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Feb 01, 2024 176 likes •442 views

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SLIDE 1

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

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SLIDE 2

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✺✳✶❪ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡ ❢♦r Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✺✳✷❪

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SLIDE 3

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❍❡r❜r❛♥❞✬s ❚❤❡♦r❡♠ ❛♥❞ r❡❢✉t❛t✐♦♥ ♣r♦❝❡❞✉r❡s

❙❛t✐s✜❛❜✐❧✐t② ♣r♦❝❡❞✉r❡s ❲❡ ❝❛♥ ❜✉✐❧❞ r❡❢✉t❛t✐♦♥ ♣r♦❝❡❞✉r❡s ❜✉✐❧❞✐♥❣ ♦♥ ❍❡r❜r❛♥❞✬s ❚❤❡♦r❡♠✳ ❋♦r ❡①❛♠♣❧❡ ●✐❧♠♦r❡✬s ♠❡t❤♦❞ ✉s✐♥❣ ❉P▲▲ ❢♦r ❝❤❡❝❦✐♥❣ s❛t✐s✜❛❜✐❧✐t②✳ ❚❤✐s r❡q✉✐r❡s t❤❡ ❣❡♥❡r❛t✐♦♥ ♦❢ s❡ts ❙′

✵✱ ❙′ ✶✱ · · · ♦❢ ❣r♦✉♥❞

❝❧❛✉s❡s✳ ❈♦♠♣✉t❛t♦♥ ✐ss✉❡✿ ❢♦r ♠♦st ❝❛s❡s t❤✐s s❡q✉❡♥❝❡ ❣r♦✇s ❡①♣♦♥❡♥t✐❛❧❧②✳

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SLIDE 4

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❈♦♠♣✉t❛t✐♦♥❛❧ ✐ss✉❡

❊①♣♦♥❡♥t✐❛❧ ❣r♦✇ ♦❢ s❡q✉❡♥❝❡ ❈♦♥s✐❞❡r ❙ =

{P(①, ❣(①), ②, ❤(①, ②), ③, ❦(①, ②, ③)), ¬P(✉, ✈, ❡(✈), ✇, ❢ (✈, ✇), ①)}

❍✵ = {❛} ❍✶ = {❛, ❣(❛), ❤(❛, ❛), ❦(❛, ❛, ❛), ❡(❛), ❢ (❛, ❛)} |❙′

✵| ❂ ✷✱ |❙′ ✶| ❂ ✶✺✶✷

❊❛r❧✐❡st ✉♥s❛t✐s✜❛❜❧❡ s❡t ✐s ❙′

✺ ✇❤✐❝❤ ❤❛s ❛♣♣r♦①✐♠❛t❡❧②

✶✵✷✺✻ ❡❧❡♠❡♥ts✦

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SLIDE 5

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❘♦❜✐♥s♦♥ ✶✾✻✺ ❆✐♠✿ ❞✐r❡❝t❧② t❡st ✉♥s❛t✐s✜❛❜✐❧✐t② ♦❢ ❛ s❡t ♦❢ ❝❧❛✉s❡s ❙ ✇✐t❤♦✉t ❣❡♥❡r❛t✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ ❛ss♦❝✐❛t❡❞ ❣r♦✉♥❞ ❝❧❛✉s❡s✳ ❇❛s✐❝ ✐❞❡❛✿ t❡st ✇❤❡t❤❡r ❙ ❝♦♥t❛✐♥s t❤❡ ❡♠♣t② ❝❧❛✉s❡

■❢ ∈ ❙ t❤❡♥ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ❖t❤❡r✇✐s❡ ♥❡❡❞ t♦ ❝❤❡❝❦ ✇❤❡t❤❡r ❙ | =

