✐❛ ❛ ❞✐❛✱ ❛ t♦❞♦ t❡♠♣♦ ❛s ♣❡ss♦❛s s❡ ✈❡❡♠ ❞✐❛♥t❡ ❞❡ s✐t✉❛çõ❡s q✉❡ ❛s ♦❜r✐❣❛♠ ❛ ❢❛③❡r r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s ❡ ✐♥❝♦♥s❝✐❡♥t❡♠❡♥t❡ ❡❧❛s ♣r❛t✐❝❛♠ ❛ ♠♦❞❡❧❛❣❡♠✳ P♦r ❡st❡ ♠♦t✐✈♦ é ✐♠♣♦rt❛♥t❡ ♣r♦♣♦r ❛ ✉t✐❧✐③❛çã♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ t❛♠❜é♠ ❞❡♥tr♦ ❞❛ ❡s❝♦❧❛✱ ♣♦✐s ❞❡st❛ ❢♦r♠❛ ❡❧❛ s❡ t♦r♥❛ ♠❛✐s s✐❣♥✐✜❝❛t✐✈❛✱ ❡st✐♠✉❧❛♥t❡ ❡ ❡♠♣♦❧❣❛♥t❡ ❛♦s ❛❧✉♥♦s✳ ❙❡❣✉♥❞♦ ❇✐❡♠❜❡♥❣✉t✱ ❬✽❪
✏♦ ❡♥s✐♥♦ ❞❛ ♠❛t❡♠át✐❝❛ ♣r❡❝✐s❛ ✈♦❧t❛r✲s❡ ♣❛r❛ ❛ ♣r♦♠♦çã♦ ❞♦ ❝♦♥❤❡❝✐✲ ♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡ ❞❛ ❤❛❜✐❧✐❞❛❞❡ ❡♠ ✉t✐❧✐③á✲❧♦✑✳
■ss♦ ♣❡r♠✐t✐rá ❛♦s ❛❧✉♥♦s q✉❡ ❢❛ç❛♠ r❡❧❛çõ❡s✱ ❞❡s❡♥✈♦❧✈❛♠ ❛ ❝r✐❛t✐✈✐❞❛❞❡ ❡ ❛♠♣❧✐❡♠ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s✱ ❛❜r❛♥❣❡♥❞♦ ♠ú❧t✐♣❧❛s ❤❛❜✐❧✐❞❛❞❡s ❡ ❝♦♠♣❡tê♥❝✐❛s✳
❇✐❡♠❜❡♥❣✉t ❬✽❪ ♣r♦♣õ❡ ✻ ❡t❛♣❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❝♦♥t❡ú❞♦ ♣r♦❣r❛♠át✐❝♦ ❛tr❛✈és ❞❛ ♠♦❞❡❧❛❣❡♠✿
✶✳ ■♥✐❝✐❛❧♠❡♥t❡✱ ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ ❞♦ t❡♠❛❀ ✷✳ ▲❡✈❛♥t❛♠❡♥t♦ ❞❡ q✉❡stõ❡s ♣❛r❛ ✐♥st✐❣❛r ♦s ❛❧✉♥♦s❀
✸✳ ❙❡❧❡❝✐♦♥❛✲s❡ ❡ ❢♦r♠✉❧❛✲s❡ q✉❡stõ❡s ❛ ✜♠ ❞❡ ❧❡✈❛r ♦s ❛❧✉♥♦s ❛ ♣r♦♣♦r r❡s♣♦st❛s ✭s❡ ♥❡❝❡ssár✐♦ ♣♦❞❡ ♣r♦♣♦r ✉♠❛ ♣❡sq✉✐s❛ ♣❛r❛ ♦s ❛❧✉♥♦s✮❀
✹✳ ❆♦ s✉s❝✐t❛r ✉♠ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦✱ ✐♥t❡rr♦♠♣❡✲s❡ ❛ ❡①♣♦s✐çã♦ ❡ ❞❡s❡♥✈♦❧✈❡✲s❡ ❛ ♠❛t❡♠át✐❝❛ ♥❡❝❡ssár✐❛❀
✺✳ Pr♦♣♦r ❡①❡♠♣❧♦s ❛♥á❧♦❣♦s ❡ r❡s♦❧✉çã♦ ❞❡ ❡①❡r❝í❝✐♦s❀
✻✳ ❘❡t♦r♥❛✲s❡ ❛ q✉❡stã♦ q✉❡ ❣❡r♦✉ ♦ ♣r♦❝❡ss♦ ❛♣r❡s❡♥t❛♥❞♦ ❛ s♦❧✉çã♦ ❞❛ q✉❡stã♦✳ ❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ♣r♦♣♦st♦ ♣♦r ❇✐❡♠❜❡♥❣✉t ❬✽❪✱ ♦s ❛❧✉♥♦s t❛♠❜é♠ t❡rã♦ ♦ ❝♦♥t❛t♦ ❝♦♠ ♦s ♣r♦❝❡❞✐♠❡♥t♦s ✉t✐❧✐③❛❞♦s ❡♠ ✉♠❛ ♣❡sq✉✐s❛ ❝✐❡♥tí✜❝❛✱ t❛✐s ❝♦♠♦ ❛ ❜✉s❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s✱ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞♦ t❡♠❛✱ ❡①♣❡r✐♠❡♥t❛çõ❡s✱ ❡t❝✳
◆ã♦ é ♣♦ssí✈❡❧ ❞❡✐①❛r ❞❡ ❝✐t❛r q✉❡ ❛♦ ❞❡s❡♥✈♦❧✈❡r ❛t✐✈✐❞❛❞❡s ❞❡ ♠♦❞❡❧❛❣❡♠ ♥❛s ❞✐✈❡rs❛s ❡t❛♣❛s ❞♦ ❡♥s✐♥♦✱ ♦ ♣r♦❢❡ss♦r ♣r❡❝✐s❛ ♠❡❞✐❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ s❡❥❛ ♦❜t✐❞♦ ✐♥t❡❣r❛❧♠❡♥t❡✳ P❛r❛ ✐ss♦✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡✱ ❞✉r❛♥t❡ q✉❛❧q✉❡r ♠♦♠❡♥t♦ ❞♦ ♣r♦❝❡ss♦✱ r❡t♦♠❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s ♥❡❝❡s✲ sár✐♦s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛t✐✈✐❞❛❞❡ ❡ ❛té ♠❡s♠♦ ♦❢❡r❡❝❡r ❛♦s ❛❧✉♥♦s ❛❧❣✉♠❛ ❛t✐✈✐❞❛❞❡ ❝♦♠♣❧❡♠❡♥t❛r ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛❣✉ç❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ s♦❜r❡ ❞❡t❡r♠✐♥❛❞♦ t❡♠❛✳ ❙❡❣✉♥❞♦ ❆❧♠❡✐❞❛ ❬✼❪
✏♦ ❢♦❝♦ ❡stá ♥♦s ❡♥❝❛♠✐♥❤❛♠❡♥t♦s ❡ ♣r♦❝❡❞✐♠❡♥t♦s q✉❡ ♠❡❞❡✐❛♠ ❛ tr❛♥✲ s✐çã♦ ❞❛ s✐t✉❛çã♦ ✐♥✐❝✐❛❧ ♣❛r❛ ❛ s✐t✉❛çã♦ ✜♥❛❧✑
❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✷✺
