Todays Lecture 3909

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Todays Lecture3/9/09

• Last 5 Equivalence Rules
• Practice with Proofs

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equivalence rules

• Equivalence rules can be applied to whole lines
  and to parts of lines.
• Contrast with Implicational rules (like MP) ,
  which can only be applied to whole lines.

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1st Five Equivalence Rules

• Double Negation (DN)
• Commutation (Com)
• Association (As)
• DeMorgan's Laws (DeM)
• Contraposition (Cont)

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2nd Five Equivalence Rules

• Exportation (Ex)
• Distribution (Dist)
• Redundancy (Re)
• Material Equivalence (ME)
• Material Implication (MI)

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Exportation (Ex)

• (p ? q) ? r p ? (q ? r)
• Example
• 1. (A ? B) ? C ? D
• 2. A
• 3. D ? B ? C
• 4. (A ? B) ? C 1, 3 DS
• 5. A ? (B ? C) 4, Ex
• B ? C 2,5 MP
• -- --

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Distribution (Dist)

• p ? (q ? r) (p ? q) ? (p ? r)
• p ? (q ? r) (p ? q) ? (p ? r)
• Example
• 1. X ? (Y ? Z)
• 2. (X ? Z) ? X ? Y
• 3. (X ? Y) ? (X ? Z) 1, Dist
• 4. X ? Y 2,3 DS

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Redundancy (Re)

• p ? p p
• p ? p p
• Example
• 1. A ? B
• 2. A ? Z
• 3. B ? Z ? Z
• Z ? Z 1,2,3 CD
• Z 4, Re
• -- -- --

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Material Equivalence (ME)

• p ? q (p ? q) ? (q ? p)
• p ? q (p ? q) ? (p ? q)
• Example
• H ? (F ? G)
• H
• 3. G ? F
• 4. F ? G 1,2 MP
• 5. (F ? G) ? (G ? F) 4, ME
• 6. F ? G 5, Simp
• F 3,6 MT
• -- -- --

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Material Implication (MI)

• p ? q p ? q
• Example
• A ? (B ? C)
• B
• C ? A
• 4. B ? C 2,3 Conj
• 5. (B ? C) 4, DN
• ( B ? C) 5, DeM
• ( B ? C) 6, DN, MI
• A 1,7 MT
• -- -- --

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Practice HW problems

• p 388C 4, 11, 14, 20, 25D 1, 7, 13,16, 22

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HW

• p 363C 8, 16, 18 D 5, 10, 15, 20, 25p
  379C 9, 12, 21D 8, 12, 17, 22, 24p 388C
  4, 11, 14, 20, 25D 1, 7, 13,16, 22