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SLIDE 6

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❙❡♠❛t✐❝ tr❡❡s

❘❡s✳ Pr✐♥❝✐♣❧❡ ❛♥❞ ❙❡♠✳ ❚r❡❡s ❘❡❝❛❧❧✿ ❜② ❍❡r❜r❛♥❞✬s ❚❤❡♦r❡♠ ✭✈❡rs✐♦♥ ■✮ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ✐✛ t❤❡r❡ ✐s ❛ ✜♥✐t❡ ❝❧♦s❡❞ s❡♠❛♥t✐❝ tr❡❡ ❚ ❢♦r ❙✳ ❙ ❝♦♥t❛✐♥s ✐✛ t❤❡ ❝♦rr❡s♣♦♥s✐♥❣ ❝❧♦s❡❞ s❡♠❛♥t✐❝ tr❡❡ ❚ ❝♦♥t❛✐♥s ♦♥❧② t❤❡ r♦♦t ♥♦❞❡✳ ■❢ ❙ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ t❤❡♥ ❚ ♠✉st ❝♦♥t❛✐♥ ♠♦r❡ t❤❛♥ ♦♥❡ ♥♦❞❡✳ ■❢ ✇❡ ❝❛♥ r❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ✐♥ ❚ t❤❡♥ ✇❡ ❝❛♥ ❢♦r❝❡ t♦ ❛♣♣❡❛r✳

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SLIDE 7

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

■♥❢❡r❡♥❝❡ ❘✉❧❡s

■♥❢❡r❡♥❝❡ ❘✉❧❡s ❚❤❡ r❡s♦❧✉t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐s ❛♥ ■♥❢❡r❡♥❝❡ ❘✉❧❡ ■♥❢❡r❡♥❝❡ ❘✉❧❡✿ ❛ r✉❧❡ t❤❛t ❣❡♥❡r❛t❡s ♥❡✇ ❝❧❛✉s❡s ✇❤✐❝❤ ❛r❡ ❛ ❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ s♦♠❡ ♦❢ t❤❡ ❡①✐st✐♥❣ ❝❧❛✉s❡s ◆❡✇ ❝❧❛✉s❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ t✉r♥ s♦♠❡ ♦❢ t❤❡ ♥♦❞❡s ✐♥ ❚ t♦ ❢❛✐❧✉r❡ ♥♦❞❡s✳ ❚❤✉s ♥✉♠❜❡r ♦❢ ♥♦❞❡s ✐♥ ❚ ❛r❡ r❡❞✉❝❡❞ ❛♥❞ ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❛♣♣❡❛r✳ ❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡ ❛♥❞ ❙❡♠✳ ❚r❡❡s✮ ❚❤❡ s❡♠❛♥t✐❝ tr❡❡ ❢♦r ❙ = {¬P ∨ ◗, P, ¬◗} ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ❜② ❛❞❞✐♥❣ {¬P} t♦ ❙✳

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SLIDE 8

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❘❡s♦❧✉t✐♦♥ ♣r✐♥❝✐♣❧❡ ❢♦r Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝

❘❡s♦❧✉t✐♦♥ ❛♥❞ ❖♥❡✲▲✐t❡r❛❧ r✉❧❡ ❊①t❡♥s✐♦♥ ♦❢ ❖♥❡✲▲✐t❡r❛❧ r✉❧❡ ♦❢ ❉P▲▲ t♦ ❛♥② ♣❛✐r ♦❢ ❝❧❛✉s❡s ❋♦❝✉s ♦♥ ❛ ✉♥✐t ❝❧❛✉s❡ ❝♦♥t❛✐♥✐♥❣ ❛ ❧✐t❡r❛❧ ▲ ❛♥❞ ❧♦♦❦ ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ▲ ✐♥ ❛♥♦t❤❡r ❝❧❛✉s❡✳ ❖❜t❛✐♥ ❛ ♥❡✇ ❝❧❛✉s❡ ❞❡❧❡t✐♥❣ t❤❡ ❖♥❡✲▲✐t❡r❛❧ ❝❧❛✉s❡✱ ❛♥❞ t❤❡ ❝♦♠♣❧❡♠❡♥t ❧✐t❡r❛❧ ❢r♦♠ t❤❡ ♦t❤❡r ❝❧❛✉s❡✳ ❊①❛♠♣❧❡ ✭❖♥❡✲▲✐t❡r❛❧ ❛♥❞ r❡s♦❧✉t✐♦♥✮ ❈✶ = P✱ ❈✷ = ¬P ∨ ◗ ❆♣♣❧②✐♥❣ t❤❡ ❖♥❡✲▲✐t❡r❛❧ r✉❧❡ ♦❢ ❉P▲▲ t♦ {❈✶, ❈✷} ✇❡ ♦❜t❛✐♥ ❈✸ = ◗