▼✉✐t♦s tr❛❜❛❧❤♦s ❝✐❡♥tí✜❝♦s ♥❛ ár❡❛ ❞❡ ❊❞✉❝❛çã♦ ▼❛t❡♠át✐❝❛ ❢♦r❛♠✱ ❡ ❡stã♦ s❡♥❞♦✱ ❞❡s❡♥✈♦❧✈✐❞♦s s♦❜r❡ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❞❡✈✐❞♦ ❛♦ s❡✉ ❣r❛♥❞❡ s✐❣♥✐✜❝❛❞♦ ❡ ♦❜t❡♥✲ çã♦ ❞❡ r❡s✉❧t❛❞♦s ♣♦s✐t✐✈♦s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❛tr❛✈és ❞❡ s✉❛ ❛❜♦r❞❛❣❡♠✳ ◆♦rt❡❛❞♦r❡s ❛t✉❛✐s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✮ ❬✾❪ ❡ ❛s ♦r✐❡♥t❛çõ❡s ❝♦♠♣❧❡♠❡♥t❛r❡s ❛ ❡ss❡s ♣❛râ♠❡tr♦s✱ ♦s P❈◆✰ ❬✶✵❪ ✐♥❝❡♥t✐✈❛♠ ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ❡♠ t♦❞♦s ♦s ♥í✈❡✐s ❞❡ ❡♥s✐♥♦✱ ❝♦♠ ✐ss♦ ♠✉✐t♦s ❛✉t♦r❡s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s ✐♥❝❧✉✐r❛♠ ❡♠ s✉❛s ♣r♦♣♦st❛s ❡st❛ ❛❜♦r❞❛❣❡♠✳
❋✐③❡♠♦s ❛ ❛♥á❧✐s❡ ❞❡ ❞♦✐s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ✈❡r✐✜❝❛r s❡ ❛ ♠♦❞❡❧❛❣❡♠ é s✉❣❡r✐❞❛ ❝♦♠♦ ❡str❛té❣✐❛ ❞❡ ❡♥s✐♥♦ ♣❛r❛ ❋✉♥çõ❡s✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ ❋✉♥çõ❡s ❊①♣♦♥❡♥❝✐❛✐s ❡ ▲♦❣❛rít♠✐❝❛s✳ ❆❧é♠ ❞✐ss♦✱ ❛♥❛✲ ❧✐s❛♠♦s ♦ ♠❛♥✉❛❧ ❞♦ ♣r♦❢❡ss♦r ❞❡st❡s ❧✐✈r♦s ❞✐❞át✐❝♦s ❜✉s❝❛♥❞♦ ❛s ♦r✐❡♥t❛çõ❡s ❞❛❞❛s ♣❡❧♦s ❛✉t♦r❡s s♦❜r❡ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳
❆q✉✐ ❞❡s✐❣♥❛r❡♠♦s ❝❛❞❛ ❧✐✈r♦ ❞✐❞át✐❝♦ ♣❡❧❛ s✐❣❧❛ ▲❉ ❝♦♥❢♦r♠❡ ❛ t❛❜❡❧❛ ✷✳✶✿ ▲❉ ✶ ❈♦♥❡①õ❡s ❝♦♠ ❛ ▼❛t❡♠át✐❝❛ ❬✶❪
▲❉ ✷ ▼❛t❡♠át✐❝❛✿ ❈♦♥t❡①t♦ ❡ ❆♣❧✐❝❛çõ❡s ❬✷❪
❚❛❜❡❧❛ ✷✳✶✿ ▲✐✈r♦s ❞✐❞át✐❝♦s ❛♥❛❧✐s❛❞♦s
❆♥á❧✐s❡ ▲❉✶
■♥✐❝✐❛❧♠❡♥t❡ ❛♥❛❧✐s❛♠♦s ♦ ▲❉ ✶ ❡ ✈❡r✐✜❝❛♠♦s q✉❡ ♥♦s ❈❛♣ít✉❧♦s ✸ ✭❋✉♥çõ❡s✮✱ ✼ ✭❋✉♥✲ çã♦ ❊①♣♦♥❡♥❝✐❛❧✮ ❡ ✽ ✭❋✉♥çã♦ ▲♦❣❛rít♠✐❝❛✮ ♥ã♦ ❤á ♥❡♥❤✉♠❛ ♣r♦♣♦st❛ ❞❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ❆❧❣✉♠❛s s✐t✉❛çõ❡s ♣r♦❜❧❡♠❛ ❛❜♦r❞❛❞❛s ♣❡❧♦ ❧✐✈r♦ ♣♦❞❡r✐❛♠ s❡r ♠♦❞❡❧❛✲ ❞❛s ♣❡❧♦s ❛❧✉♥♦s✱ ♣♦ré♠ ♦ ♣ró♣r✐♦ ❧✐✈r♦ ♦❢❡r❡❝❡ ♦ ❡①❡♠♣❧♦ ❡ ❡❢❡t✉❛ ❛ ♠♦❞❡❧❛❣❡♠ ❡ ❞❡st❛ ❢♦r♠❛ ♦ ❛❧✉♥♦ s❡ t♦r♥❛ ❛♣❡♥❛s ❧❡✐t♦r ❡ ♥ã♦ ❛❣❡♥t❡ ♣❛rt✐❝✐♣❛t✐✈♦ ❞❛ ♠♦❞❡❧❛❣❡♠✳ ◆❡♠ ♠❡s♠♦ ♥♦s ❡①❡r❝í❝✐♦s ♣r♦♣♦st♦s ♦✉ ❝♦♠♣❧❡♠❡♥t❛r❡s ❢♦r❛♠ ❡♥❝♦♥tr❛❞❛s ♦♣♦rt✉♥✐❞❛❞❡s ❞❡ ♠♦❞❡❧❛❣❡♠ ❛ s❡r❡♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❡❧♦s ❛❧✉♥♦s✳
❱❡❥❛♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ❛❜♦r❞❛❣❡♠ ❞♦ ▲❉✶✿
❙✐t✉❛çã♦ ✶✿ ❈❛♣ít✉❧♦ ✸ ✭❋✉♥çõ❡s✮
✷✻ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
❋✐❣✉r❛ ✷✳✷✿ ❙✐t✉❛çã♦ ✶✳ ❋♦♥t❡✿ ❬✶❪ ✲ ♣❛❣ ✺✽
❙✐t✉❛çã♦ ✷✿ ❈❛♣ít✉❧♦ ✽ ✭❋✉♥çã♦ ▲♦❣❛rít♠✐❝❛✮
❆ s✐t✉❛çã♦ ✷✱ r❡♣r❡s❡♥t❛❞❛ ♥❛ ❋✐❣✉r❛ ✷✳✸✱ t❛♠❜é♠ tr❛③ ❝♦♥❞✐çõ❡s ❞❡ ♠♦❞❡❧❛❣❡♠✱ ❝♦♠ ✉♠ t❡♠❛ q✉❡ ♣❡r♠✐t❡ r❡❧❛❝✐♦♥❛r ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❧é♠ ❞❡ ❞❡s❡♥✈♦❧✈❡r ❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛✱ ♣♦ré♠ ♦ ❧✐✈r♦✱ ♥♦✈❛♠❡♥t❡✱ ♥ã♦ ♣❡r♠✐t❡ ❛ ♣❛r✲ t✐❝✐♣❛çã♦ ❛t✐✈❛ ❞♦ ❛❧✉♥♦ ♥❛ ❝♦♥str✉çã♦ ❞♦ ♠♦❞❡❧♦ q✉❡ ♠❡❧❤♦r r❡♣r❡s❡♥t❛ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❝é❧✉❧❛s✳ ❆ s✐t✉❛çã♦ ❛♣❛r❡❝❡ ♥♦ ❢♦r♠❛t♦ ❞❡ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ❢✉♥✲ çã♦ ❧♦❣❛rít♠✐❝❛ ❡ ♥ã♦ ♣❡r♠✐t❡ ❛ ♣❛rt✐❝✐♣❛çã♦ ❞♦ ❛❧✉♥♦ ♥❛ ♠❛t❡♠❛t✐③❛çã♦✳ ❱❡❥❛♠♦s ❛ ❋✐❣✉r❛ ✷✳✸ q✉❡ ❛♣r❡s❡♥t❛ ✉♠❛ r❡♣r♦❞✉çã♦ ❞♦ ❡①❡♠♣❧♦ ❡♥❝♦♥tr❛❞♦ ♥♦ ▲❉✶✿
❆♦ ❛♥❛❧✐s❛r ♦ ✏●✉✐❛ ❞♦ ♣r♦❢❡ss♦r✑ ❞♦ ▲❉ ✶ ♥ã♦ ❡♥❝♦♥tr❛♠♦s ♥❡♥❤✉♠❛ ♠❡♥çã♦ ❡①♣❧í❝✐t❛ s♦❜r❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ❝♦♠♦ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ❡♥s✐♥♦ ♦✉ ❛❜♦r❞❛❣❡♠ ❞❡ ❝♦♥t❡ú❞♦s✳
❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✷✼
❋✐❣✉r❛ ✷✳✸✿ ❙✐t✉❛çã♦ ✷✳ ❋♦♥t❡✿ ❬✶❪ ✲ ♣❛❣ ✶✽✷
❆♥á❧✐s❡ ▲❉✷
❊♠ s❡❣✉✐❞❛✱ ❛♥❛❧✐s❛♠♦s ♦ ▲❉ ✷ ❡ ♥♦s ❈❛♣ít✉❧♦s ✷ ✭❋✉♥çõ❡s✮✱ ✺ ✭❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧✮ ❡ ✻ ✭▲♦❣❛r✐t♠♦ ❡ ❋✉♥çã♦ ▲♦❣❛rít♠✐❝❛✮ ❢♦r❛♠ ❡♥❝♦♥tr❛❞❛s ❞✐✈❡rs❛s ❛❜♦r❞❛❣❡♥s q✉❡ ♣❡r♠✐t❡♠ ❛ r❡❛❧✐③❛çã♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ❢♦r♠❛ ❞❡ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ ✏❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛✑ q✉❡ ♠❡❧❤♦r r❡♣r❡s❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛✱ ❝♦♠♦ r❡♣r❡s❡♥t❛❞♦ ♥❛ s✐t✉❛çã♦ ✸✳
❙✐t✉❛çã♦ ✸✿ ❈❛♣ít✉❧♦ ✷ ✭❋✉♥çõ❡s✮
❆ s✐t✉❛çã♦ ✸✱ r❡♣r❡s❡♥t❛❞❛ ♥❛ ❋✐❣✉r❛ ✷✳✹✱ ♦❢❡r❡❝❡ ✉♠❛ t❛❜❡❧❛ r❡❧❛❝✐♦♥❛♥❞♦ ♦ ♥ú✲ ♠❡r♦ ❞❡ ♣❡ç❛s ❞❡ ✐♥❢♦r♠át✐❝❛ ♣r♦❞✉③✐❞❛s ❡ ♦ ❝✉st♦ ❞❡ss❛s ♣❡ç❛s✳ ❆s ♣❡r❣✉♥t❛s ❛✮ ❡ ❜✮ ♣❡r♠✐t❡♠ ❛♦ ❛❧✉♥♦ ❛♥❛❧✐s❛r ❛ s✐t✉❛çã♦ ✐♥t✉✐t✐✈❛♠❡♥t❡ ❜✉s❝❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✳ ❆ ♣❡r❣✉♥t❛ ❝✮ s♦❧✐❝✐t❛ ❛♦ ❛❧✉♥♦ q✉❡ ♠♦❞❡❧❡ ❛ s✐t✉❛çã♦✱ ❜✉s❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ q✉❡ ❞á ♦ ❝✉st♦ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ç❛s ❡ ❛s ♣❡r❣✉♥t❛s ❞✮ ❡ ❡✮ ❧❡✈❛♥t❛♠ q✉❡st✐♦✲ ♥❛♠❡♥t♦s ❞❡ ✐♥t❡r♣r❡t❛çã♦ ❡ ✉t✐❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ♦❜t✐❞♦ ♥♦ ✐t❡♠ ❝✮✳ ❊st❛ ❛❜♦r❞❛❣❡♠ ♣❡r♠✐t❡ q✉❡ ♦ ❛❧✉♥♦ ✐♥t❡r♣r❡t❡ ❡ ♠♦❞❡❧❡ ♦ ♣r♦❜❧❡♠❛ ❛❧é♠ ❞❡ ✉t✐❧✐③❛r ❡ss❛s ✐♥❢♦r♠❛çõ❡s ❝♦♠ ♥♦✈♦s ❡ ♣♦ssí✈❡✐s ✈❛❧♦r❡s✳ ❱❡❥❛♠♦s ❛ ❋✐❣✉r❛ ✷✳✹ q✉❡ ❛♣r❡s❡♥t❛ ✉♠❛ r❡♣r♦❞✉çã♦ ❞♦ ❡①❡r❝í❝✐♦ ❡♥❝♦♥tr❛❞♦ ♥♦ ▲❉✷✿
❆❧é♠ ❞♦ ❢♦r♠❛t♦ r❡♣r❡s❡♥t❛❞♦ ♥❛ s✐t✉❛çã♦ ✸ t❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s ❛❜♦r❞❛❣❡♥s q✉❡ ♣❡r♠✐t❡♠ ❛ ♠♦❞❡❧❛❣❡♠ ❝♦♠ ♣❛rt✐❝✐♣❛çã♦ ❝rít✐❝❛ ❡ ❡♥✈♦❧✈❡♥t❡ ❞♦s ❛❧✉♥♦s s❡ ❢♦r ❜❡♠ ♠❡❞✐❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r✳
❙✐t✉❛çã♦ ✹✿ ❈❛♣ít✉❧♦ ✺ ✭❋✉♥çã♦ ❊①♣♦♥❡♥❝✐❛❧✮
✷✽ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛
❋✐❣✉r❛ ✷✳✹✿ ❙✐t✉❛çã♦ ✸✳ ❋♦♥t❡✿ ❬✷❪ ✲ ♣❛❣ ✹✹
❛♣r❡s❡♥t❛❞❛ ❞❡ ✉♠❛ ❢♦r♠❛ q✉❡ ❛ ♠♦❞❡❧❛❣❡♠ s✉r❣❡ ♥❛t✉r❛❧ ❡ ✐♥t✉✐t✐✈❛♠❡♥t❡✳
◆❛ s✐t✉❛çã♦ ✹✱ q✉❡ ❡stá r❡♣r❡s❡♥t❛❞❛ ♥❛ ❋✐❣✉r❛ ✷✳✺✱ é ♣♦ssí✈❡❧ ❞❡t❡❝t❛r ❡❧❡♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠♦❞❡❧❛❣❡♠✱ ❝♦♠♦ ❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ s♦❜r❡ ♦ t❡♠❛ ❡ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ q✉❡stõ❡s q✉❡ ✐♥st✐❣❛♠ ♦s ❛❧✉♥♦s✳ ◆❡st❛ s✐t✉❛çã♦ ❛ ♠♦❞❡❧❛❣❡♠ s✉r❣❡ ♥❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛✱ ♥❛ ♦❜t❡♥çã♦ ❞❛ ❧❡✐ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ q✉❡ ❞❡s❝r❡✈❡ ❛ s✐t✉❛çã♦ ❡ ♥❛ ♣♦ssí✈❡❧ ♣❡r❝❡♣çã♦ ❞❡ q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥ã♦ s❡rá ✉♠❛ r❡t❛✳ ❱❡❥❛♠♦s ❛ ❋✐❣✉r❛ ✷✳✺ q✉❡ ❛♣r❡s❡♥t❛ ✉♠❛ r❡♣r♦❞✉çã♦ ❞♦ ❡①❡r❝í❝✐♦ ❡♥❝♦♥tr❛❞♦ ♥♦ ▲❉✷✿
❆♦ ❛♥❛❧✐s❛r ♦ ✏▼❛♥✉❛❧ ❞♦ Pr♦❢❡ss♦r✑ ❞♦ ▲❉ ✷ ✈❡r✐✜❝❛♠♦s ✉♠❛ s❡çã♦ s♦❜r❡ ❊t♥♦✲ ♠❛t❡♠át✐❝❛ ❡ ▼♦❞❡❧❛❣❡♠ ❡ ❡♥tr❡ ❛s ❞❡s❝r✐çõ❡s ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛ ❝♦❧❡çã♦ ❤á ✉♠ tr❡❝❤♦ q✉❡ ❞✐③✿
