Homework1 - is in X . if and are in X , then is in X . if...

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Homework 1 PHIL 151 Due Wed. Jan 12th, in class at 10am 1. (5 points) Prove by induction that Σ n i =0 2 i = 2 n +1 - 1 2. (5 points) Write down explicit definitions of the functions f and g on the natural numbers, where (a) f is defined recursively by f (0) = 1 ,f ( n + 1) = 2 f ( n ). (b) g is defined recursively by g (0) = 1 ,g ( n + 1) = ( n + 1) g ( n ). 3. (8 points) Prove by induction that p 1 ( ∧∨ ( is not a member of PROP . 4. Read the definition for a formation sequence on page 9 of Van Dalen (definition 1.1.4). Then do (a) through (d). (a) (3 points) Give formation sequences for p 1 ( ¬ p 4 ( p 3 p 9 )) and ( p 2 ↔ ¬ p 5 ) ( p 6 ( p 3 p 2 )) (b) (2 points) Is ( p 1 ,p 2 ,p 3 ,p 1 p 3 ,p 4 ,p 5 ) a formation sequence for p 5 ? (c) (8 points) Prove by induction that if s is a formation sequence for ϕ , then every subformula of ϕ appears in s . (d) (8 points) Prove by induction that for every ϕ there is a formation sequence for ϕ that contains only subformulas of ϕ . 5. Define a set PROP* as the smallest set X such that: p i is in X for every i and is in X . if ϕ is in X , then ¬ ϕ is in X . if ϕ and ψ are in X , then ϕψ
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Unformatted text preview: is in X . if and are in X , then is in X . if and are in X , then is in X . if and are in X , then is in X . 1 PROP* can be understood to contain the propositional formulas of PROP written in a dierent notation. In this notation the connectives come in front of arguments, instead of in between them. For example, one writes p 1 p 2 instead of p 1 p 2 . Notice that with this notation no parentheses are used. (a) (2 points) Convert p 1 p 4 p 7 p 9 to a regular propositional formula in PROP . (b) (2 points) Convert (( p 1 p 4 ) (( p 7 ) p 9 )) to a formula in PROP* . (c) (6 points) Dene a function recursively that maps the proposi-tional formulas of PROP* to PROP . (d) (6 points) Dene a function recursively that maps the proposi-tional formulas of PROP to PROP* . 2...
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Homework1 - is in X . if and are in X , then is in X . if...

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