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SLIDE 9

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡ ❋♦r ❛♥② t✇♦ ❝❧❛✉s❡s ❈✶ ❛♥❞ ❈✷ ✐❢ t❤❡r❡ ✐s ❛ ❧✐t❡r❛❧ ▲✶ ✐♥ ❈✶ t❤❛t ✐s ❝♦♠♣❧❡♠❡♥t❛r② t♦ ❛ ❧✐t❡r❛❧ ▲✷ ✐♥ ❈✷ t❤❡♥ ❞❡❧❡t❡ ▲✶ ❛♥❞ ▲✷ ❢r♦♠ ❈✶ ❛♥❞ ❈✷ ❛♥❞ ❣❡♥❡r❛t❡ ❛ ♥❡✇ ❝❧❛✉s❡ ❈✸ ❛s t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ❝❧❛✉s❡s✳ ❈✸ ✐s ❛ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✿ ■♥❢❡r❡♥❝❡ r✉❧❡ ▲✶ ∨ ❈ ′

✶

¬▲✶ ∨ ❈ ′

✷

❈ ′

✶ ∨ ❈ ′ ✷

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SLIDE 10

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = P ∨ ❘ ❛♥❞ ❈✷ = ¬P ∨ ◗ P ∨ ❘ ¬P ∨ ◗ ❘ ◗ ❈✸ ❘ ◗ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 11

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = P ∨ ❘ ❛♥❞ ❈✷ = ¬P ∨ ◗ P ∨ ❘ ¬P ∨ ◗ ❘ ∨ ◗ ❈✸ ❘ ◗ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 12

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = P ∨ ❘ ❛♥❞ ❈✷ = ¬P ∨ ◗ P ∨ ❘ ¬P ∨ ◗ ❘ ∨ ◗ ❈✸ = ❘ ∨ ◗ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 13

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = ¬P ∨ ◗ ∨ ❘ ❛♥❞ ❈✷ = ¬◗ ∨ ❙ ¬P ∨ ◗ ∨ ❘ ¬◗ ∨ ❙ P ❘ ❙ ❈✸ P ❘ ❙ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 14

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = ¬P ∨ ◗ ∨ ❘ ❛♥❞ ❈✷ = ¬◗ ∨ ❙ ¬P ∨ ◗ ∨ ❘ ¬◗ ∨ ❙ ¬P ∨ ❘ ∨ ❙ ❈✸ P ❘ ❙ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 15

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = ¬P ∨ ◗ ∨ ❘ ❛♥❞ ❈✷ = ¬◗ ∨ ❙ ¬P ∨ ◗ ∨ ❘ ¬◗ ∨ ❙ ¬P ∨ ❘ ∨ ❙ ❈✸ = ¬P ∨ ❘ ∨ ❙ ✐s t❤❡ r❡s♦❧✈❡♥t ❢♦r ❈✶ ❛♥❞ ❈✷✳

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SLIDE 16

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■■

❊①❛♠♣❧❡ ✭❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡s ❈✶ = ¬P ∨ ◗ ❛♥❞ ❈✷ = ¬P ∨ ❙ ❚❤❡r❡ ✐s ♥♦ r❡s♦❧✈❡♥t ✐♥ t❤✐s ❝❛s❡ ❛s ♥♦ ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐r ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ ❝❧❛✉s❡s✳

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SLIDE 17

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

Pr♦♣❡rt② ♦❢ ❘❡s♦❧✉t✐♦♥

▲♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡

  • ✐✈❡♥ t✇♦ ❝❧❛✉s❡s ❈✶ ❛♥❞ ❈✷✱ ❛♥❞ t❤❡✐r r❡s♦❧✈❡♥t ❈✱ ❈✶∧❈✷ |

= ❈ ✭❈ ✐s ❛ ❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❈✶ ❛♥❞ ❈✷✮✳ Pr♦♦❢✳ ▲❡t ❈✶ = ▲ ∨ ❈ ′