✏❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ é ❢❡✐t❛ ♣❡❧❛ ♣r♦❝✉r❛ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ❛ ♣❛rt✐r ❞❡ ♣r♦❜❧❡♠❛s r❡❛✐s✑✳
❉❡st❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ▲❉ ✷ tr❛③ ❜♦❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ♠♦❞❡❧❛❣❡♠ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ❞á ✉♠ s✉♣♦rt❡ ❛♦ ♣r♦❢❡ss♦r q✉❡ ♠❡❞✐❛rá ❡ss❡ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦✳
❆ ▼♦❞❡❧❛❣❡♠ ❡ ♦ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ✷✾
❋✐❣✉r❛ ✷✳✺✿ ❙✐t✉❛çã♦ ✹✳ ❋♦♥t❡✿ ❬✷❪ ✲ ♣❛❣ ✶✹✼
✸ ❋✉♥çõ❡s
◆❛ ♠❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ❡♠ ♠✉✐t❛s ♦✉tr❛s ❝✐ê♥❝✐❛s✱ ♠✉✐t❛s ✈❡③❡s é ♥❡❝❡ssár✐♦ ❡st❛❜❡❧❡❝❡r ❛❧❣✉♠❛ r❡❧❛çã♦ ♦✉ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ❝♦♥❥✉♥t♦s ♦✉ ❣r❛♥❞❡③❛s✳ ❆ ♥♦çã♦ ❞❡ ❢✉♥çã♦ ♣❛rt❡ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ✈❛r✐á✈❡✐s❀ ✐♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ r❡❧❛çã♦ ♣❛r❛ ♣♦st❡r✐♦r♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❞❡ r❡❧❛çã♦ ❞❡✜♥✐r♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çõ❡s✳ ❆❜♦r❞❛r❡♠♦s t❛♠❜é♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❡❧❡♠❡♥t❛r❡s ❞❛s ❢✉♥çõ❡s✱ ❞❡✜♥✐r❡♠♦s ❣rá✜❝♦ ❞❡ ❢✉♥çõ❡s tr❛③❡♥❞♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ♠✉✐t♦ s✐❣♥✐✜❝❛t✐✈♦s ❞❡ ❢✉♥çõ❡s ❡ s❡✉s ❣rá✜❝♦s✳ ❆✐♥❞❛ ♥❡st❡ ❝❛♣ít✉❧♦ tr❛t❛r❡♠♦s s♦❜r❡ ❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✱ ❞❡✜♥✐♥❞♦✲❛s ❛tr❛✈és ❞♦ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥t❡r♠❡❞✐ár✐♦ ❡ ♠♦str❛♥❞♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s✳ P❛r❛ ❡♥❝❡rr❛r✱ ✈❛♠♦s ❞❡✜♥✐r ❢✉♥çã♦ ✐♥✈❡rs❛ ❡ ♠♦str❛r ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ s❡✉ ❣rá✜❝♦✳ ❆s ❞❡✜♥✐çõ❡s✱ ❡①❡♠♣❧♦s ❡ r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ r❡t✐r❛❞❛s ❞❛s r❡❢❡rê♥❝✐❛s ❬✸❪✱ ❬✶✶❪✱ ❬✶✷❪ ❡ ❬✶✸❪✳
✸✳✶ ❆ ❉❡✜♥✐çã♦ ❞❡ ❋✉♥çã♦
❉❡✜♥✐çã♦ ✸✳✶✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s X✱ Y✱ ✉♠❛ r❡❧❛çã♦ ❞❡X ❡♠ Y ✭♦✉
❡♥tr❡ ❳ ❡ ❨✱ ♥❡ss❛ ♦r❞❡♠✮ é ✉♠ s✉❜❝♦♥❥✉♥t♦ R ❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ X×Y✱ ✐✳❡✳✱ R
é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞♦ t✐♣♦ (x, y)✱ ❝♦♠ x∈X ❡ y∈Y✳
❙❡R é ✉♠❛ r❡❧❛çã♦ ❞❡ X ❡♠ Y✱ ❡♥tã♦ R ⊂X×Y ♣♦r ❞❡✜♥✐çã♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱
❡s❝♦❧❤✐❞♦ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (x, y)✱ t❡♠♦s q✉❡ (x, y) ∈ R ♦✉ (x, y) 6∈ R ✭✐✳é✱ q✉❡ x ❡ y
s❡❥❛♠ r❡❧❛❝✐♦♥❛❞♦s ♣♦r R ♦✉ ♥ã♦✮✳
❉❡✜♥✐çã♦ ✸✳✷✳ ❉❛❞♦s ❞♦✐s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s X ❡ Y✱ ✉♠❛ r❡❧❛çã♦ q✉❡ ❛ ❝❛❞❛
❡❧❡♠❡♥t♦x∈X ❛ss♦❝✐❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ yt❛❧ q✉❡ (x, y)∈f é ❞❡♥♦♠✐♥❛❞❛ ❢✉♥çã♦✳
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ∀x∈X,∃ ✉♠ ú♥✐❝♦ y∈Y t❛❧ q✉❡ (x, y)∈f✳
