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.
modelling computing systems faron moller georg struth
logic for computer science
computer technology logic
Connected book
Book Title:
Author(s):
Edition:
ISBN:
Edition:
More summaries for
Samenvatting alle stof logica voor informatica
Samenvatting Logica voor Informatica (INFOB1LI)
Samenvatting Logica voor informatica
All for this textbook (14)
Written for
Universiteit Utrecht (UU)
Informatica
Logica (INFOB1LI)
All documents for this subject (6)
1
review
By: bobkreugel • 3 year ago
Seller
Follow
luukvaa
Reviews received
Content preview
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 :
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller luukvaa. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.22. You're not tied to anything after your purchase.