✶✱ ❈✷ = ¬▲ ∨ ❈ ′ ✷✱ ❈ = ❈ ′ ✶ ∨ ❈ ′ ✷ ✇❤❡r❡ ❈ ′ ✶ ❛♥❞ ❈ ′ ✷

❛r❡ ❞✐s❥✉♥❝t✐♦♥s ♦❢ ❧✐t❡r❛❧s✳ ❙✉♣♣♦s❡ ■ | = ❈✶ ∧ ❈✷✱ ✇❡ ✇❛♥t t♦ s❤♦✇ t❤❛t ■ | = ❈✳ ◆♦t❡ t❤❛t ❡✐t❤❡r ■ | = ▲ ♦r ■ | = ¬▲✳ ❆ss✉♠❡ ■ | = ¬▲ ❚❤❡♥ s✐♥❝❡ ■ | = ❈✶✱ ❈ ′

✶ = ❛♥❞ ■ |

= ❈ ′

✶✳

❚❤❡r❡❢♦r❡ s✐♥❝❡ ❈ = ❈ ′

✶ ∨ ❈ ′ ✷ ✇❡ ❤❛✈❡ t❤❛t ■ |

= ❈✳ ❙✐♠✐❧❛r ❝♦♥s✐❞❡r❛t✐♦♥s ❤♦❧❞ ❢♦r ■ | = ▲✳

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SLIDE 18

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❡♠♣t② ❝❧❛✉s❡

❘❡s♦❧✉t✐♦♥ ❛♥❞ s❛t✐s✜❛❜✐❧✐t② ■❢ ❈✶ ❛♥❞ ❈✷ ❛r❡ ✉♥✐t ❝❧❛✉s❡s t❤❡♥✱ ✐❢ t❤❡r❡ ✐s r❡s♦❧✈❡♥t✱ t❤❛t r❡s♦❧✈❡♥t ✇✐❧❧ ♥❡❝❡ss❛r② ❜❡ ✳ ■❢ ✇❡ ❝❛♥ ❞❡r✐✈❡ t❤❡ ❡♠♣t② ❝❧❛✉s❡ ❢r♦♠ ❙✱ t❤❡♥ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ✭❝♦rr❡❝t♥❡ss✮ ■❢ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ✉s✐♥❣ r❡s♦❧✉t✐♦♥ ✇❡ ❝❛♥ ❛❧✇❛②s ❞❡r✐✈❡ t❤❡ ❡♠♣t② ❝❧❛✉s❡ ✭❝♦♠♣❧❡t❡♥❡ss✮

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SLIDE 19

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❉❡❞✉❝t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❉❡❞✉❝t✐♦♥✮

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡s ❙ ❛ ✭r❡s♦❧✉t✐♦♥✮ ❞❡❞✉❝t✐♦♥ ♦❢ ❈ ❢r♦♠ ❙ ✐s

❛ ✜♥✐t❡ s❡q✉❡♥❝❡ ❈✶, ❈✷, · · · , ❈❦ ♦❢ ❝❧❛✉s❡s s✉❝❤ t❤❛t ❡❛❝❤ ❈✐ ✐s ❡✐t❤❡r ❛ ❝❧❛✉s❡ ✐♥ ❙ ♦r ❛ r❡s♦❧✈❡♥t ♦❢ ❝❧❛✉s❡s ♣r❡❝❡❞✐♥❣ ❈✐✱ ❛♥❞ ❈❦ = ❈✳

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SLIDE 20

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■✿ ❉❡❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✭❞❡❞✉❝t✐♦♥✮ ❈♦♥s✐❞❡r ❙ = {❈✶, ❈✷, ❈✸}✱ ✇❤❡r❡ ❈✶ = ¬P ∨ ◗ ❈✷ = P ❛♥❞ ❈✸ = ¬◗✳ ❆♣♣❧②✐♥❣ r❡s♦❧✉t✐♦♥ t♦ ❈✶ ❛♥❞ ❈✷ ✇❡ ❤❛✈❡✿ ¬P ∨ ◗, P ◗ ❚❤❡♥ ❛♣♣❧②✐♥❣ ¬◗, ◗