◆♦t❛çã♦✿ f :X →Y ❡ y=f(x)✳
❉❡♥♦♠✐♥❛♠♦s ♦ ❝♦♥❥✉♥t♦ X ❞❡ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❡ ❞❡♥♦t❛♠♦s X =Dom(f)✱ ♦
❝♦♥❥✉♥t♦ Y ❞❡ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❡ ♦ ❡❧❡♠❡♥t♦ y=f(x)∈Y ❞❡ ✐♠❛❣❡♠ ❞❡ x∈X ♣❡❧❛ ❢✉♥çã♦ f✳
●❡r❛❧♠❡♥t❡✱ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦f :X →Y✱ ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡
❛ ✐♠❛❣❡♠ ❞❛ ❢✉♥çã♦ f é ♦ ❝♦♥❥✉♥t♦ Im(f)✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s sã♦ ❛s ✐♠❛❣❡♥sf(x)∈Y
✸✷ ❋✉♥çõ❡s
❞♦s ❡❧❡♠❡♥t♦s x ∈ X✱ ♦✉ s❡❥❛✱ Im(f) = {f(x) ∈ Y;x ∈ X}✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠♦s
s❡♠♣r❡ Im(f)⊂Y ❡ ♣♦❞❡ ♦❝♦rr❡r q✉❡ Im(f)6=Y✳
➱ ♠✉✐t♦ ❝♦♠✉♠ ✉t✐❧✐③❛r ❞✐❛❣r❛♠❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çõ❡s✱ ❛ss✐♠ ❝♦♠♦ ❢❛r❡♠♦s ♥❛ ❋✐❣✉r❛ ✸✳✶✱ ♦♥❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦x ❞♦ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦X é r❡❧❛❝✐♦✲
♥❛❞♦ ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ y ❞♦ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ Y✳
❋✐❣✉r❛ ✸✳✶✿ ❉✐❛❣r❛♠❛ ❞❡ ✢❡❝❤❛s ❡①❡♠♣❧✐✜❝❛♥❞♦ ❛ ❢✉♥çã♦ f ❞❡ X ❡♠ Y✳
◆❛ ✜❣✉r❛ ✸✳✶ t❡♠♦s X = {1,2,3,4}✱ Y = {a, b, c, d, e, f} ❡ f(1) = a✱ f(2) = c✱ f(3) =c ❡ f(4) =e✱ ❧♦❣♦ a é ❛ ✐♠❛❣❡♠ ❞❡ ✶✱ c é ❛ ✐♠❛❣❡♠ ❞❡ ✷ ❡ ❞❡ ✸ ❡ e ❛ ✐♠❛❣❡♠
❞❡ ✹ ♣♦rf✳ ❆❧é♠ ❞✐ss♦ ❛ ❢✉♥çã♦f ❢♦r♠❛r✐❛ ♦s s❡❣✉✐♥t❡s ♣❛r❡s ♦r❞❡♥❛❞♦s✿ (1, a)✱(2, c)✱ (3, c)✱ (4, e)✳
❊①❡♠♣❧♦ ✸✳✶✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f :R→R ❞❛❞❛ ♣♦r f(x) = x2+ 1✳
❚❡♠♦s ❛q✉✐ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ f(x) = x2 + 1 q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛
❡❧❡♠❡♥t♦x∈Rs❡✉ q✉❛❞r❛❞♦ ❛❞✐❝✐♦♥❛❞♦ ❞❡ ✶ ✉♥✐❞❛❞❡✳ ◆❡st❡ ❡①❡♠♣❧♦ t❛♥t♦ ♦ ❞♦♠í♥✐♦
❝♦♠♦ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ sã♦ ♦s ♥ú♠❡r♦s r❡❛✐s R ❡ ❛ ✐♠❛❣❡♠✱ sã♦ ♦s ♥ú♠❡r♦s
r❡❛✐s ♣♦s✐t✐✈♦s R+✳
❊①❡♠♣❧♦ ✸✳✷✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f :R→R ❞❛❞❛ ♣♦r✿
f(x) =
(
x+ 1, x >0
x2, x≤0 ✭✸✳✶✮
❚❡♠♦s ❛q✉✐ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r ❞✉❛s ❡①♣r❡ssõ❡sf(x) = x+1❡f(x) =x2❀ ♥❡st❡
❝❛s♦ ❞✐③❡♠♦s q✉❡f ❡stá ❞❡✜♥✐❞❛ ♣♦r ♣❛rt❡s✱ ❞❡ ♠♦❞♦ q✉❡ q✉❛♥❞♦x∈(−∞,0]✱f(x)
♦ r❡❧❛❝✐♦♥❛ ❛♦ s❡✉ q✉❛❞r❛❞♦ ❡✱ q✉❛♥❞♦ x ∈ (0,∞)✱ f(x) ♦ r❡❧❛❝✐♦♥❛ ❛♦ s❡✉ s✉❝❡ss♦r✳
◆❡st❡ ❡①❡♠♣❧♦ ♦ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ sã♦ ♦s ♥ú♠❡r♦s r❡❛✐s R ❡ ❛ ✐♠❛❣❡♠ sã♦ ♦s
♥ú♠❡r♦s r❡❛✐s ♣♦s✐t✐✈♦s ❡ ♦ ③❡r♦ R+∪ {0}✳
◆❡st❡ t❡①t♦ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❛s ❢✉♥çõ❡s f :X →Y ❞❡ t❛❧ ❢♦r♠❛ q✉❡ X, Y ⊂R✳
❉❡st❛ ❢♦r♠❛✱ ❛ ❢✉♥çã♦ f :X →R t❛❧ q✉❡ X ⊂ R é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ r❡❛❧ ❞❡ ✉♠❛
✈❛r✐á✈❡❧ r❡❛❧✱ ❥á q✉❡ f ❛ss✉♠❡ ✈❛❧♦r❡s r❡❛✐s ✭♦✉ s❡❥❛✱ s❡✉ ❝♦♥tr❛❞♦♠í♥✐♦ é R✮ ❡ ❝❛❞❛
❆ ❉❡✜♥✐çã♦ ❞❡ ❋✉♥çã♦ ✸✸
◆♦ ❝♦♥t❡①t♦ ❞❡ ❢✉♥çõ❡s r❡❛✐s✱ t❡♠♦s ❝♦♠♦ ❝♦♥str✉✐r ♥♦✈❛s ❢✉♥çõ❡s ❛ ♣❛rt✐r ❞❡ ♦✉✲ tr❛s ❥á ❝♦♥❤❡❝✐❞❛s✱ ✉t✐❧✐③❛♥❞♦ ❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦ R❀ ❛ss✐♠
