Lista_de_Exercicios_1_Solucoes (1)

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▲✐st❛ ❞❡ ❊①❡r❝í❝✐♦s ✶✿ ❙♦❧✉çõ❡s
❋✉♥❞❛♠❡♥t♦s ❞❛ ▲ó❣✐❝❛ ✕ ▲ó❣✐❝❛ Pr♦♣♦s✐❝✐♦♥❛❧
❯❋▼●✴■❈❊①✴❉❈❈ ❉❈❈✶✶✶ ✕ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛
❈✐ê♥❝✐❛s ❊①❛t❛s ✫ ❊♥❣❡♥❤❛r✐❛s ✶♦ ❙❡♠❡str❡ ❞❡ ✷✵✶✼
✶✳ ❈♦♥str✉❛ ❛ t❛❜❡❧❛ ❞❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿ E = (p ∨ (¬p ∨ q)) ∧ ¬(q ∧ ¬r)
❘❡s♣♦st❛✿
p q r (p ∨ (¬p ∨ q)) ¬(q ∧ ¬r) E
❱ ❱ ❱ ❱ ❱ ❱
❱ ❱ ❋ ❱ ❋ ❋
❱ ❋ ❱ ❱ ❱ ❱
❱ ❋ ❋ ❱ ❱ ❱
❋ ❱ ❱ ❱ ❱ ❱
❋ ❱ ❋ ❱ ❋ ❋
❋ ❋ ❱ ❱ ❱ ❱
❋ ❋ ❋ ❱ ❱ ❱
✷✳ ▼♦str❡ s❡ ❛s ❡①♣r❡ssõ❡s E1 ❡ E2 sã♦ ❡q✉✐✈❛❧❡♥t❡s ❧♦❣✐❝❛♠❡♥t❡✿
E1 = (s→ (p ∧ ¬r)) ∧ ((p→ (r ∨ q)) ∧ s)
E2 = (p ∧ q ∧ ¬r ∧ s) ∨ ¬(p ∨ s)
❘❡s♣♦st❛✿
p q r s (s → (p ∧ ¬r)) ((p → (r ∨ q)) ∧ s) E1 (p ∧ q ∧ ¬r ∧ s) ¬(p ∨ s) E2
❱ ❱ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❱ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❋ ❋
❱ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱
❱ ❱ ❋ ❋ ❱ ❋ ❋ ❋ ❋ ❋
❱ ❋ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❱ ❋ ❱ ❋ ❱ ❋ ❋ ❋ ❋ ❋
❱ ❋ ❋ ❱ ❱ ❋ ❋ ❋ ❋ ❋
❱ ❋ ❋ ❋ ❱ ❋ ❋ ❋ ❋ ❋
❋ ❱ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❋ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❱ ❱
❋ ❱ ❋ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❋ ❱ ❋ ❋ ❱ ❋ ❋ ❋ ❱ ❱
❋ ❋ ❱ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❋ ❋ ❱ ❋ ❱ ❋ ❋ ❋ ❱ ❱
❋ ❋ ❋ ❱ ❋ ❱ ❋ ❋ ❋ ❋
❋ ❋ ❋ ❋ ❱ ❋ ❋ ❋ ❱ ❱
❈♦♠♦ ❛s ❞✉❛s t❛❜❡❧❛s ❞❛ ✈❡r❞❛❞❡ ♥ã♦ sã♦ ✐❞ê♥t✐❝❛s✱ ❛s ❡①♣r❡ssõ❡s E1 ❡ E2 ♥ã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❧♦❣✐❝❛♠❡♥t❡✳
✶
✸✳ ❋❛ç❛ ❛ s✐♠♣❧✐✜❝❛çã♦ ❧ó❣✐❝❛ ❞❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦ ✉s❛♥❞♦ ❛♣❡♥❛s ❛s ❧❡✐s ❞❛ ❧ó❣✐❝❛✿
(p ∧ (¬(¬p ∨ q))) ∨ (p ∧ q)
❘❡s♣♦st❛✿
(p ∧ (¬(¬p ∨ q))) ∨ (p ∧ q) ❉❡ ▼♦r❣❛♥ s♦❜r❡ ¬(¬p ∨ q)✳
(p ∧ (p ∧ ¬q)) ∨ (p ∧ q) ❆ss♦❝✐❛t✐✈✐❞❛❞❡ s♦❜r❡ p ∧ (p ∧ ¬q)✳
((p ∧ p) ∧ ¬q) ∨ (p ∧ q) ■❞❡♠♣♦tê♥❝✐❛ s♦❜r❡ p ∧ p✳
(p ∧ ¬q) ∨ (p ∧ q) ❉✐str✐❜✉t✐✈✐❞❛❞❡ s♦❜r❡ ❛ ❡①♣r❡ssã♦✳
p ∧ (¬q ∨ q) ◆❡❣❛çã♦ s♦❜r❡ ¬q ∨ q✳
p ∧ t ■❞❡♥t✐❞❛❞❡ ❝♦♠ ❛ t❛✉t♦❧♦❣✐❛ t✳
p
✹✳ ▼♦str❡ s❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ♥ã♦ ✉s❛♥❞♦ ❛s ❢♦r♠❛s ✈á❧✐❞❛s ❞❡ ❛r❣✉♠❡♥t♦s✳ ❊♠ ❝❛❞❛ ♣❛ss♦✱
✐❞❡♥t✐✜q✉❡ ❛ r❛③ã♦ ♣❛r❛ s❡ ♦❜t❡r ❛ ❝♦♥❝❧✉sã♦✿
✭❛✮ ¬p→ r ∧ ¬s
✭❜✮ t→ s
✭❝✮ u→ ¬p
✭❞✮ ¬w
✭❡✮ u ∨ w
✭❢✮ ✳✳✳ ¬t ∨ w
❘❡s♣♦st❛✿
❉❡❞✉çõ❡s✿
✭✐✮ u ∨ w ✭❡✮
¬w ✭❞✮
✳✳✳ u ❙✐❧♦❣✐s♠♦ ❉✐s❥✉♥t✐✈♦
✭✐✐✮ u→ ¬p ✭❝✮
u ✭✐✮
✳✳✳ ¬p ▼♦❞✉s P♦♥❡♥s
✭✐✐✐✮ ¬p→ r ∧ ¬s ✭❛✮
¬p ✭✐✐✮
✳✳✳ r ∧ ¬s ▼♦❞✉s P♦♥❡♥s
✭✐✈✮ r ∧ ¬s ✭✐✐✐✮
✳✳✳ ¬s ❙✐♠♣❧✐✜❝❛çã♦ ❈♦♥❥✉♥t✐✈❛
✭✈✮ t→ s ✭❜✮
¬s ✭✐✈✮
✳✳✳ ¬t ▼♦❞✉s ❚♦❧❧❡♥s
✭✈✐✮ ¬t ✭✈✮
✳✳✳ ¬t ∨ w ❆❞✐çã♦ ❉✐s❥✉♥t✐✈❛
❆ ❢♦r♠❛ ❞❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞❛✳
✺✳ ❊st❡ ❡①❡r❝í❝✐♦ é ✏♣r❛t✐❝❛♠❡♥t❡ ✐❞ê♥t✐❝♦✑ ❛♦ ❛♥t❡r✐♦r ❡①❝❡t♦ q✉❡ ❛ ❝♦♥❝❧✉sã♦ é ✉♠ ✏♣♦✉❝♦ ❞✐❢❡r❡♥t❡✑✳ ▼♦str❡
s❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ♥ã♦ ✉s❛♥❞♦ ❛s ❢♦r♠❛s ✈á❧✐❞❛s ❞❡ ❛r❣✉♠❡♥t♦s✳ ❊♠ ❝❛❞❛ ♣❛ss♦✱ ✐❞❡♥t✐✜q✉❡
❛ r❛③ã♦ ♣❛r❛ s❡ ♦❜t❡r ❛ ❝♦♥❝❧✉sã♦✿
✭❛✮ ∼p→ r ∧ ∼s
✷
✭❜✮ t→ s
✭❝✮ u→ ∼s
✭❞✮ ∼w
✭❡✮ u ∨ w
✭❢✮ ✳✳✳ ∼t ∧ w
❘❡s♣♦st❛✿
❉❡❞✉çõ❡s✿
✭✐✮ u ∨ v ✭❡✮
∼w ✭❞✮
✳✳✳ u ❙✐❧♦❣✐s♠♦ ❉✐s❥✉♥t✐✈♦
✭✐✐✮ u→ ∼s ✭❝✮
u ✭✐✮
✳✳✳ ∼s ▼♦❞✉s P♦♥❡♥s
✭✐✐✐✮ t→ s ✭❜✮
∼s ✭✐✐✮
✳✳✳ ∼t ▼♦❞✉s ❚♦❧❧❡♥s
❆ ♣r❡♠✐ss❛ ✭❛✮ ♣♦ss✉✐ ❛s ✈❛r✐á✈❡✐s p ❡ r q✉❡ ♥ã♦ ❡stã♦ ♣r❡s❡♥t❡s ♥❛s ♦✉tr❛s ♣r❡♠✐ss❛s✳ ❆ss✐♠✱ ♥❛❞❛ ♣♦❞❡ s❡r
❞✐t♦✳ ❆ ❝♦♥❝❧✉sã♦ ❞♦ ❛r❣✉♠❡♥t♦ ✭❢✮ ♣♦ss✉✐ ❞♦✐s t❡r♠♦s✿ ∼t✱ q✉❡ s❛❜❡✲s❡ s❡r ✈❡r❞❛❞❡✐r♦ ✭✐✐✐✮✱ ❡ w✱ q✉❡ ♥❡❣❛
❛ ♣r❡♠✐ss❛ ✭❞✮✳ ❆ss✐♠✱ ❛ ❝♦♥❝❧✉sã♦ ❡stá ❡rr❛❞❛ ❡ ❛ ❢♦r♠❛ ❞❡ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞❛✳
❊ss❛ ❝♦♥❝❧✉sã♦ t❛♠❜é♠ ♣♦❞❡ s❡r ✈✐st❛ ✉s❛♥❞♦ ❛ t❛❜❡❧❛ ❞❛ ✈❡r❞❛❞❡ ❛❜❛✐①♦✳ ❊①✐st❡♠ três ❧✐♥❤❛s ❝rít✐❝❛s✿ ✶✹✱
✸✵ ❡ ✹✻✱ q✉❡ ♣♦ss✉❡♠ ❝♦♥❝❧✉sã♦ ❢❛❧s❛✳ ■ss♦ ❞❡♠♦♥str❛ q✉❡ ❛ ❢♦r♠❛ ❞❡ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞❛✳
