3.6 Workshop 5 (Reflection)

Task 3.21.

Before you begin your usual tasks for the Reflection workshop, we’d like you to reflect on how you’ve been finding the course so far and let us know by taking a few minutes to complete the IMU Mid-semester feedback form. You can also access the form by scanning the QR code.

This form will close at 5pm on Friday 17th November 2025.

Refer to caption — Figure 3.3: QR code for IMU Mid-semester feedback form

In this workshop, you’ll receive feedback from your tutor on the proof you submitted for this block. Your tutor will give your group general feedback on the key areas for improvement and commendation that they noticed whilst marking your group’s proofs and they will also have a one-to-one conversation with you about your specific feedback. It is important that you understand the written feedback your tutor has given you to improve, especially if your proof has not been awarded a pass yet. So make sure you take this opportunity to ask about any part of the feedback you’re unsure of.

When you’re not discussing feedback with your tutor, work with your group on the summary task below. This task has been chosen to draw on a variety of concepts you’ve been exploring throughout this block and is deliberately open ended to give you more practice at forming conjectures of your own.

Task 3.22.

Summary Task

Let $A_{1},\cdots,A_{n}$ and $B$ be sets.

1.

Use induction to prove that set intersection distributes over set unions i.e.

$\bigcup_{i=1}^{n}(A_{i}\cap B)=\left(\bigcup_{i=1}^{n}A_{i}\right)\cap B$

for every $n\in\mathbb{N}$ .
2.

Prove a similar result about the distributivity of the set union over set intersection in $\displaystyle\bigcap_{i=1}^{n}(A_{i}\cup B)$ .
3.

Does set union distribute over set difference? What about set intersection? Prove your claims.

Task 3.23 (EXTENSION).

Consider a $2^{n}\times 2^{n}$ square grid, where $n\in\mathbb{N}$ . An L-shaped tile consists of 3 squares, each of the same dimension as the squares in the grid, which form the shape of the letter L.

An $L$ -shaped tile looks like the following:

How much of the grid can be covered completely by L-shaped tiles (allowing for rotation) which do not overlap? Prove any claims you make.