❞❡✜♥✐♠♦s ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❢✉♥çõ❡s f ❡g ❡ ♦ ♣r♦❞✉t♦ c·f ❞❡ ✉♠ ♥ú♠❡r♦
r❡❛❧ c ♣❡❧❛ ❢✉♥çã♦f✳
❉❡✜♥✐çã♦ ✸✳✸✳ ❉❛❞♦s ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ X ⊂ R✱ ✉♠ ♥ú♠❡r♦ r❡❛❧ c ❡ ❢✉♥çõ❡s
r❡❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ f, g : X → R ✭f ❡ g ❞❡ ♠❡s♠♦ ❞♦♠í♥✐♦✮✱ ❞❡✜♥✐♠♦s ❛s
❢✉♥çõ❡s✿
• f +g :X →R ❝♦♠♦ (f+g)(x) =f(x) +g(x)❀ ♣❛r❛ t♦❞♦ x∈X✳
• f −g : X → R ❝♦♠♦ (f −g)(x) = f(x) + (−g(x)) = f(x)−g(x)❀ ♣❛r❛ t♦❞♦
x∈X✳
• f ·g :X →R ❝♦♠♦ (f ·g)(x) = f(x)·g(x)❀ ♣❛r❛ t♦❞♦ x∈X✳
• f /g :X →R ❝♦♠♦ (f /g)(x) = f(x)/g(x)❀ ♣❛r❛ t♦❞♦ x∈X✳
• c·f :X →R ❝♦♠♦ (c·f)(x) = c·f(x)❀ ♣❛r❛ t♦❞♦ x∈X✳
❖❜s❡r✈❛çã♦✿ P❛r❛ ❛s ❢✉♥çõ❡sf+g✱f−g✱f·g✱ c·f✱ ♦ ❞♦♠í♥✐♦ ♣❡r♠❛♥❡❝❡rá s❡♥❞♦ X ✈✐st♦ q✉❡ X∩X =X ❡ ♣❛r❛ f /g t❡♠♦s ❛ r❡str✐çã♦ ❞❡g(x)6= 0 ❡ s❡✉ ❞♦♠í♥✐♦ s❡rá
{x∈X, g(x)6= 0}✳
❊①❡♠♣❧♦ ✸✳✸✳ ❙❡♥❞♦ c = 4✱ f ❡ g ❛s ❢✉♥çõ❡s ❞❡ R ❡♠ R ❞❛❞❛s ♣♦r f(x) = x−3 ❡
g(x) = x
x+ 3✱ t❡♠♦s✿
(f+g)(x) = f(x) +g(x)
= (x−3) + x
x+ 3
= (x+ 3)(x−3) +x
x+ 3
= x
2−3x+ 3x−9 +x
x+ 3
= x
2+x−9
x+ 3 ;
(f ·g)(x) = f(x)·g(x) = (x−3)· x
x+ 3
= (x−3)(x)
x+ 3
= x
2−3x
✸✹ ❋✉♥çõ❡s
(c·f)(x) =c·f(x) = 4·(x−3) = 4x−12.
✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❊❧❡♠❡♥t❛r❡s ❞❛s ❋✉♥çõ❡s
❉❡✜♥✐çã♦ ✸✳✹✳ ❯♠❛ ❢✉♥çã♦ f :X →Y é ❞✐t❛✿• ■♥❥❡t♦r❛✱ ♦✉ ✐♥❥❡t✐✈❛ s❡ ♣❛r❛ t♦❞♦ y∈Y✱ ❡①✐st✐r ♥♦ ♠á①✐♠♦ ✉♠x ∈X t❛❧ q✉❡ f(x) = y✳
• ❙♦❜r❡❥❡t♦r❛✱ ♦✉ s♦❜r❡❥❡t✐✈❛ s❡ s✉❛ ✐♠❛❣❡♠ ❢♦r t♦❞♦ ♦ ❝♦♥❥✉♥t♦Y✱ ✐✳é✳✱ s❡✱ ♣❛r❛
t♦❞♦ y∈Y✱ ❡①✐st✐r ♣❡❧♦ ♠❡♥♦s ✉♠ x∈X✱ t❛❧ q✉❡ y=f(x)✳
• ❇✐❥❡t♦r❛✱ ♦✉ ❜✐❥❡t✐✈❛✱ s❡ ❢♦r ❛♦ ♠❡s♠♦ t❡♠♣♦ ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✳
❯♠❛ ♠❛♥❡✐r❛ ❡✜❝❛③ ❞❡ ✈❡r✐✜❝❛r s❡ ✉♠❛ ❢✉♥çã♦ f : X → Y é ✐♥❥❡t♦r❛ é ✈❡r✐✜❝❛r s❡
❛ ✐♠♣❧✐❝❛çã♦f(x1) =f(x2)⇒x1 =x2 é s❛t✐s❢❡✐t❛✱ ♣❛r❛ t♦❞♦s x1, x2 ∈X✳
❊①❡♠♣❧♦ ✸✳✹✳ ❆ ❢✉♥çã♦ f : R→ R ❞❛❞❛ ♣♦r f(x) = 3x é ✐♥❥❡t♦r❛✱ ♣♦✐s ❢❛③ ❝♦rr❡s✲
♣♦♥❞❡r ❛ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ x s❡✉ tr✐♣❧♦✱ ❡ ♥ã♦ ❡①✐st❡♠ ❞♦✐s ♥ú♠❡r♦s r❡❛✐s ❞✐❢❡r❡♥t❡s
q✉❡ t❡♥❤❛♠ ♦ ♠❡s♠♦ tr✐♣❧♦✱ ♦✉ s❡❥❛✿
P❛r❛ q✉❛✐sq✉❡r x1, x2 ∈R, f(x1) = f(x2)⇒3x1 = 3x2 ⇒x1 =x2
❊①❡♠♣❧♦ ✸✳✺✳ ❆ ❢✉♥çã♦ f : R → R+ ❞❛❞❛ ♣♦r f(x) = x2 é s♦❜r❡❥❡t♦r❛✱ ♣♦✐s t♦❞♦
❡❧❡♠❡♥t♦ ❞❡R+ é ✐♠❛❣❡♠ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ❞❡R ♣❡❧❛ ❢✉♥çã♦x=±p
f(x)✳
❖❜s❡r✈❡✿
• f(x) = 9 é ✐♠❛❣❡♠ ❞❡ x= 3 ❡ ❞❡ x=−3✱ (±√9)
• f(x) = 0 é ✐♠❛❣❡♠ ❞❡ x= 0✱ (±√0)
• f(x) = 2 é ✐♠❛❣❡♠ ❞❡ x=√2 ❡ ❞❡ x=−√2✱ (±√2)
❊①❡♠♣❧♦ ✸✳✻✳ ❆ ❢✉♥çã♦ f : R → R ❞❛❞❛ ♣♦r f(x) = 3x é ❜✐❥❡t♦r❛✱ ♣♦✐s ❡❧❛ é
s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✱ ❝❛❞❛ ♥ú♠❡r♦ r❡❛❧ ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦ R t❡♠
❝♦♠♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♥♦ ❞♦♠í♥✐♦ ❛ s✉❛ t❡rç❛ ♣❛rt❡ q✉❡ s❡♠♣r❡ ❡①✐st❡ ❡ é ú♥✐❝❛✳ ❉❡✜♥✐çã♦ ✸✳✺✳ ❙❡❥❛ X ⊂ R✱ ❞✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : X → R é ♣❛r s❡ f(x) =