✸
p r s t u w ∼p → r ∧ ∼s t → s u → ∼s ∼w u ∨ w ∼t ∧ w
✶✳ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❋
✷✳ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❱ ❋ ❱ ❱ ❋
✸✳ ❱ ❱ ❱ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋
✹✳ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✺✳ ❱ ❱ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❱
✻✳ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❱ ❋
✼✳ ❱ ❱ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❱
✽✳ ❱ ❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✾✳ ❱ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✶✵✳ ❱ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❱ ❋
✶✶✳ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✶✷✳ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❋
✶✸✳ ❱ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❱
→ ✶✹✳ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❋
✶✺✳ ❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❱
✶✻✳ ❱ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✶✼✳ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❋
✶✽✳ ❱ ❋ ❱ ❱ ❱ ❋ ❱ ❱ ❋ ❱ ❱ ❋
✶✾✳ ❱ ❋ ❱ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋
✷✵✳ ❱ ❋ ❱ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✷✶✳ ❱ ❋ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❱
✷✷✳ ❱ ❋ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❱ ❋
✷✸✳ ❱ ❋ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❱
✷✹✳ ❱ ❋ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✷✺✳ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✷✻✳ ❱ ❋ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❱ ❋
✷✼✳ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✷✽✳ ❱ ❋ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❋
✷✾✳ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❱
→ ✸✵✳ ❱ ❋ ❋ ❋ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❋
✸✶✳ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❱
✸✷✳ ❱ ❋ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✸✸✳ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❋
✸✹✳ ❋ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋
✸✺✳ ❋ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋
✸✻✳ ❋ ❱ ❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋
✸✼✳ ❋ ❱ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❱
✸✽✳ ❋ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋
✸✾✳ ❋ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❱
✹✵✳ ❋ ❱ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋
✹✶✳ ❋ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✹✷✳ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❱ ❋
✹✸✳ ❋ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❋
✹✹✳ ❋ ❱ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❋
✹✺✳ ❋ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❱
→ ✹✻✳ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❋
✹✼✳ ❋ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❱
✹✽✳ ❋ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❋
✹✾✳ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❋
✺✵✳ ❋ ❋ ❱ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋
✺✶✳ ❋ ❋ ❱ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋
✺✷✳ ❋ ❋ ❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋
✺✸✳ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❋ ❋ ❱ ❱
✺✹✳ ❋ ❋ ❱ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋
✺✺✳ ❋ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❱
✺✻✳ ❋ ❋ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋
✺✼✳ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❋
✺✽✳ ❋ ❋ ❋ ❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❋
✺✾✳ ❋ ❋ ❋ ❱ ❋ ❱ ❋ ❋ ❱ ❋ ❱ ❋
✻✵✳ ❋ ❋ ❋ ❱ ❋ ❋ ❋ ❋ ❱ ❱ ❋ ❋
