Assignment 1 - Sets and Functions
Please submit using the Fire system
Question 1
Here are three properties that two sets A and B may fulfill:
- \(A \cup B = \{1, 2, 4, 5, 6, 7\}\)
- \(A \cap B = \{2, 5\}\)
- \(A \setminus B = \{1, 4, 7\}\)
(a.) 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.
Question 2
Find a function
\[ f : \mathbb{Z} \rightarrow \mathbb{N} \]
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: 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.
Question 3
Take a look at the operator \(\star\), defined as follows:
\[ \star : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\\ x \star y = x + 3y \]
(a.) Is the operator \(\star\) commutative? Explain.
(b.) Is the operator \(\star\) associative? Explain.
(c.) Does the operator \(\star\) have an identity \(e\)? Explain.
(d.) Can you find two integers a and b that commute ("kommuterar") for \(\star\)? Explain.
Submission
Submission of this assignment should be done electronically through the Fire system.
The submission deadline is Wed, Sep 11, 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 Mon, Sep 23, at midnight to submit a completely correct version.
You can submit your answers (in Swedish or English) 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 a camera or 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.