f(−x)✱ ♣❛r❛ t♦❞♦ x ∈ X✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ✉♠❛ ❢✉♥çã♦ f : X → R é í♠♣❛r s❡ f(x) =
−f(−x)✱ ♣❛r❛ t♦❞♦ x∈X✳
❊①❡♠♣❧♦ ✸✳✼✳ ❆ ❢✉♥çã♦ f :R→R ❞❛❞❛ ♣♦rf(x) =x2 é ♣❛r✱ ♣♦✐s ♣❛r❛ t♦❞♦ x∈R✱
●rá✜❝♦s ❞❡ ❋✉♥çõ❡s ✸✺
❊①❡♠♣❧♦ ✸✳✽✳ ❆ ❢✉♥çã♦ f : R → R ❞❛❞❛ ♣♦r f(x) = 5x é í♠♣❛r✱ ♣♦✐s ♣❛r❛ t♦❞♦ x∈R✱ f(−x) = 5(−x) = −5(x) = −f(x)✳
❉❛❞❛s ❛s ❢✉♥çõ❡s f : X → Y ❡ g : Y → Z t❡♠♦s✱ ❡♠ ú❧t✐♠❛ ❛♥á❧✐s❡✱ r❡❣r❛s
❜❡♠ ❞❡✜♥✐❞❛s ♣❛r❛✱ ♣❛rt✐♥❞♦ ❞❡ x ∈ X ✈✐❛ f✱ ♦❜t❡r y = f(x) ∈ Y ❡✱ ✈✐❛ g✱ ♦❜t❡r z = g(y) ∈ Z✳ P❛r❡❝❡✱ ❡♥tã♦✱ r❛③♦á✈❡❧ q✉❡ ♣♦ss❛♠♦s ❢♦r♠❛r ✉♠❛ ❢✉♥çã♦ q✉❡ ♥♦s
♣❡r♠✐t❛ s❛✐r ❞❡ X ❡ ✐r ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ Z✳ ❊st❛ ❢✉♥çã♦ r❡s✉❧t❛♥t❡ é ❞❡♥♦♠✐♥❛❞❛
❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ f ❡g✳
❉❡✜♥✐çã♦ ✸✳✻✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f :X →Y ❡ g :Y →Z✱ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ ❞❡ f
❡ g ✭♥❡st❛ ♦r❞❡♠✮ é ❛ ❢✉♥çã♦ g◦f :X →Z ❞❡✜♥✐❞❛✱ ♣❛r❛ ❝❛❞❛ x∈X✱ ♣♦r✿
(g◦f)(x) = g(f(x)).
❙❡♥❞♦ ❛ss✐♠✱ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ✐♠❛❣❡♠ ❞❡x∈X♣♦r(g◦f)✱ ❜❛st❛ ❡♥❝♦♥tr❛r♠♦s
❛ ✐♠❛❣❡♠ ❞❡ f(x)∈Y ♣♦r g✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ❢❛③❡r ❛ ❝♦♠♣♦s✐çã♦ é ♥❡❝❡ssár✐♦ q✉❡ ❛
✐♠❛❣❡♠ ❞❡ f ❡st❡❥❛ ❝♦♥t✐❞❛ ♥♦ ❞♦♠í♥✐♦ ❞❡ g✳
❊①❡♠♣❧♦ ✸✳✾✳ ❉❛❞❛s ❛s ❢✉♥çõ❡s f, g : R → R ❞❡✜♥✐❞❛s ♣♦r f(x) = x+ 1 ❡ g(x) =
x2+x+ 1✱ ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛ h= (g◦f)s❡rá✿
h(x) = (g◦f)(x) =g(f(x)) = [f(x)2+f(x)+1] = (x+1)2+(x+1)+1 =x2+3x+3✳
✸✳✸ ●rá✜❝♦s ❞❡ ❋✉♥çõ❡s
◆❡st❛ s❡çã♦ ❛❜♦r❞❛r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❢♦r♠❛❧ ❞♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❛❝♦♠♣❛♥❤❛❞♦ ❞❡ ❡①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s ❜❛st❛♥t❡ s✐❣♥✐✜❝❛t✐✈❛s ❡ s❡✉s ❣rá✜❝♦s✳
❉❡✜♥✐çã♦ ✸✳✼✳ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f : X → Y✱ ♦ ❣rá✜❝♦ ❞❡ f é ♦ s✉❜❝♦♥❥✉♥t♦ Gf
❞♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ X × Y ❞❡✜♥✐❞♦ ♣♦r Gf = {(x, y) ∈ X ×Y;y = f(x)} ♦✉
Gf ={(x, f(x));x∈X}✳
P❛r❛ q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ Gf ⊂X×Y s❡❥❛ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f :X →Y é
♥❡❝❡ssár✐♦ q✉❡ Gf ❝✉♠♣r❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
• P❛r❛ t♦❞♦ x ∈X ❡①✐st❡ ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (x, y) ∈Gf ❝✉❥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛
é x✳
• ❙❡ P(x, y) ❡ P′(x, y′) sã♦ ♣❛r❡s ♣❡rt❡♥❝❡♥t❡s ❛ Gf ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛ x
❡♥tã♦ y =y′✳
◗✉❛♥❞♦ f : X → Y ❢♦r ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ ❝♦♠ X ⊂ R ✉♠❛
✉♥✐ã♦ ✜♥✐t❛ ❞❡ ✐♥t❡r✈❛❧♦s ✭♣♦ss✐✈❡❧♠❡♥t❡ X =R✮ ♦ ❣rá✜❝♦ ❞❡f ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦
✸✻ ❋✉♥çõ❡s
❖s ❡❧❡♠❡♥t♦s (x, y) ❞❡ R2 sã♦✱ ♥❛t✉r❛❧♠❡♥t❡✱ ♦s ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ♥ú♠❡r♦s r❡✲
