How to submit:

Submit before actual tutorial time for it to be graded. There are 2 ways to do this:
1. There is a submission box on Canvas for you to submit your document. Either .docx, .pdf, or a picture of your written solutions are acceptable as long as we can read your attempts.
2. Submit your written attempts in-person during our tutorial.

Official due date for submission : 04-Mar-2025, 23:59 or during tutorial itself.

Collaboration Policy:

You may discuss high-level ideas with your classmates or friends. You should list your collaborators if you do so.
Do not share your solutions.
ChatGPT (and other LLMs) are not allowed.
Your submission must be of your own write-up.

Late Policy:

< 1 week after submission deadline: 50% penalty
< 2 weeks after submission deadline: 75% penalty
No submissions after 2 weeks.

Overview

This tutorial gives practice questions to be discussed during the relevant tutorial in person. This particular tutorial sheet corresponds to Unit 2 and Unit 3. It is recommended to either watch the lectures or read the notes for each respective parts before attempting the tutorial sheet.

Questions 1 through 4 are related to set theory.
Questions 5 through 8 are related to relations.

After week 5’s content, you should be able to attempt questions 1 through 4. After week 6’s content, you should be able to attempt questions 5 through 8.

Questions 2, 4, 5, 7 are graded for participation.

That said, we encourage you to try all the questions, this way when you come for tutorial we can best make use of your time since you can either verify your solutions, or understand the discussions when our tutors go through the solutions.

Question 1

Let $A = {0}$ , and $B = P ({0, 1})$ .

Find $P (A)$ and $P (B)$ .

Question 2 [Graded for Participation]

Consider the sets $A = {x \in Z : x^{2} \leq 16}$ and $B = {x \in N : (x \geq 1) \land (\exists k \in Z [x = 3 k + 1])}$ , $C = {1, 2}$ .

Find the following sets:

$A \cap B$
$A \cup B$
$A ∖ C$
$A \times C$

Note: Some of these can be written using Set Roster Notation, these are the ones where you can list them out one by one. For the rest, you can write them using set builder notation.

Question 3

Convert the following from set builder notation to set roster notation:

$A = {x \in Z : x = 5 \lor x = 10}$
$B = {x \in N : (x > 1) \land (x < 10) \land e v e n (x)}$
$C = {x \in Z : \exists k \in Z [(x = 7 k) \land (x \geq 0) \land (x < 60)}$ .
$D = {x \in Z : \forall a, b \in N [(x < 10) \land (x > 1) \land ((x = ab) \to (a = 1 \land b = x) \lor (a = x \land b = 1))]}$

Hint for $D$ : It might be helpful to first think about what it’s trying to do. That might speed things up for you.

Question 4 [Graded for Participation]

Prove the set equality $(A ∖ B) ∖ C = (A ∖ C) ∖ (B \cup C)$ .

Prove the set equality $(A ∖ B) \cup C = (A \cup C) ∖ (B ∖ C)$ .

Hint: Based on logical equivalences shows a method that is helpful.

Question 5 [Graded for Participation]

Let $R$ be the following relation:

R = {(1, 2), (3, 3), (2, 1), (5, 6), (6, 7), (7, 8)}

Compute $R^{- 1}$ . Compute $R; R$ .

It might be helpful to refer to Operations on Relations.

Question 6

Let $A$ be the following set: $A = {0, 1, 2}$ . Let $R = A \times A$ .