midterm-solW08 - Math 1090 W08 Midterm Exam Solutions(1...

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Unformatted text preview: Math 1090 W08 Midterm Exam Solutions (1) Prove ` ( A ∧ B β†’ C ) ≑ ( A β†’ ( B β†’ C )). (You cannot use Post’s theorem, but you can use any of the proof methods covered so far.) Solution. A β†’ ( B β†’ C ) ⇔ (2.4.11) Β¬ A ∨ ( B β†’ C ) ⇔ (2.4.11) Β¬ A ∨ Β¬ B ∨ C ⇔ (2.4.4) ¬¬ ( Β¬ A ∨ Β¬ B ) ∨ C ⇔ (2.4.17) Β¬ ( A ∧ B ) ∨ C ⇔ (2.4.11) A ∧ B β†’ C (2) Prove ` ( A ∨ B ∨ C ) ∧ ( A β†’ D ) β†’ (( B β†’ D ) ∧ ( C β†’ D ) β†’ D ) (You cannot use Post’s theorem, but you can use any of the proof methods covered so far.) Solution. Assume A ∨ B ∨ C and A β†’ D . By the Deduction Theorem, it suffices to prove ( B β†’ D ) ∧ ( C β†’ D ) β†’ D . Assume B β†’ D and C β†’ D . By the Deduction Theorem, it suffices to prove D . (1) A β†’ D (assumption) (2) B β†’ D (assumption) (3) C β†’ D (assumption) (4) A ∨ B ∨ C (assumption) (5) A ∨ B β†’ D ∨ D (1), (2), (2.5.10) (6) A ∨ B β†’ D (5), axiom (7) (7) A ∨ B ∨ C β†’ D ∨ D (3), (6), (2.5.10) (8)...
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This note was uploaded on 05/09/2010 for the course LAPS CSE 1090 taught by Professor Peter during the Winter '10 term at York University.

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midterm-solW08 - Math 1090 W08 Midterm Exam Solutions(1...

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