✻✶✳ ❋ ❋ ❋ ❋ ❱ ❱ ❋ ❱ ❱ ❋ ❱ ❱
✻✷✳ ❋ ❋ ❋ ❋ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋
✻✸✳ ❋ ❋ ❋ ❋ ❋ ❱ ❋ ❱ ❱ ❋ ❱ ❱
✻✹✳ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❋
✻✳ ❖ ❢❛♠♦s♦ ❞❡t❡t✐✈❡ P❡r❝✉❧❡ ❍♦✐r♦t ❢♦✐ ❝❤❛♠❛❞♦ ♣❛r❛ r❡s♦❧✈❡r ✉♠ ❛ss❛ss✐♥❛t♦ ♠✐st❡r✐♦s♦✳ ❊❧❡ ❞❡t❡r♠✐♥♦✉ ♦s
s❡❣✉✐♥t❡s ❢❛t♦s✿
✭❛✮ ▲♦r❞ ❈❤❛r❧❡s✱ ♦ ❤♦♠❡♠ ❛ss❛ss✐♥❛❞♦✱ ❢♦✐ ♠♦rt♦ ❝♦♠ ✉♠❛ ♣❛♥❝❛❞❛ ♥❛ ❝❛❜❡ç❛ ❝♦♠ ✉♠ ❝❛st✐ç❛❧✳
✭❜✮ ❖✉ ▲❛❞② ❈❛♠✐❧❛ ♦✉ ❛ ❡♠♣r❡❣❛❞❛ ❙❛r❛ ❡st❛✈❛♠ ♥❛ s❛❧❛ ❞❡ ❥❛♥t❛r ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✳
✹
✭❝✮ ❙❡ ♦ ❝♦③✐♥❤❡✐r♦ ❡st❛✈❛ ♥❛ ❝♦③✐♥❤❛ ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✱ ❡♥tã♦ ♦ ❛ç♦✉❣✉❡✐r♦ ♠❛t♦✉ ▲♦r❞ ❈❤❛r❧❡s
❝♦♠ ✉♠❛ ❞♦s❡ ❢❛t❛❧ ❞❡ ❛rsê♥✐❝♦✳
✭❞✮ ❙❡ ▲❛❞② ❈❛♠✐❧❛ ❡st❛✈❛ ♥❛ s❛❧❛ ❞❡ ❥❛♥t❛r ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✱ ❡♥tã♦ ♦ ♠♦t♦r✐st❛ ♠❛t♦✉ ▲♦r❞
❈❤❛r❧❡s✳
✭❡✮ ❙❡ ♦ ❝♦③✐♥❤❡✐r♦ ♥ã♦ ❡st❛✈❛ ♥❛ ❝♦③✐♥❤❛ ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✱ ❡♥tã♦ ❙❛r❛ ♥ã♦ ❡st❛✈❛ ♥❛ s❛❧❛ ❞❡
❥❛♥t❛r q✉❛♥❞♦ ♦ ❛ss❛ss✐♥❛t♦ ♦❝♦rr❡✉✳
✭❢✮ ❙❡ ❙❛r❛ ❡st❛✈❛ ♥❛ s❛❧❛ ❞❡ ❥❛♥t❛r ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✱ ❡♥tã♦ ♦ ❛❥✉❞❛♥t❡ ♣❡ss♦❛❧ ❞❡ ▲♦r❞ ❈❤❛r❧❡s
♦ ♠❛t♦✉✳
➱ ♣♦ssí✈❡❧ ♣❛r❛ ♦ ❞❡t❡t✐✈❡ P❡r❝✉❧❡ ❍♦✐r♦t ❞❡❞✉③✐r q✉❡♠ ♠❛t♦✉ ▲♦r❞❡ ❈❤❛r❧❡s❄ ❙❡ s✐♠✱ q✉❡♠ é ♦ ❛ss❛ss✐♥♦❄
❘❡s♣♦st❛✿
❙❡❥❛♠ ♦s s❡❣✉✐♥t❡s ❛r❣✉♠❡♥t♦s✿
p ❂ ▲♦r❞ ❈❤❛r❧❡s ❢♦✐ ♠♦rt♦ ❝♦♠ ✉♠❛ ♣❛♥❝❛❞❛ ♥❛ ❝❛❜❡ç❛ ❝♦♠ ✉♠ ❝❛st✐ç❛❧✳
q ❂ ▲❛❞② ❈❛♠✐❧❛ ❡st❛✈❛ ♥❛ s❛❧❛ ❞❡ ❥❛♥t❛r ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✳
r ❂ ❙❛r❛ ❡st❛✈❛ ♥❛ s❛❧❛ ❞❡ ❥❛♥t❛r ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✳
s ❂ ❈♦③✐♥❤❡✐r♦ ❡st❛✈❛ ♥❛ ❝♦③✐♥❤❛ ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✳
t ❂ ❆ç♦✉❣✉❡✐r♦ ♠❛t♦✉ ▲♦r❞ ❈❤❛r❧❡s ❝♦♠ ✉♠❛ ❞♦s❡ ❢❛t❛❧ ❞❡ ❛rsê♥✐❝♦✳
u ❂ ▼♦t♦r✐st❛ ♠❛t♦✉ ▲♦r❞ ❈❤❛r❧❡s✳
v ❂ ❆❥✉❞❛♥t❡ ♣❡ss♦❛❧ ❞❡ ▲♦r❞ ❈❤❛r❧❡s ♦ ♠❛t♦✉✳
❚r❛❞✉çã♦ ❞♦s ❢❛t♦s ♣❛r❛ ❛s ♣r♦♣♦s✐çõ❡s✿
✭❛✮ p
✭❜✮ q ∨ r
✭❝✮ s→ t
✭❞✮ q → u
✭❡✮ ¬s→ ¬r
✭❢✮ r → v
❉❡❞✉çõ❡s✿
✭✐✮ ❙✉♣♦♥❤❛ q✉❡ s s❡❥❛ ❱✱ ✐✳❡✳✱ ♦ ❝♦③✐♥❤❡✐r♦ ❡st❛✈❛ ♥❛ ❝♦③✐♥❤❛ ♥♦ ♠♦♠❡♥t♦ ❞♦ ❛ss❛ss✐♥❛t♦✳
s→ t ✭❝✮
s ✭❙✉♣♦s✐çã♦✮
✳✳✳ t ▼♦❞✉s P♦♥❡♥s
✭✐✐✮ ◆♦ ❡♥t❛♥t♦✱ t→ ✏❝♦♥tr❛❞✐çã♦✑✱ ❥á q✉❡ ▲♦r❞ ❈❤❛r❧❡s ♥ã♦ ❢♦✐ ♠♦rt♦ ❝♦♠ ❛rsê♥✐❝♦ ❡ s✐♠ ♣♦r ✉♠❛ ♣❛♥❝❛❞❛
♥❛ ❝❛❜❡ç❛ ✭❛✮✳ ▲♦❣♦✱ ❛ s✉♣♦s✐çã♦ q✉❡ ♦ ❝♦③✐♥❤❡✐r♦ ❡st❛✈❛ ♥❛ ❝♦③✐♥❤❛ é ❢❛❧s❛✳ ❆ss✐♠✱ t❡♠♦s ¬s✳
s→ t ✭❝✮
t→ ✏❝♦♥tr❛❞✐çã♦✑ ✭✐✮
✳✳✳ ¬s ❙✐❧♦❣✐s♠♦ ❍✐♣♦tét✐❝♦ ❡ ❈♦♥tr❛❞✐çã♦
✭✐✐✐✮ ¬s→ ¬r ✭❡✮
¬s ✭✐✐✮
✳✳✳ ¬r ▼♦❞✉s P♦♥❡♥s
✭✐✈✮ q ∨ r ✭❜✮
¬r ✭✐✐✐✮
✳✳✳ q ❙✐❧♦❣✐s♠♦ ❉✐s❥✉♥t✐✈♦
✭✈✮ q → u ✭❞✮
q ✭✐✈✮
✳✳✳ u ▼♦❞✉s P♦♥❡♥s
✺
❉❡ ♦♥❞❡ s❡ ❝♦♥❝❧✉✐ q✉❡ ♦ ♠♦t♦r✐st❛ ♠❛t♦✉ ▲♦r❞ ❈❤❛r❧❡s✳
✼✳ ❈♦♥str✉❛ ❛ t❛❜❡❧❛ ❞❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿ E = (p⊕ q) ∧ (p⊕ ¬q)
❘❡s♣♦st❛✿
p q (p⊕ q) (p⊕ ¬q) E
❱ ❱ ❋ ❱ ❋
❱ ❋ ❱ ❋ ❋
❋ ❱ ❱ ❋ ❋
❋ ❋ ❋ ❱ ❋
✽✳ ❈♦♥str✉❛ ❛ t❛❜❡❧❛ ❞❛ ✈❡r❞❛❞❡ ♣❛r❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿ E = (¬p↔ ¬q)→ (p→ ¬r)
❘❡s♣♦st❛✿
p q r ¬p ¬q ¬r (¬p ↔ ¬q) (p → ¬r) E
❱ ❱ ❱ ❋ ❋ ❋ ❱ ❋ ❋
❱ ❱ ❋ ❋ ❋ ❱ ❱ ❱ ❱
❱ ❋ ❱ ❋ ❱ ❋ ❋ ❋ ❱
❱ ❋ ❋ ❋ ❱ ❱ ❋ ❱ ❱
❋ ❱ ❱ ❱ ❋ ❋ ❋ ❱ ❱
❋ ❱ ❋ ❱ ❋ ❱ ❋ ❱ ❱
❋ ❋ ❱ ❱ ❱ ❋ ❱ ❱ ❱
❋ ❋ ❋ ❱ ❱ ❱ ❱ ❱ ❱
✾✳ ❙❡❥❛ ❛ t❛❜❡❧❛ ❞❛ ✈❡r❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r ⊙✿
p q p ⊙ q
❱ ❱ ❱
❱ ❋ ❋
❋ ❱ ❋
❋ ❋ ❱
✭❛✮ ❖ ♦♣❡r❛❞♦r ⊙ s❡❣✉❡ ❛ ❧❡✐ ❞❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❝♦♠ ♦ ♦♣❡r❛❞♦r ∧✱ ✐✳❡✳✱ (x ⊙ y) ∧ z
?
≡ x ⊙ (y ∧ z).
❘❡s♣♦st❛✿
x y z (x ⊙ y) ∧ z x ⊙ (y ∧ z)
❱ ❱ ❱ ❱ ❱
❱ ❱ ❋ ❋ ❋
❱ ❋ ❱ ❋ ❋
❱ ❋ ❋ ❋ ❋
❋ ❱ ❱ ❋ ❋
❋ ❱ ❋ ❋ ❱
❋ ❋ ❱ ❱ ❱
❋ ❋ ❋ ❋ ❱
❈♦♠♦ ❛s ❞✉❛s ❝♦❧✉♥❛s ❞❛ ❞✐r❡✐t❛ ♥ã♦ sã♦ ✐❞ê♥t✐❝❛s✱ ♦ ♦♣❡r❛❞♦r ⊙ ♥ã♦ s❡❣✉❡ ❛ ❧❡✐ ❞❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡
❝♦♠ ♦ ♦♣❡r❛❞♦r ∧✳
✭❜✮ ❖ ♦♣❡r❛❞♦r ⊙ s❡❣✉❡ ❛ ❧❡✐ ❞❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡ ❝♦♠ ♦ ♦♣❡r❛❞♦r ∧✱ ✐✳❡✳✱ x ⊙ (y ∧ z)
?
≡ (x ⊙ y) ∧(x ⊙ z).