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SLIDE 21

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❉❡❞✉❝✐♥❣ t❤❡ ❡♠♣t② ❝❧❛✉s❡

❊♠♣t② ❝❧❛✉s❡✱ ❉❡❞✉❝t✐♦♥ ❛♥❞ ❯♥s❛t✐s✜❛❜✐❧✐t②

  • ✐✈❡♥ ❙✱ s✉♣♣♦s❡ ✇❡ ❞❡r✐✈❡ ✉s✐♥❣ r❡s♦❧✉t✐♦♥❀

⇒ ✐s ❛ ❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❙❀ ❙✐♥❝❡ ❙ | = t❤❡♥ ∀■ ✐❢ ■ | = ❙ t❤❡♥ ■ | = ❀ ❇✉t t❤❡r❡ ✐s ♥♦ ■ t❤❛t ❝❛♥ ✈❡r✐❢② ❀ ⇒ ✐❢ ✇❡ ❞❡r✐✈❡ ❢r♦♠ ❙ ✉s✐♥❣ r❡❢✉t❛t✐♦♥ t❤❡♥ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡✳ ▲❛t❡r ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t ✐❢ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ t❤❡♥ ✇❡ ❝❛♥ ❛❧✇❛②s ❞❡r✐✈❡ ✉s✐♥❣ r❡s♦❧✉t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✭❘❡❢✉t❛t✐♦♥✮ ❆ ❞❡❞✉❝t✐♦♥ ♦❢ ✐s ❝❛❧❧❡❞ ❛ r❡❢✉t❛t✐♦♥ ✭♦r ❛ ♣r♦♦❢✮ ♦❢ ❙

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SLIDE 22

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■✿ ❉❡❞✉❝t✐♦♥

❊①❛♠♣❧❡ ✭❞❡❞✉❝t✐♦♥✮

  • ✐✈❡♥ ❙ = {❈✶, ❈✷, ❈✸, ❈✹} ❛♥❞ ❈✶ = {P ∨ ◗}✱

❈✷ = {¬P ∨ ◗}✱ ❈✸ = {P ∨ ¬◗} ❛♥❞ ❈✹ = {¬P ∨ ¬◗}✳ ❆♣♣❧② r❡s♦❧✉t✐♦♥ t♦ ❈✶ ❛♥❞ ❈✷ ❛♥❞ ♦❜t❛✐♥ ❈ ′ = {◗}✳ ❆♣♣❧② r❡s♦❧✉t✐♦♥ t♦ ❈✸ ❛♥❞ ❈✹ ❛♥❞ ♦❜t❛✐♥ ❈ ′′ = {¬◗}✳ ❆♣♣❧② r❡s♦❧✉t✐♦♥ t♦ ❈ ′ ❛♥❞ ❈ ′′ ❛♥❞ ♦❜t❛✐♥ ✳ ❍❡♥❝❡ ❙ ✐s ✉♥s❛t✳

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SLIDE 23

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❛♠♣❧❡ ■■✿ ❉❡❞✉❝t✐♦♥ ❚r❡❡

❊①❛♠♣❧❡ ✭❞❡❞✉❝t✐♦♥✮ ❈♦♥s✐❞❡r ❙ ❢r♦♠ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ❞❡❞✉❝t✐♦♥ st❡♣s✳ ❚❤❡ ❞❡❞✉❝t✐♦♥ tr❡❡ ✐s✿

P∨◗, ¬P∨◗ ◗ P∨¬◗, ¬P∨¬◗ ¬◗

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SLIDE 24

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❡r❝✐s❡

❊①❡r❝✐s❡ Pr♦✈❡ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛s ❛r❡ ✉♥s❛t✳ ✉s✐♥❣ t❤❡ r❡s♦❧✉t✐♦♥ ♣r✐♥❝✐♣❧❡

✶ ¬q ∨ ♣✱ ¬♣ ∨ ¬q✱ q ∨ r✱ ¬q ∨ ¬r✱ ¬♣ ∨ ¬r✱ ♣ ∨ ¬r ✷ P(❛), ¬❉(❛)∨▲(❛, ❛), ¬P(❛)∨¬◗(❛)∨¬▲(❛, ❛), ❉(❛), ◗(❛)

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SLIDE 25

❚❤❡ ❘❡s♦❧✉t✐♦♥ Pr✐♥❝✐♣❧❡

❊①❡r❝✐s❡ ■■

❊①❡r❝✐s❡

✶ ❋✶ P → (¬◗ ∨ (❘ ∧ ❙)) ✷ ❋✷ P ✸ ❋✸ ¬❙ ✹ ● = ¬◗

Pr♦✈❡ ✉s✐♥❣ t❤❡ r❡s♦❧✉t✐♦♥ ♣r✐♥❝✐♣❧❡ t❤❛t ❋✶ ∧ ❋✷ ∧ ❋✸ | = ●