• Stanford Introduction To Logic 3.6

  Here are my answers to Stanford Introduction To Logic 3.6.

  Logical equivalence, logical entailment, and logical consistency are related to each other in interesting ways, but they are not identical. Answer the following true or false questions about the relationships between these concepts.

  a. If φ is equivalent to ψ, then φ entails ψ.

  If φ is equivalent to ψ, then φ can be written wherever ψ appears so:

  φ entails φ

  which (by the deduction theorem) is true if:

  φ → φ

  is valid. It is valid and so a. is true.

  b. If φ is equivalent to ψ, then φ is consistent with ψ.

  If the theory {φ, ψ} is consistent, then:

  φ ∧ ψ

  is satisfiable. If φ is equivalent to ψ, then φ can be written wherever ψ appears so:

  φ ∧ φ

  since φ ∧ φ is equivalent to φ, we can say that the theory is only consistent if φ is satisfiable. So b. is false.

  c. If φ entails ψ, then φ is equivalent to ψ.

  Disprove by counterexample. If φ entails ψ then:

  φ → ψ

  is valid. If φ is (P ∧ ¬P) and ψ = P then (φ → ψ) is valid and so (φ entails ψ) is true. But (P ∧ ¬P) is not equivalent to P, so φ is not equivalent to ψ, and so c. is false.

  d. If φ entails ψ, then φ is consistent with ψ.

  Disprove by counterexample. If φ entails ψ then (φ → ψ) is valid, and if φ is consistent with ψ then (φ ∧ ψ) is satisfiable. If φ is (P ∧ ¬P) and ψ is P then (φ → ψ) is valid, but (φ ∧ ψ) is not satisfiable. Therefore d. is false.

  e. If φ is consistent with ψ, then φ is equivalent to ψ.

  Disprove by counterexample. If φ is consistent with ψ then (φ ∧ ψ) is satisfiable. Say If φ is (P ∨ ¬P) and ψ is P, then (φ ∧ ψ) is satisfiable, but φ is not equivalent to ψ, so e. is false.

  f. If φ is consistent with ψ, then φ entails ψ.

  Disprove by counterexample. If φ is consistent with ψ then (φ ∧ ψ) is satisfiable. If φ entails ψ then (φ → ψ) is valid. But if φ is (P ∨ ¬P) and ψ is P, then (φ ∧ ψ) is satisfiable and (φ → ψ) is not valid, so f. is false.

• Stanford Introduction To Logic: 3.5

  Here are my answers to Stanford Introduction To Logic: 3.5.

  In each of the following cases, determine whether the given individual sentence is consistent with the given set of sentences.

  a. {p ∨ q, p ∨ ¬q, ¬p ∨ q} and (¬p ∨ ¬q)

  Using a truth table:

  p q p ∨ q p …

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• Stanford Introduction To Logic: 3.4

  I'm learning about logic by going through the excellent Stanford Introduction to Logic, as part of writing a maths tutorial. Here I'm working through Exercise 3.4 - Propositional Entailment. Let me know if you disagree with any of my answers:

  Let Γ and Δ be sets of sentences in Propositional …

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• Haskell wikibook: Higher-order functions

  Write insensitive, such that quickSort' insensitive dictionary gives ["a", "for", "have", "I", "Linux", "thing"].

  insensitive x y = compare (map toLower x) (map toLower y)

  Exercises
  (Challenging) The following exercise combines what you have learned about higher order functions, recursion and I/O. We are going to recreate what is known …

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• Haskell Wikibook: More on functions

  The questions for this chapter were:

  Exercises

  • Lambdas are a nice way to avoid defining unnecessary separate functions. Convert the following let- or where-bindings to lambdas:
    • map f xs where f x = x * 2 + 3

  map (\ x -> x * 2 + 3) xs

  • • let f x y = read x + y in foldr …

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• Haskell Wikibook: Control Structures

  Well I can see that I haven't done any learning of Haskell for ages. Now that I look back at it, it all looks like gobbledigook to me. Well I probably need to get on with doing some questions:

  Exercises

  1. Redo the "Haskell greeting" exercise in Simple input and output …

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• Haskell Wikibook: Pattern Matching

  Here are my answers to the exercises in the Haskell Wikibook: Pattern Matching chapter:

  1. Test the flawed h function above in GHCi, with arguments equal to and different from 1. Then, explain what goes wrong.

     The following code:

     k = 1
     
     h :: Int -> Bool
     h k = True
     h _ = False
     

     won't …

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• Haskell Wikibook: Type declarations

  Here are my answers to the Haskell Wikibook: Type declarations chapter questions:

  Reread the function definitions above. Then look closer at the showDate helper function. We said it was provided "for the sake of code clarity", but there is a certain clumsiness in the way it is used. You have …

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• Haskell Wikibook: Lists III

  Reading the Haskell Wikibook, Lists III chapter, I can't get my head round the idea that since Haskell has lazy evaluation, a foldr can be used on infinite lists. Here's foldr:

  foldr            :: (a -> b -> b) -> b -> [a] -> b
  foldr f acc []     = acc
  foldr f acc (x:xs) = f x (foldr …

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• Haskell Wikibook: Lists II

  Here are my answers to the Haskell Wikibook: Lists II questions:

  Write the following functions and test them out. Don't forget the type signatures.

  1. takeInt returns the first n items in a list. So, takeInt 4 [11,21,31,41,51,61] returns [11,21,31,41].

     takeInt :: Integer -> [Integer …

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