❘❡s♣♦st❛✿
✻
x y z x ⊙ (y ∧ z) (x ⊙ y) ∧ (x ⊙ z)
❱ ❱ ❱ ❱ ❱
❱ ❱ ❋ ❋ ❋
❱ ❋ ❱ ❋ ❋
❱ ❋ ❋ ❋ ❋
❋ ❱ ❱ ❋ ❋
❋ ❱ ❋ ❱ ❋
❋ ❋ ❱ ❱ ❋
❋ ❋ ❋ ❱ ❱
❈♦♠♦ ❛s ❞✉❛s ❝♦❧✉♥❛s ❞❛ ❞✐r❡✐t❛ ♥ã♦ sã♦ ✐❞ê♥t✐❝❛s✱ ♦ ♦♣❡r❛❞♦r ⊙ ♥ã♦ s❡❣✉❡ ❛ ❧❡✐ ❞❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡
❝♦♠ ♦ ♦♣❡r❛❞♦r ∧✳
✶✵✳ ▼♦str❡ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❧ó❣✐❝❛ ❞❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ✉s❛♥❞♦ ❛♣❡♥❛s ❛s ❧❡✐s ❞❛ ❧ó❣✐❝❛✿
(p→ r) ∨ (q → r) ≡ (p ∧ q)→ r
❘❡s♣♦st❛✿
(p→ r) ∨ (q → r) ≡ (p ∧ q)→ r
(¬p ∨ r) ∨ (¬q ∨ r) ≡
¬p ∨ ¬q ∨ r ≡
(¬p ∨ ¬q) ∨ r ≡
¬(¬p ∨ ¬q)→ r ≡
(p ∧ q)→ r ≡
✶✶✳ ▼♦str❡ s❡ ♦s s❡❣✉✐♥t❡s r❡q✉✐s✐t♦s sã♦ ❝♦♥s✐st❡♥t❡s ♦✉ ♥ã♦✳ ❈❛s♦ s❡❥❛♠✱ ♣❛r❛ q✉❡ ✈❛❧♦r❡s ❡ss❡s r❡q✉✐s✐t♦s sã♦
❝♦♥s✐st❡♥t❡s✳
❙❡ ♦ s✐st❡♠❛ ❞❡ ❛rq✉✐✈♦s ♥ã♦ ❡stá tr❛✈❛❞♦✱ ❡♥tã♦ ♥♦✈❛s ♠❡♥s❛❣❡♥s s❡rã♦ ❡♥✜❧❡✐r❛❞❛s✳ ❙❡ ♦ s✐st❡♠❛
❞❡ ❛rq✉✐✈♦s ♥ã♦ ❡stá tr❛✈❛❞♦✱ ❡♥tã♦ ♦ s✐st❡♠❛ ❡stá ❢✉♥❝✐♦♥❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡ ❡ ✈✐❝❡✲✈❡rs❛✳ ❙❡
♥♦✈❛s ♠❡♥s❛❣❡♥s ♥ã♦ sã♦ ❡♥✜❧❡✐r❛❞❛s✱ ❡♥tã♦ ❡❧❛s s❡rã♦ ❡♥✈✐❛❞❛s ♣❛r❛ ♦ ❜✉✛❡r ❞❡ ♠❡♥s❛❣❡♥s✳ ❙❡
♦ s✐st❡♠❛ ❞❡ ❛rq✉✐✈♦s ♥ã♦ ❡stá tr❛✈❛❞♦✱ ❡♥tã♦ ♥♦✈❛s ♠❡♥s❛❣❡♥s s❡rã♦ ❡♥✈✐❛❞❛s ♣❛r❛ ♦ ❜✉✛❡r ❞❡
♠❡♥s❛❣❡♥s✳ ◆♦✈❛s ♠❡♥s❛❣❡♥s ♥ã♦ s❡rã♦ ❡♥✈✐❛❞❛s ♣❛r❛ ♦ ❜✉✛❡r ❞❡ ♠❡♥s❛❣❡♥s✳
❘❡s♣♦st❛✿
❙❡❥❛♠ ♦s s❡❣✉✐♥t❡s ❛r❣✉♠❡♥t♦s✿
p = ❖ s✐st❡♠❛ ❞❡ ❛rq✉✐✈♦s ♥ã♦ ❡stá tr❛✈❛❞♦✳
q = ◆♦✈❛s ♠❡♥s❛❣❡♥s s❡rã♦ ❡♥✜❧❡✐r❛❞❛s✳
r = ❖ s✐st❡♠❛ ❡stá ❢✉♥❝✐♦♥❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡✳
s = ▼❡♥s❛❣❡♥s s❡rã♦ ❡♥✈✐❛❞❛s ♣❛r❛ ♦ ❜✉✛❡r ❞❡ ♠❡♥s❛❣❡♥s✳
❚r❛❞✉çã♦ ❞♦s ❢❛t♦s ♣❛r❛ ❛s ♣r♦♣♦s✐çõ❡s✿
✭❛✮ p→ q
✭❜✮ p→ r
✭❝✮ r → p
✭❞✮ ¬q → s
✭❡✮ p→ s
✭❢✮ ¬s
❉❡❞✉çõ❡s✿
✭✐✮ ¬q → s ✭❞✮
✼
¬s ✭❢✮
✳
✳
✳ q ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✐✮ p → s ✭❡✮
¬s ✭❢✮
✳
✳
✳ ¬p ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✐✐✮ r → p ✭❝✮
¬p ✭✐✐✮
✳
✳
✳ ¬r ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✈✮ p → r ✭❜✮
¬r ✭✐✐✐✮
✳
✳
✳ ¬p ▼♦❞✉s ❚♦❧❧❡♥s
P❡❧❛s ❞❡❞✉çõ❡s ❛❝✐♠❛ t❡♠♦s q✉❡ ♦s r❡q✉✐s✐t♦s sã♦ ❝♦♥s✐st❡♥t❡s ♣❛r❛ ♦s ✈❛❧♦r❡s✿
¬p = V ✿ ❖ s✐st❡♠❛ ❞❡ ❛rq✉✐✈♦s ❡stá tr❛✈❛❞♦✳
q = V ✿ ◆♦✈❛s ♠❡♥s❛❣❡♥s s❡rã♦ ❡♥✜❧❡✐r❛❞❛s✳
¬r = V ✿ ❖ s✐st❡♠❛ ♥ã♦ ❡stá ❢✉♥❝✐♦♥❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡✳
¬s = V ✿ ▼❡♥s❛❣❡♥s ♥ã♦ s❡rã♦ ❡♥✈✐❛❞❛s ♣❛r❛ ♦ ❜✉✛❡r ❞❡ ♠❡♥s❛❣❡♥s✳
✶✷✳ ▼♦str❡ s❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ♥ã♦✿
p ∧ ¬q → r
p ∨ q
q → p
✳✳✳ r
❘❡s♣♦st❛✿
❱❛r✐á✈❡✐s ❆✉① Pr❡♠✐ss❛s ❈♦♥❝❧✉sã♦
p q r p ∧ ¬q p ∧ ¬q → r p ∨ q q → p r
✶✳ → ❱ ❱ ❱ ❋ ❱ ❱ ❱ ❱
✷✳ → ❱ ❱ ❋ ❋ ❱ ❱ ❱ ❋
✸✳ → ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❱
✹✳ ❱ ❋ ❋ ❱ ❋ ❱ ❱
✺✳ ❋ ❱ ❱ ❋ ❱ ❱ ❋
✻✳ ❋ ❱ ❋ ❋ ❱ ❱ ❋
✼✳ ❋ ❋ ❱ ❋ ❱ ❋ ❱
✽✳ ❋ ❋ ❋ ❋ ❱ ❋ ❱
➜ ❆ ❝♦♥❝❧✉sã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❛s ❧✐♥❤❛s ✶ ❡ ✸ ❡ ❢❛❧s❛ ♣❛r❛ ❛ ❧✐♥❤❛ ✷✳ ❖✉ s❡❥❛✱ ❛ ❝♦♥❝❧✉sã♦ ♥ã♦ é ✈❡r❞❛❞❡✐r❛
♣❛r❛ t♦❞❛s ❛s ❧✐♥❤❛s ❝rít✐❝❛s✳ ▲♦❣♦✱ ♦ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞♦✳
✶✸✳ ▼♦str❡ s❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ♥ã♦ ✉s❛♥❞♦ ❛s ❢♦r♠❛s ✈á❧✐❞❛s ❞❡ ❛r❣✉♠❡♥t♦s✳ ❊♠ ❝❛❞❛ ♣❛ss♦✱
✐❞❡♥t✐✜q✉❡ ❛ r❛③ã♦ ♣❛r❛ s❡ ♦❜t❡r ❛ ❝♦♥❝❧✉sã♦✿
✭❛✮ p→ q
✭❜✮ r ∨ s
✭❝✮ ¬s→ ¬t
✭❞✮ ¬q ∨ s
✭❡✮ ¬s
✭❢✮ ¬p ∧ r → u
✭❣✮ w ∨ t
✭❤✮ ✳✳✳ u ∧ w
✽
❘❡s♣♦st❛✿
✭✐✮ r ∨ s ✭❜✮
¬s ✭❡✮
✳✳✳ r ❙✐❧♦❣✐s♠♦ ❞✐s❥✉♥t✐✈♦
✭✐✐✮ ¬q ∨ s ✭❞✮
¬s ✭❡✮
✳✳✳ ¬q ❙✐❧♦❣✐s♠♦ ❞✐s❥✉♥t✐✈♦
✭✐✐✐✮ p→ q ✭❛✮
¬q ✭✐✐✮
✳✳✳ ¬p ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✈✮ ¬p ∧ r → u ✭❢✮