❛✐s✳ ❊❧❡s s✉r❣❡♠ ❝♦♠♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ❞❡ ✉♠ ♣♦♥t♦ P ❞♦ ♣❧❛♥♦ Π (x =
abscissa, y =ordenada) q✉❛♥❞♦ s❡ ✜①❛ ♥❡ss❡ ♣❧❛♥♦ ✉♠ ♣❛r ❞❡ ❡✐①♦s ♦rt♦❣♦♥❛✐sOX ❡ OY✱ q✉❡ s❡ ✐♥t❡rs❡♣t❛♠ ♥♦ ♣♦♥t♦ O✱ ❝❤❛♠❛❞♦ ❞❡ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱
❝♦♥❢♦r♠❡ ❛ ✜❣✉r❛ ✸✳✷✳
❋✐❣✉r❛ ✸✳✷✿ P♦♥t♦ P✭①✱②✮ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s✳
❉❛❞♦ ♦ ♣♦♥t♦ P ∈Π✱ ❛ ❛❜s❝✐ss❛ ❞❡ P é ♦ ♥ú♠❡r♦ x✱ ❝♦♦r❞❡♥❛❞❛ ❞♦ ♣é ❞❛ ♣❡r♣❡♥✲
❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ P s♦❜r❡ ♦ ❡✐①♦ OX✱ ❡♥q✉❛♥t♦ ❛ ♦r❞❡♥❛❞❛ ❞❡ P é ❛ ❝♦♦r❞❡♥❛❞❛ y
❞♦ ♣é ❞❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❜❛✐①❛❞❛ ❞❡ P s♦❜r❡ ♦ ❡✐①♦OY✳ ❉✐③✲s❡ ❡♥tã♦ q✉❡ (x, y) é ♦ ♣❛r
❞❡ ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ P r❡❧❛t✐✈❛♠❡♥t❡ ❛♦s ❡✐①♦s OXY✳
❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧f :X →Ré ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ R2✱ ❧♦❣♦ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞♦ ✭♣❡❧♦ ♠❡♥♦s ♥♦s ❝❛s♦s ♠❛✐s s✐♠♣❧❡s✮ ❝♦♠♦ ✉♠❛ ❧✐♥❤❛✱
❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s (x, y)✱ q✉❛♥❞♦ x ✈❛r✐❛ ♥♦ ❝♦♥❥✉♥t♦ X✳ ❱❡❥❛♠♦s
❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s ❡ s❡✉s ❣rá✜❝♦s✿
❊①❡♠♣❧♦ ✸✳✶✵✳ ❙❡❥❛ f : R→R ❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✶ ❡ ✐❣✉❛❧ ❛ c✱ ♦✉ s❡❥❛✱ f(x) = c✳
❖ ❣rá✜❝♦ ❞❡ f é ♦ s✉❜❝♦♥❥✉♥t♦ Gf = {(x, y);x ∈ R ❡ y = c} = {(x, c);x ∈ R}✱
r❡♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛ ✸✳✸✳
❊①❡♠♣❧♦ ✸✳✶✶✳ ❙❡❥❛ IdR : R → R ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡✷ IdR(x) = x ♣❛r❛ t♦❞♦
x∈R✳ ❙❡✉ ❣rá✜❝♦ é GId ={(x, y);x∈R ❡ y =x}={(x, x);x∈ R}✱ r❡♣r❡s❡♥t❛❞♦ ♥❛
❋✐❣✉r❛ ✸✳✹✳
✶P♦r ❞❡✜♥✐çã♦ t❡♠♦s✿ ❉❛❞♦s ♦s ❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦sX ❡Y ❡ ✜①❛❞♦ ✉♠ ❡❧❡♠❡♥t♦c∈Y✱ ❛ ❢✉♥çã♦
❝♦♥st❛♥t❡c ❞❡X ❡♠Y é ❛ ❢✉♥çã♦f :X →Y t❛❧ q✉❡f(x) =c ♣❛r❛ t♦❞♦x∈X✳
✷P♦r ❞❡✜♥✐çã♦ t❡♠♦s✿ ❉❛❞♦s ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ X✱ ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ X é ❛ ❢✉♥çã♦
●rá✜❝♦s ❞❡ ❋✉♥çõ❡s ✸✼
❋✐❣✉r❛ ✸✳✸✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ f(x) = c,
❋✐❣✉r❛ ✸✳✹✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ IdR✳
❊①❡♠♣❧♦ ✸✳✶✷✳ ❙❡❥❛f :R→R✉♠❛ ❢✉♥çã♦ ❛✜♠✸✐❣✉❛❧ ❛f(x) =ax+b✳ ❖ ❣rá✜❝♦ ❞❡ f é ♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦Gf ={(x, y);x, y ∈R❡y=ax+b}r❡♣r❡s❡♥t❛❞♦
♥❛ ❋✐❣✉r❛ ✸✳✺✳
❊①❡♠♣❧♦ ✸✳✶✸✳ ❙❡❥❛f :R→R✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✹✐❣✉❛❧ ❛f(x) =ax2+bx+c✳
❖ ❣rá✜❝♦ ❞❡ f é ✉♠❛ ♣❛rá❜♦❧❛ ❞❡✜♥✐❞❛ ♣❡❧♦ t❡♦r❡♠❛ q✉❡ s❡❣✉❡ ❡ ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛ ✸❯♠❛ ❢✉♥çã♦ ❛✜♠ é ✉♠❛ ❢✉♥çã♦ f :R→R✱ t❛❧ q✉❡f(x) =ax+b ♣❛r❛ t♦❞♦xr❡❛❧✱ ♦♥❞❡ a❡b
sã♦ ♥ú♠❡r♦s r❡❛✐s ❞❛❞♦s✱ ❝♦♠ a6= 0✳
✹❯♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ✉♠❛ ❢✉♥çã♦f :R→R✱ q✉❛♥❞♦ sã♦ ❞❛❞♦sa✱b✱c✱ ❝♦♠a6= 0✱ t❛✐s q✉❡
