Math+150++F09-+Hwk+4+Solutions

Math+150++F09-+Hwk+4+Solutions - SOLUTIONS TO HOMEWORK 4...

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SOLUTIONS TO HOMEWORK 4 MATH 150, FALL 09 Problem 1. Section 2.1/ Exercise 1 Part (a): x ( Nx 0 < x ) Part (b): x ( Nx ( Ix I 0)) Part (c): x ( Nx → ¬ ( x < 0)) Part (d): x ( Nx ( ¬ Ix ( y ( Ny ( y < x Iy )) Ix ))) Part (e): x ( Nx → ¬∀ y ( Ny y < x )) Part (f): x ( Nx → ¬∀ y ( Ny → ¬ ( y < x ))) Problem 2. Section 2.1/ Exercise 3 ¬∀ x ( a x ) ∧ ¬∀ x ( b x ) Problem 3. Section 2.1/ Exercise 5 Part (a): There are two possible interpretations and respective transla- tions: (1) “There is a person, which can be fooled at all times” x ( Px ∧ ∀ y ( Ty Fxy )) iff ¬∀ x ( ¬ ( Px ∧ ∀ y ( Ty Fxy )) iff ¬∀ x ( Px → ¬∀ y ( Ty Fxy )) (2) “At all times you can find someone to fool” y ( Ty → ∃ x ( Px Fxy )) iff y ( Ty → ¬∀ x ( Px → ¬ Fxy )) Part (b): There are two possible interpretations and respective transla- tions: (1) “Every person can be fooled some time” x ( Px → ∃ y ( Ty Fxy )) iff x ( Px → ¬∀ y ( Ty → ¬ Fxy ))
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