¬p ∧ r ❆❞✐çã♦ ❝♦♥❥✉♥t✐✈❛ ❞❡ ✭✐✐✐✮ ❡ ✭✐✮
✳✳✳ u ▼♦❞✉s P♦♥❡♥s
✭✈✮ ¬s→ ¬t ✭❝✮
¬s ✭❡✮
✳✳✳ ¬t ▼♦❞✉s P♦♥❡♥s
✭✈✐✮ w ∨ t ✭❣✮
¬t ✭✈✮
✳✳✳ w ❙✐❧♦❣✐s♠♦ ❞✐s❥✉♥t✐✈♦
✭✈✐✐✮ u ✭✐✈✮
w ✭✈✐✮
✳✳✳ u ∧ w ❆❞✐çã♦ ❝♦♥❥✉♥t✐✈❛
➜ ❖ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦✳
✶✹✳ ❙❡❥❛♠ ❞✉❛s ✈❛r✐á✈❡✐s ❧ó❣✐❝❛s x ❡ y✱ ♦✉ s❡❥❛ ✈❛r✐á✈❡✐s q✉❡ ♣♦❞❡♠ ❛ss✉♠✐r ♦ ✈❛❧♦r ✈❡r❞❛❞❡✐r♦ ✭V ✮ ♦✉ ❢❛❧s♦ ✭F ✮✳
❙❡❥❛ ← ♦ ❝♦♠❛♥❞♦ ❞❡ ❛tr✐❜✉✐çã♦ ❡①✐st❡♥t❡ ❡♠ ❧✐♥❣✉❛❣❡♥s ❞❡ ♣r♦❣r❛♠❛çã♦ ❝♦♠♦ ❈ ❡ ⊙ ♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦
♥♦ ❡①❡r❝í❝✐♦ ✾✳ ◗✉❛❧ é ♦ ✈❛❧♦r ❞❡ss❛s ✈❛r✐á✈❡✐s ❛♦ ✜♥❛❧ ❞❛ ❡①❡❝✉çã♦ s❡q✉❡♥❝✐❛❧ ❞♦s três ❝♦♠❛♥❞♦s ❛❜❛✐①♦✳
❆♣r❡s❡♥t❡ ❛ s✉❛ r❡s♣♦st❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ❞❡ x ❡ y✳
x← ¬(x⊙ y)
y ← ¬(x⊙ y)
x← ¬(x⊙ y)
❘❡s♣♦st❛✿
❘❡❡s❝r❡✈❡♥❞♦ ♦s ❝♦♠❛♥❞♦s ❛❝✐♠❛ ❝♦♠ ♦ ♦♣❡r❛❞♦r ✏♦✉ ❡①❝❧✉s✐✈♦✑ t❡♠♦s ✭✈❡❥❛ ❡①❡r❝í❝✐♦ ✾✮✿
x← (x⊕ y)
y ← (x⊕ y)
x← (x⊕ y)
❊①❡❝✉t❛♥❞♦ ♦s ❝♦♠❛♥❞♦s ❡ ❧❡♠❜r❛♥❞♦ q✉❡ ♦ ♦♣❡r❛❞♦r ✏♦✉ ❡①❝❧✉s✐✈♦✑ s❡❣✉❡ ❛ ❧❡✐ ❞❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡ q✉❡ 0
é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ t❡♠♦s✿
x← (x⊕ y) ✭✶✮
y ← (x⊕ y) ≡ ((x⊕ y)⊕ y) ≡ (x⊕ (y ⊕ y)) ≡ (x⊕ 0) ≡ x ✭✷✮
x← (x⊕ y) ≡ ((x⊕ y)⊕ y) ≡ ((x⊕ y)⊕ x) ≡ ((x⊕ x)⊕ y) ≡ (0⊕ y) ≡ y ✭✸✮
✾
◆♦ ❝♦♠❛♥❞♦ ✭✶✮✱ ❛ ✈❛r✐á✈❡❧ x r❡❝❡❜❡ ♦ r❡s✉❧t❛❞♦ ❞❡ x⊕ y✳ ◆♦ ❝♦♠❛♥❞♦ ✭✷✮✱ ❛ ✈❛r✐á✈❡❧ y r❡❝❡❜❡ ♦ r❡s✉❧t❛❞♦
❞❛ ♦♣❡r❛çã♦ ❞♦ ✏♦✉ ❡①❝❧✉s✐✈♦✑ ❡♥tr❡ ❛ ✈❛r✐á✈❡❧ x✱ q✉❡ ❝♦♥té♠ ♦ r❡s✉❧t❛❞♦ ❞❡ x⊕y✱ ❡ ❛ ✈❛r✐á✈❡❧ y✳ ❖ r❡s✉❧t❛❞♦
❞❡ss❛ ♦♣❡r❛çã♦ é x✳ ◆♦ ❝♦♠❛♥❞♦ ✭✸✮✱ ❛ ✈❛r✐á✈❡❧ x r❡❝❡❜❡ ♦ r❡s✉❧t❛❞♦ ❞❛ ♦♣❡r❛çã♦ ❞♦ ✏♦✉ ❡①❝❧✉s✐✈♦✑ ❡♥tr❡ ❛
✈❛r✐á✈❡❧ x✱ q✉❡ ❝♦♥té♠ ♦ r❡s✉❧t❛❞♦ ❞❡ x⊕ y✱ ❡ ❛ ✈❛r✐á✈❡❧ y✱ ❝✉❥♦ ✈❛❧♦r é x✳ ❖ r❡s✉❧t❛❞♦ ❞❡ss❛ ♦♣❡r❛çã♦ é y✳
❖✉ s❡❥❛✱ ❛♣ós ❛ ❡①❡❝✉çã♦ ❞❡ss❡s três ❝♦♠❛♥❞♦s ❛ ✈❛r✐á✈❡❧ x t❡♠ ♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ y ❡ ❛ ✈❛r✐á✈❡❧ y t❡♠ ♦
✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ x✳
❖ t❡①t♦✱ ❛ s❡❣✉✐r✱ ❢♦✐ r❡t✐r❛❞♦ ❞♦ ❧✐✈r♦ t❡①t♦✱ ♣á❣✐♥❛ ✶✾✿ ✏▲ó❣✐❝❛s ❋✉③③② sã♦ ✉t✐❧✐③❛❞❛s ❡♠ ✐♥t❡❧✐❣ê♥❝✐❛ ❛rt✐✜❝✐❛❧✳
◆❛ ❧ó❣✐❝❛ ❢✉③③②✱ ❛ ♣r♦♣♦s✐çã♦ t❡♠ ✉♠ ✈❛❧♦r✲✈❡r❞❛❞❡ q✉❡ é ✉♠ ♥ú♠❡r♦ ❡♥tr❡ ✵ ❡ ✶ ✐♥❝❧✉s✐✈❡✳ ❯♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠
✈❛❧♦r✲✈❡r❞❛❞❡ ✵ é ❢❛❧s❛ ❡ ✉♠❛ ❝♦♠ ✈❛❧♦r✲✈❡r❞❛❞❡ ✶ é ✈❡r❞❛❞❡✐r❛✳ ❱❛❧♦r❡s ❡♥tr❡ ✵ ❡ ✶ ✐♥❞✐❝❛♠ ✈❛r✐❛♥t❡s ❞❡ ❣r❛✉
❞❡ ✈❡r❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✱ ♦ ✈❛❧♦r✲✈❡r❞❛❞❡ ✵✱✽ ♣♦❞❡ s❡r ✐♥❞✐❝❛❞♦ ♣❛r❛ ✉♠❛ ♣r♦♣♦s✐çã♦ ✧❋r❡❞ é ❢❡❧✐③✧✱ ♣♦rq✉❡ ❡❧❡ é
❢❡❧✐③ ♥❛ ♠❛✐♦r ♣❛rt❡ ❞♦ t❡♠♣♦❀ ❡ ♦ ✈❛❧♦r✲✈❡r❞❛❞❡ ✵✱✹ ♣♦❞❡ s❡r ✐♥❞✐❝❛❞♦ ♣❛r❛ ❛ ♣r♦♣♦s✐çã♦ ✧❏♦❤♥ é ❢❡❧✐③✧✱ ♣♦rq✉❡
❡❧❡ é ❢❡❧✐③ ♠❡♥♦s q✉❡ ❛ ♠❡t❛❞❡ ❞♦ t❡♠♣♦✑✳
✶✺✳ ❖ ✈❛❧♦r✲✈❡r❞❛❞❡ ❞❛ ♥❡❣❛çã♦ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ❡♠ ❧ó❣✐❝❛ ❢✉③③② é ✶ ♠❡♥♦s ♦ ✈❛❧♦r✲✈❡r❞❛❞❡ ❞❛ ♣r♦♣♦s✐çã♦✳
◗✉❛✐s sã♦ ♦s ✈❛❧♦r❡s✲✈❡r❞❛❞❡ ❞❛s ♣r♦♣♦s✐çõ❡s ✧❋r❡❞ ♥ã♦ é ❢❡❧✐③✧ ❡ ✧❏♦❤♥ ♥ã♦ é ❢❡❧✐③✧❄
❘❡s♣♦st❛✿
pf = 0, 8✿ ✧❋r❡❞ é ❢❡❧✐③✧❀
pj = 0, 4✿ ✧❏♦❤♥ é ❢❡❧✐③✧✳
¬pf = 1− pf = 1− 0, 8 = 0, 2✿ ✧❋r❡❞ ♥ã♦ é ❢❡❧✐③✧❀
¬pj = 1− pj = 1− 0, 4 = 0, 6✿ ✧❏♦❤♥ ♥ã♦ é ❢❡❧✐③✧❀
✶✻✳ ❖ ✈❛❧♦r✲✈❡r❞❛❞❡ ❞❛ ❞✐s❥✉♥çã♦ ❞❡ ❞✉❛s ♣r♦♣♦s✐çõ❡s ❡♠ ❧ó❣✐❝❛ ❢✉③③② é ♦ ♠á①✐♠♦ ❞♦s ✈❛❧♦r❡s✲✈❡r❞❛❞❡ ❞❡ ❞✉❛s
♣r♦♣♦s✐çõ❡s✳ ◗✉❛✐s sã♦ ♦s ✈❛❧♦r❡s✲✈❡r❞❛❞❡ ❞❛s ♣r♦♣♦s✐çõ❡s ✧❋r❡❞ é ❢❡❧✐③ ♦✉ ❏♦❤♥ é ❢❡❧✐③✧ ❡ ✧❋r❡❞ ♥ã♦ é ❢❡❧✐③
♦✉ ❏♦❤♥ ♥ã♦ é ❢❡❧✐③✧❄
❘❡s♣♦st❛✿
❙❡❥❛♠ p ❡ q ✈❛r✐á✈❡✐s ❞❛ ▲ó❣✐❝❛ ❋✉③③②✳ ❆ ❞✐s❥✉♥çã♦ p ∨ q = max(p ; q)✳
✭❛✮ ✧❋r❡❞ é ❢❡❧✐③ ♦✉ ❏♦❤♥ é ❢❡❧✐③✧✳
pf ∨ pj = max(0, 8; 0, 4) = 0, 8.
