Discrete Mathematics for Computer Scientists -- Assignment Week 1

DIT980, HT 2015

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Assignment Week 1 - Sets and Functions

Please submit using the Fire system

Assignments

(1.) Imagine two sets A and B that satisfy the following properties:

A U B = {1, 2, 4, 5, 6, 7}
A ∩ B = {2, 5}
A \ B = {1, 4, 7}

(a.) Please find two sets A and B that satisfy all of the above properties (at the same time).

(b.) Is your solution the only solution? Explain.

(2.) Find a function

f : ℤ -> ℕ

from the integers ("heltal") to the natural numbers, that is surjective but not injective.

Explain why your function is surjective. Explain why it is not injective.

Note: In a previous version of this assignment, the above sentence had a mistake, which is now fixed.

Note: This is not the same assignment as one of the exercises from week 1; this time the function takes integers ("heltal") as arguments, not natural numbers.

(3.) Take a look at the operator #, defined as follows:

# : ℤ x ℤ -> ℤ
x # y = x + 3y

(a.) Is the operator # commutative? Explain.

(b.) Is the operator # associative? Explain.

(c.) Does the operator # have an identity e? Explain.

(d.) Can you find two integers a and b that commute ("kommuterar") for #? Explain.

Submission

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

The submission deadline is Wednesday, September 9, 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 September 21 (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.