Discrete Mathematics for Computer Scientists -- Assignment 3

Discrete Mathematics for Computer Scientists -- Assignment 3 - Induction

DIT980, HT 2015

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Assignment Week 3 - Induction

Please submit using the Fire system

Assignments

(1.) Show that n³ ≤ 3ⁿ, for all n ≥ 0.

Hint: Show the cases n=0, n=1, n=2 separately first, and then prove it for n ≥ 3 by induction.

(2.) Consider the following recursively defined function F : ℕ -> ℕ:

F(0) = 1

_n-1

F(n) = ∑ F(i) , for n ≥ 1

ⁱ⁼⁰

In other words, F(n) for n ≥ 1 is defined as the sum of all previous values of F.

(a.) Show the values of F(0),F(1),F(2),F(3),F(4),F(5) and how to compute them using the definition above.

(b.) For which of the F(0),...,F(5) do we have that F(n) = 2^n-1?

(c.) Prove by induction over n that F(n) = 2^n-1 for n ≥ 1.

Submission

Submission of this assignment should be done electronically through the Fire system.

The submission deadline is Wednesday, September 23, at 13:00. At this time, you should have submitted a serious attempt to solve the assignment. A serious attempt is either an answer you believe to be correct, or a partial answer plus a detailed explanation of what you have tried to come up with a full answer. An empty document is not a serious attempt.

After submitting, you have until October 5 (midnight) to submit a completely correct version.

You can submit your answers in any of the following formats:

A simple text file, ending in .txt. You can make text files in any text editor, and then upload it to the Fire system.
(When using a text file, you may sometimes need to "invent" notation. Please be clear about what your notation means. A list of suggested notation for text files is provided by us. You may also use Unicode.)
A picture you have made in a painting program, ending in .gif, .jpg, or .png. You can use for example MS Paint, or Google Docs to make a picture, and upload it to the Fire system.
A PDF-file, ending in .pdf. You can make PDF files by for example using OpenOffice, LibreOffice, Microsoft Word, or Google Docs, and choosing export as PDF. Then, upload the PDF-file to the Fire system.
A scanned in document, ending in .pdf or .jpg. You can write your answer on a piece of paper, and use a scanner to scan it in and convert to a PDF-file. Or you can use your mobile phone to take pictures of the papers, and upload these to the Fire system.

If you submit multiple files, please name and/or number them such that the order in which we should read them is obvious. You can also write a text file, and have it refer to pictures that you upload separately.