✭❜✮ ✧❋r❡❞ ♥ã♦ é ❢❡❧✐③ ♦✉ ❏♦❤♥ ♥ã♦ é ❢❡❧✐③✧
¬pf ∨ ¬pj = max(0, 2; 0, 6) = 0, 6.
✶✼✳ ❈❛❞❛ ❤❛❜✐t❛♥t❡ ❞❡ ✉♠❛ ✈✐❧❛ ❧♦♥❣í♥q✉❛ s❡♠♣r❡ ❞✐③ ❛ ✈❡r❞❛❞❡ ♦✉ s❡♠♣r❡ ♠❡♥t❡✳ ❯♠ ❤❛❜✐t❛♥t❡ ❞❡❧❛ ❞❛rá
❛♣❡♥❛s ❝♦♠♦ r❡s♣♦st❛ ✉♠ s✐♠ ♦✉ ✉♠ ♥ã♦ ♣❛r❛ ❛ ♣❡r❣✉♥t❛ q✉❡ ✉♠ t✉r✐st❛ ✜③❡r✳ ❙✉♣♦♥❤❛ q✉❡ ✈♦❝ê s❡❥❛ ✉♠
t✉r✐st❛ q✉❡ ✈✐s✐t❛ ❡ss❛ ár❡❛ ❡ q✉❡ ❝❤❡❣✉❡ ❛ ✉♠❛ ❜✐❢✉r❝❛çã♦ ♥❛ ❡str❛❞❛✳ ❯♠ ❧❛❞♦ ❧❡✈❛ ❛té às r✉í♥❛s q✉❡ ✈♦❝ê
q✉❡r ✈✐s✐t❛r❀ ♦ ♦✉tr♦✱ às ♣r♦❢✉♥❞❡③❛s ❞❡ ✉♠❛ ✢♦r❡st❛✳ ❯♠ ❤❛❜✐t❛♥t❡ ❞❡ss❛ ✈✐❧❛ ❡stá ♣❛r❛❞♦ ♥❡ss❛ ❜✐❢✉r❝❛çã♦✳
◗✉❡ ♣❡r❣✉♥t❛ ✈♦❝ê ♣♦❞❡ ❢❛③❡r ❛♦ ❤❛❜✐t❛♥t❡ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛❧ ❧❛❞♦ s❡❣✉✐r❄
❘❡s♣♦st❛✿
❙❡ ❢♦r ♣❡r❣✉♥t❛❞♦ s✐♠♣❧❡s♠❡♥t❡ q✉❛❧ ❡str❛❞❛ ❧❡✈❛ às r✉í♥❛s ❡ ♦ t✉r✐st❛ ♥ã♦ s♦✉❜❡r s❡ é ✉♠ ❤❛❜✐t❛♥t❡ q✉❡ ❢❛❧❛
❛ ✈❡r❞❛❞❡ ♦✉ ♠❡♥t❡✱ ❡♥tã♦ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r s❡ ❛ r❡s♣♦st❛ é ❝♦rr❡t❛ ♦✉ ♥ã♦✳ ❆ ♣❡r❣✉♥t❛ ♥ã♦ ♣♦❞❡
❞❡♣❡♥❞❡r ❞♦ t✐♣♦ ❞❡ ❤❛❜✐t❛♥t❡ ❡✱ ♣❛r❛ ✐ss♦✱ é ♥❡❝❡ssár✐♦ ❢♦r♠✉❧❛r ✉♠❛ ♣❡r❣✉♥t❛ q✉❡ ❡♥✈♦❧✈❛ ❞✉❛s q✉❡stõ❡s✳
❆ss✐♠✱ ❛♦ ✐♥✈és ❞❡ ♣❡r❣✉♥t❛r ✏❛ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ❧❡✈❛ às r✉í♥❛s❄✑ ✭♣❡r❣✉♥t❛ P✶✮✱ q✉❡ ♥ã♦ r❡s♦❧✈❡ ♦
♣r♦❜❧❡♠❛✱ ❞❡✈❡✲s❡ ♣❡r❣✉♥t❛r ❛❧❣♦ s✐♠✐❧❛r ❛ ✏s❡ ❡✉ ❢♦ss❡ ❧❤❡ ♣❡r❣✉♥t❛r s❡ ❛ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ♠❡ ❧❡✈❛
às r✉í♥❛s✱ ✈♦❝ê r❡s♣♦♥❞❡r✐❛ s✐♠❄✑ ✭♣❡r❣✉♥t❛ P✷✮✳ ◆❛t✉r❛❧♠❡♥t❡ ❛ ♠❡s♠❛ ❡str❛té❣✐❛ ♣♦❞❡ s❡r ❢❡✐t❛ ♣❛r❛ ❛
❜✐❢✉r❝❛çã♦ ❞❛ ❡sq✉❡r❞❛✳ ❱❡❥❛ q✉❡ ❛ ♣❡r❣✉♥t❛ P✷ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦✿ ✏s❡ ❡✉ ❢♦ss❡ ❧❤❡ ♣❡r❣✉♥t❛r s❡ P✶✱ ✈♦❝ê
r❡s♣♦♥❞❡r✐❛ s✐♠❄✑✱ ♦ q✉❡ ❝❧❛r❛♠❡♥t❡ ♠♦str❛ q✉❡ ❛ ♣❡r❣✉♥t❛ P✷ tr❛t❛ ❞❡ ❞✉❛s q✉❡stõ❡s✳
❖ ❤❛❜✐t❛♥t❡ q✉❡ ❢❛❧❛ ❛ ✈❡r❞❛❞❡ r❡s♣♦♥❞❡ ❝♦rr❡t❛♠❡♥t❡✳ ❖ ❤❛❜✐t❛♥t❡ q✉❡ ❢❛❧❛ ♠❡♥t✐r❛ ❞❡✈❡ ✏♠❡♥t✐r ❞✉❛s
✈❡③❡s✑ ♥♦ s❡♥t✐❞♦ q✉❡ s❡ ❛ ♣❡r❣✉♥t❛ ❢♦ss❡ s✐♠♣❧❡s♠❡♥t❡ ✏❛ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ❧❡✈❛ às r✉í♥❛s❄✑✱ ✐st♦ é✱ P✶✱
❡❧❡ t❡r✐❛ q✉❡ ♠❡♥t✐r✳ ◆♦ ❡♥t❛♥t♦✱ ❝♦♠♦ ❤á ✉♠❛ s❡❣✉♥❞❛ q✉❡stã♦ ❡♥✈♦❧✈✐❞❛ ✏s❡ ❡✉ ❢♦ss❡ ❧❤❡ ♣❡r❣✉♥t❛r s❡ P✶✱
✈♦❝ê r❡s♣♦♥❞❡r✐❛ s✐♠❄✑ ❡♥tã♦ ❡ss❡ ❤❛❜✐t❛♥t❡ ❞❡✈❡ ♠❡♥t✐r ♥♦✈❛♠❡♥t❡✳ ❊ss❡ r❛❝✐♦❝í♥✐♦ ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛
t❛❜❡❧❛ ❛❜❛✐①♦✳
✶✵
P✶✿ ❆ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ❧❡✈❛ às r✉í♥❛s❄
P✷✿ ❙❡ ❡✉ ❢♦ss❡ ❧❤❡ ♣❡r❣✉♥t❛r s❡ ❛ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ♠❡ ❧❡✈❛ às r✉í♥❛s✱ ✈♦❝ê r❡s♣♦♥❞❡r✐❛ s✐♠❄
❚✐♣♦ ❞❡ ❤❛❜✐t❛♥t❡ ❇✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛
❧❡✈❛ às r✉í♥❛s❄
P✶ P✷
❋❛❧❛ ✈❡r❞❛❞❡
❙✐♠ ❙✐♠ ❙✐♠
◆ã♦ ◆ã♦ ◆ã♦
❋❛❧❛ ♠❡♥t✐r❛
❙✐♠ ◆ã♦ ❙✐♠
◆ã♦ ❙✐♠ ◆ã♦
❙❡ ♦ ❤❛❜✐t❛♥t❡ s❡♠♣r❡ ❢❛❧❛ ❛ ✈❡r❞❛❞❡✱ ❡❧❡ ❞✐rá ❛♦ t✉r✐st❛ s❡ ❞❡✈❡ s❡❣✉✐r♣❡❧❛ ❜✐❢✉r❝❛çã♦ ❞❛ ❞✐r❡✐t❛ ♦✉ ❡sq✉❡r❞❛✳
❙❡ ♦ ❤❛❜✐t❛♥t❡ s❡♠♣r❡ ❢❛❧❛ ♠❡♥t✐r❛✱ ❛♦ r❡s♣♦♥❞❡r ❛ P✷ ❡❧❡ s❛❜❡ q✉❡ ♠❡♥t✐✉ ❛♦ r❡s♣♦♥❞❡r P✶ ✭♣❛rt❡ ❞❡ss❛
♣❡r❣✉♥t❛✮✳ ◆♦ ❡♥t❛♥t♦✱ ❡❧❡ ♠❡♥t❡ ♥♦✈❛♠❡♥t❡ ❡✱ ♥♦ ✜♥❛❧✱ ❡❧❡ r❡s♣♦♥❞❡ ❝♦rr❡t❛♠❡♥t❡ ❛ ❜✐❢✉r❝❛çã♦ ❛ s❡r
s❡❣✉✐❞❛✳ ❆ ❡str❛té❣✐❛ é ✉s❛r ✉♠❛ ❞✉♣❧❛ ♥❡❣❛t✐✈❛✳
