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Summary Modelling Computing Systems Chapter 9 Faron Moller & Georg Struth

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  • Modelling Computing Systems

Logic for Computer Science/Logic for Computer Technology Chapter 9 Summary of the book Modelling Computing Systems written by Faron Moller and Georg Struth. Summary written in English. Using examples and pictures, the substance and theory are clarified. Given at Utrecht University.

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  • No
  • Hoofdstuk 9
  • December 22, 2020
  • December 22, 2020
  • 5
  • 2020/2021
  • Summary

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  • university u
  • modelling computing systems
  • modelling computing systems chapter 9
  • modelling computing systems faron moller georg struth
  • logic for computer science
  • computer technology logic

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  • Universiteit Utrecht (UU)
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  • Logica (INFOB1LI)
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Hoorcollege 10(Hoofdstuk 9):

We can then define a functions over N by induction. For example, we may want to compute the sum
of the first n numbers: 1 + 2 + 3 + … + n. We can do so using an inductive definition:

sum(0) = 0

sum(n + 1) = (n + 1) + sum(n).



Claim: For all n, we can show that sum(n) = n×(n+1) 2 . How to prove this? Let’s check that the
equality holds for the first few numbers:

 if n = 0, we have that sum(0) = 0 = (0×1) / 2 .
 if n = 1, we have that sum(1) = 0 + 1 = 1 = (1×2) / 2 .
 if n = 2, we have that sum(2) = 0 + 1 + 2 = 3 = (2×3) / 2 . But we need proof.

Proof by induction:

We defined the set of natural numbers using the following two clauses:

 0∈N
 for any n ∈ N, the number (n + 1) ∈ N.

To show that some property P holds for all natural numbers, it suffices to show:

 P(0)
 for all n, if we assume that P(n) we need to show that P(n + 1)



Example proof by induction where we will proof the base case and inductive case as well:

Claim: For all n, we can show that sum(n) = n×(n+1) 2 . Proof: We prove this statement by induction
on n.

 if n = 0, we need to show that sum(0) = (0×1) / 2 .
 Suppose that n = k + 1 and that sum(k) = (k×(k+1)) / 2 .

We need to show sum(k + 1) = (k+1)(k+2) / 2 .

Base Case proof: If n = 0, we need to show that sum(0) = (0×1) / 2 . Using the definition of sum, we
know that sum(0) = 0 = (0×1) / 2 as required. This completes the base case.

Inductive case proof: Suppose that that sum(k) = (k×(k+1)) / 2 . We need to show sum(k + 1) = ((k+1)
(k+2)) / 2 :

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