hw3-soln - CISC 404/604 Homework 3 Solutions 1a. (( ∀ x )...

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Unformatted text preview: CISC 404/604 Homework 3 Solutions 1a. (( ∀ x ) P ( x ) ⇒ ( ∀ x ) Q ( x )) ⇒ ( ∀ x )( P ( x ) ⇒ Q ( x )) To satisfy: Q M = D M Example: D = { , 1 ,... } (henceforth referred to as N ), P M = { , 2 , 4 ,... } , and Q M = D M . To falsify: D M 6 = P M 6 = Q M Example: D = N , P M = { , 2 , 4 ,... } , and Q M = { 1 , 3 , 5 ,... } . 1b. ( ∀ x )( P ( x ) ⇒ P ( a )) To satisfy: P M = D M Example: D = N , a M = 5, P M = { , 1 ,... } . To falsify: P M 6 = ∅ , a / ∈ P M Example: D = N , a M = 5, P M = { , 2 , 4 ,... } . 1c. ( ∃ x )( ∃ y )[ P ( x,y ) ∧ ( ∀ z )( ¬ ( P ( x,z ) ∧ P ( z,y )))] To satisfy: P M = { <α,β> } where α 6 = β and <α,β> is the only member of P M Example: D = N , P M = { < 1 , 2 > } ). To falsify: P M = ∅ . 1d. ( ∀ x ) ¬ P ( x,x ) ∧ ( ∀ x )( ∀ y )( ∀ z )[( P ( x,y ) ∧ P ( y,z )) ⇒ P ( x,z )] ∧ ( ∀ x )( ∃ y ) P ( x,y ) To satisfy: (see example) Example: D = { ...,- 1 , , 1 ,... } (henceforth referred to as Z ), P M = {...
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hw3-soln - CISC 404/604 Homework 3 Solutions 1a. (( ∀ x )...

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