✶✽✳ ❊st❡ s✐st❡♠❛ ❞❡ ❡s♣❡❝✐✜❝❛çõ❡s é ❝♦♥s✐st❡♥t❡❄ ✏❖ s✐st❡♠❛ ❡stá ❡♠ ✉♠ ❡st❛❞♦ ❞❡ ♠✉❧t✐✉s♦ s❡ ❡ s♦♠❡♥t❡ s❡
❡st✐✈❡r ♦♣❡r❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡✳ ❙❡ ♦ s✐st❡♠❛ ❡stá ♦♣❡r❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡✱ ♦ ♥ú❝❧❡♦ ❞♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧
✭❦❡r♥❡❧✮ ❡stá ❢✉♥❝✐♦♥❛♥❞♦✳ ❖ ❦❡r♥❡❧ ♥ã♦ ❡stá ❢✉♥❝✐♦♥❛♥❞♦ ♦✉ ♦ s✐st❡♠❛ ❡stá ♥♦ ♠♦❞♦ ❞❡ ✐♥t❡rr✉♣çã♦✳ ❙❡ ♦
s✐st❡♠❛ ♥ã♦ ❡stá ❡♠ ✉♠ ❡st❛❞♦ ❞❡ ♠✉❧t✐✉s♦✱ ❡♥tã♦ ❡stá ❡♠ ✉♠ ♠♦❞♦ ❞❡ ✐♥t❡rr✉♣çã♦✳ ❖ s✐st❡♠❛ ♥ã♦ ❡stá ♥♦
♠♦❞♦ ❞❡ ✐♥t❡rr✉♣çã♦✳✑
❘❡s♣♦st❛✿
❙❡❥❛♠ ♦s s❡❣✉✐♥t❡s ❛r❣✉♠❡♥t♦s✿
p = ❖ s✐st❡♠❛ ❡stá ❡♠ ✉♠ ❡st❛❞♦ ❞❡ ♠✉❧t✐✉s♦✳
q = ❖ s✐st❡♠❛ ❡stá ♦♣❡r❛♥❞♦ ♥♦r♠❛❧♠❡♥t❡✳
r = ❖ ♥ú❝❧❡♦ ❞♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ✭❦❡r♥❡❧✮ ❡stá ❢✉♥❝✐♦♥❛♥❞♦✳
s = ❖ s✐st❡♠❛ ❡stá ♥♦ ♠♦❞♦ ❞❡ ✐♥t❡rr✉♣çã♦✳
❚r❛❞✉çã♦ ❞♦s ❢❛t♦s ♣❛r❛ ❛s ♣r♦♣♦s✐çõ❡s✿
✭❛✮ p↔ q ≡ (p→ q) ∧ (q → p)
✭❜✮ q → r
✭❝✮ ¬r ∨ s
✭❞✮ ¬p→ s
✭❡✮ ¬s
❉❡❞✉çõ❡s✿
✭✐✮ ¬r ∨ s ✭❝✮
¬s ✭❡✮
✳
✳
✳ ¬r ❙✐❧♦❣✐s♠♦ ❉✐s❥✉♥t✐✈♦
✭✐✐✮ q → r ✭❜✮
¬r ✭✐✮
✳
✳
✳ ¬q ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✐✐✮ ¬p → s ✭❞✮
¬s ✭❡✮
✳
✳
✳ p ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✈✮ (p → q) ∧ (q → p) ✭❛✮
✳
✳
✳ p → q ❙✐♠♣❧✐✜❝❛çã♦ ❈♦♥❥✉♥t✐✈❛
✭✈✮ p → q ✭✐✈✮
¬q ✭✐✐✮
✶✶
✳
✳
✳ ¬p ▼♦❞✉s ❚♦❧❧❡♥s
❆s ❝♦♥❝❧✉sõ❡s ✭✐✐✐✮ ❡ ✭✈✮ ❞✐③❡♠ q✉❡ p ❡ ¬p ❞❡✈❡♠ s❡r ✈❡r❞❛❞❡✐r♦s✱ ♦ q✉❡ ❝❧❛r❛♠❡♥t❡ é ✐♥❝♦♥s✐st❡♥t❡✳
✶✾✳ ❊st❡ s✐st❡♠❛ ❞❡ ❡s♣❡❝✐✜❝❛çõ❡s é ❝♦♥s✐st❡♥t❡❄ ✏❖ r♦t❡❛❞♦r ♣♦❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ♣❛r❛ ♦ s✐st❡♠❛ ♣r✐♥❝✐♣❛❧
s♦♠❡♥t❡ s❡ ❡❧❡ tr❛t❛r ✉♠ ♥♦✈♦ ❡s♣❛ç♦ ❞❡ ❡♥❞❡r❡ç♦✳ P❛r❛ ♦ r♦t❡❛❞♦r tr❛t❛r ♦ ♥♦✈♦ ❡s♣❛ç♦ ❞❡ ❡♥❞❡r❡ç♦✱ é
♥❡❝❡ssár✐♦ q✉❡ ❛ ú❧t✐♠❛ ✈❡rsã♦ ❞♦ s♦❢t✇❛r❡ s❡❥❛ ✐♥st❛❧❛❞❛✳ ❖ r♦t❡❛❞♦r ♣♦❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ❛♦ s✐st❡♠❛
♣r✐♥❝✐♣❛❧ s❡ ❛ ú❧t✐♠❛ ✈❡rsã♦ ❞♦ s♦❢t✇❛r❡ ❡st✐✈❡r ✐♥st❛❧❛❞❛✳ ❖ r♦t❡❛❞♦r ♥ã♦ tr❛t❛ ♦ ♥♦✈♦ ❡s♣❛ç♦✳✑
❘❡s♣♦st❛✿
❙❡❥❛♠ ♦s s❡❣✉✐♥t❡s ❛r❣✉♠❡♥t♦s✿
p = ❖ r♦t❡❛❞♦r ♣♦❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ♣❛r❛ ♦ s✐st❡♠❛ ♣r✐♥❝✐♣❛❧✳
q = ❖ r♦t❡❛❞♦r tr❛t❛ ✉♠ ♥♦✈♦ ❡s♣❛ç♦ ❞❡ ❡♥❞❡r❡ç♦✳
r = ❆ ú❧t✐♠❛ ✈❡rsã♦ ❞♦ s♦❢t✇❛r❡ s❡❥❛ ✐♥st❛❧❛❞❛✳
❚r❛❞✉çã♦ ❞♦s ❢❛t♦s ♣❛r❛ ❛s ♣r♦♣♦s✐çõ❡s✿
✭❛✮ p s♦♠❡♥t❡ s❡ q ≡ p→ q
✭❜✮ ♣❛r❛ q é ♥❡❝❡ssár✐♦ r ≡ r é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q ≡ q → r
✭❝✮ p s❡ r ≡ r → p ≡ ¬p→ ¬r
✭❞✮ ¬q
❉❡❞✉çã♦✿
✭✐✮ p → q ✭❛✮
¬q ✭❞✮
✳
✳
✳ ¬p ▼♦❞✉s ❚♦❧❧❡♥s
✭✐✐✮ ¬p → ¬r ✭❝✮
¬p ✭✐✮
✳
✳
✳ ¬r ▼♦❞✉s P♦♥❡♥s
P❡❧❛s ❞❡❞✉çõ❡s ❛❝✐♠❛✱ ❛s três ♣r♦♣♦s✐çõ❡s sã♦ ❢❛❧s❛s✿ ¬q ❞❡ ❛❝♦r❞♦ ❝♦♠ ✭❞✮✱ ¬p ❞❡ ❛❝♦r❞♦ ❝♦♠ ✭✐✮ ❡ ¬r ❞❡
❛❝♦r❞♦ ❝♦♠ ✭✐✐✮✳ ❙❡ t♦❞❛s três ♣r♦♣♦s✐çõ❡s sã♦ ❢❛❧s❛s✱ ❛s q✉❛tr♦ ❡s♣❡❝✐✜❝❛çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s ❡ ❡❧❛s sã♦
❝♦♥s✐st❡♥t❡s✳
✷✵✳ ❯♠ ❞❡t❡t✐✈❡ ❡♥tr❡✈✐st♦✉ q✉❛tr♦ t❡st❡♠✉♥❤❛s ❞❡ ✉♠ ❝r✐♠❡✳ ❆ ♣❛rt✐r ❞❛s ❤✐stór✐❛s ❞❛s t❡st❡♠✉♥❤❛s✱ ♦
❞❡t❡t✐✈❡ ❝♦♥❝❧✉✐✉ q✉❡✱ s❡ ♦ ♠♦r❞♦♠♦ ❡stá ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡✱ ❡♥tã♦ ♦ ❝♦③✐♥❤❡✐r♦ t❛♠❜é♠ ❡stá❀ ♦ ❝♦③✐♥❤❡✐r♦
