Homework1

# Homework1 - is in X • if ϕ and ψ are in X then ∨ ϕψ...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: is in X . • if ϕ and ψ are in X , then ∨ ϕψ is in X . • if ϕ and ψ are in X , then → ϕψ is in X . • if ϕ and ψ are in X , then ↔ ϕψ is in X . 1 PROP* can be understood to contain the propositional formulas of PROP written in a diﬀerent 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) Deﬁne a function recursively that maps the proposi-tional formulas of PROP* to PROP . (d) (6 points) Deﬁne a function recursively that maps the proposi-tional formulas of PROP to PROP* . 2...
View Full Document

## This document was uploaded on 02/08/2011.

### Page1 / 2

Homework1 - is in X • if ϕ and ψ are in X then ∨ ϕψ...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online