❡ ♦ ❥❛r❞✐♥❡✐r♦✱ ❛♠❜♦s✱ ♥ã♦ ♣♦❞❡♠ ❡st❛r ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡❀ ♦ ❥❛r❞✐♥❡✐r♦ ❡ ♦ ③❡❧❛❞♦r✱ ❛♠❜♦s✱ ♥ã♦ ❡stã♦
♠❡♥t✐♥❞♦❀ ❡ s❡ ♦ ③❡❧❛❞♦r ❡stá ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡✱ ❡♥tã♦ ♦ ❝♦③✐♥❤❡✐r♦ ❡stá ♠❡♥t✐♥❞♦✳ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s
q✉❛tr♦ t❡st❡♠✉♥❤❛s✱ ♦ ❞❡t❡t✐✈❡ ♣♦❞❡ ❞❡t❡r♠✐♥❛r s❡ ❛ ♣❡ss♦❛ ❡stá ♠❡♥t✐♥❞♦ ♦✉ ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡❄
❘❡s♣♦st❛✿
❆s q✉❛tr♦ t❡st❡♠✉♥❤❛s ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞❛s ♣❡❧❛s ✈❛r✐á✈❡✐s C ✭❝♦③✐♥❤❡✐r♦✮✱ J ✭❥❛r❞✐♥❡✐r♦✮✱ M ✭♠♦r❞♦♠♦✮
❡ Z ✭③❡❧❛❞♦r✮✱ q✉❡ s❡rã♦ ✉s❛❞❛s ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ❡stã♦ ❢❛❧❛♥❞♦ ❛ ✈❡r❞❛❞❡✳
❙❡❥❛♠ ♦s s❡❣✉✐♥t❡s ❛r❣✉♠❡♥t♦s✿
✭❛✮ M → C ❬s❡ ♦ ♠♦r❞♦♠♦ ❡stá ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡✱ ❡♥tã♦ ♦ ❝♦③✐♥❤❡✐r♦ t❛♠❜é♠ ❡stá❪
✭❜✮ C ⊕ J ❬♦ ❝♦③✐♥❤❡✐r♦ ❡ ♦ ❥❛r❞✐♥❡✐r♦✱ ❛♠❜♦s✱ ♥ã♦ ♣♦❞❡♠ ❡st❛r ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡❪
✭❝✮ ¬J ⊕ ¬Z ❬♦ ❝♦③✐♥❤❡✐r♦ ❡ ♦ ❥❛r❞✐♥❡✐r♦✱ ❛♠❜♦s✱ ♥ã♦ ♣♦❞❡♠ ❡st❛r ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡❪
✭❞✮ Z → ¬C ❬s❡ ♦ ③❡❧❛❞♦r ❡stá ❞✐③❡♥❞♦ ❛ ✈❡r❞❛❞❡✱ ❡♥tã♦ ♦ ❝♦③✐♥❤❡✐r♦ ❡stá ♠❡♥t✐♥❞♦❪
P❡❧♦s q✉❛tr♦ ❛r❣✉♠❡♥t♦s ❛❝✐♠❛✱ ♥ã♦ é ♣♦ssí✈❡❧ ❛♣❧✐❝❛r ✉♠❛ r❡❣r❛ ❞❡ ✐♥❢❡rê♥❝✐❛✳ ❙❡ ✜③❡r♠♦s ✉♠❛ t❛❜❡❧❛ ❞❛
✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ♦ ❝❡♥ár✐♦ ♥♦ q✉❛❧ ❛s q✉❛tr♦ ♣r❡♠✐ss❛s s✉♣♦st❛♠❡♥t❡ sã♦ ✈❡r❞❛❞❡✐r❛s✱ ❝♦♥❢♦r♠❡
♠♦str❛❞♦ ❛ s❡❣✉✐r✳
✶✷
❱❛r✐á✈❡✐s Pr❡♠✐ss❛s
C J M Z ✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮
✶✳ ❱ ❱ ❱ ❱ ❱ ❋ ❋ ❋
✷✳ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❱
✸✳ ❱ ❱ ❋ ❱ ❱ ❋ ❋ ❋
✹✳ ❱ ❱ ❋ ❋ ❱ ❋ ❱ ❱
✺✳ ❱ ❋ ❱ ❱ ❱ ❱ ❱ ❋
✻✳ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱
✼✳ ❱ ❋ ❋ ❱ ❱ ❱ ❱ ❋
✽✳ ❱ ❋ ❋ ❋ ❱ ❱ ❋ ❱
✾✳ ❋ ❱ ❱ ❱ ❋ ❱ ❋ ❱
✶✵✳ ❋ ❱ ❱ ❋ ❋ ❱ ❱ ❱
✶✶✳ ❋ ❱ ❋ ❱ ❱ ❱ ❋ ❱
✶✷✳ → ❋ ❱ ❋ ❋ ❱ ❱ ❱ ❱
✶✸✳ ❋ ❋ ❱ ❱ ❋ ❱ ❱ ❱
✶✹✳ ❋ ❋ ❱ ❋ ❋ ❋ ❋ ❱
✶✺✳ ❋ ❋ ❋ ❱ ❱ ❋ ❱ ❱
✶✻✳ ❋ ❋ ❋ ❋ ❱ ❋ ❋ ❱
➜ ❆s ♣r❡♠✐ss❛s sã♦ ✈❡r❞❛❞❡✐r❛s ♣❛r❛ ❛ ❧✐♥❤❛ ✶✷✱ ♦✉ s❡❥❛✱ ♦ ❥❛r❞✐♥❡✐r♦ ❢❛❧❛ ❛ ✈❡r❞❛❞❡ ❡ ❛s ♦✉tr❛s t❡st❡♠✉♥❤❛s
♥ã♦✳
❖ t❡①t♦✱ ❛ s❡❣✉✐r✱ ❢♦✐ r❡t✐r❛❞♦ ❞♦ ❧✐✈r♦ t❡①t♦✱ ♣á❣✐♥❛ ✷✾✿ ✏❖ ❞✉❛❧ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛ q✉❡ ❝♦♥té♠ ❛♣❡♥❛s
♦s ♦♣❡r❛❞♦r❡s ❧ó❣✐❝♦s ∨✱ ∧ ❡ ¬ é ❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛ ♦❜t✐❞❛ ♣❡❧❛ tr♦❝❛ ❞❡ ❝❛❞❛ ∨ ♣♦r ∧✱ ❝❛❞❛ ∧ ♣♦r ∨✱ ❝❛❞❛
V ♣♦r F ❡ ❝❛❞❛ F ♣♦r V ✳ ❖ ❞✉❛❧ ❞❡ s é r❡♣r❡s❡♥t❛❞♦ ♣♦r s∗✑✳
✷✶✳ ❊♥❝♦♥tr❡ ♦ ❞✉❛❧ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦st❛s✿
✭❛✮ s1 : p ∧ ¬q ∧ ¬r
❘❡s♣♦st❛✿
s∗1 : p ∨ ¬q ∨ ¬r
✭❜✮ s2 : (p ∧ q ∧ r) ∨ s
❘❡s♣♦st❛✿
s∗2 : (p ∨ q ∨ r) ∧ s
✭❝✮ s3 : (p ∨ F ) ∧ (q ∨ V )
❘❡s♣♦st❛✿
s∗3 : (p ∧ V ) ∨ (q ∧ F )
✭❞✮ ❆♣r❡s❡♥t❡ ✉♠ ❡①❡♠♣❧♦ ❞❡ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛ ❢♦r♠❛❞❛ ♣♦r ♣❡❧♦ ♠❡♥♦s ❞✉❛s ✈❛r✐á✈❡✐s t❛❧ q✉❡ s ≡ s∗✳
❘❡s♣♦st❛✿
s4✿ (p ∨ F ) ∧ (q ∨ V )
p ∧ V
p
s∗4✿ (p ∧ V ) ∨ (q ∧ F )
p ∨ F
p
✶✸