7.5 Workshop 5 (Reflection)

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.

Definition 7.19 (Coarser and Finer Relations).

Suppose $\sim$ and $\approx$ are two equivalence relations on the same set $S$ .

If $a\sim b$ implies $a\approx b$ for all $a,b\in S,$ then $\approx$ is said to be a coarser relation than $\sim$ , and $\sim$ is a finer relation than $\approx$ .

Task 7.20.

Summary Task

1.

Suppose $\sim$ and $\approx$ are two equivalence relations on the same set $S$ . If every equivalence class of $\sim$ is a subset of an equivalence class of $\approx$ , is $\sim$ finer or coarser than $\approx$ ? Prove your claim.
2.

Suppose that $\sim$ and $\approx$ are two equivalence relations on the same set $S$ where $\sim$ is finer than $\approx$ . Let $P_{\sim}$ and $P_{\approx}$ be the partitions corresponding to $\sim$ and $\approx$ respectively.

What can we conclude about $P_{\sim}$ and $P_{\approx}$ ? Prove any conjectures you make.
3.

We’ve seen that the relation

$\{(x,x):x\in S\}$

of equality is an equivalence relation on any set $S$ . Does there exist a relation on $S$ that is finer than equality? Prove your claim.
4.

Let the universal relation $U$ on a set $X$ be given by $xUy$ if $x\in X$ and $y\in X$ . Does there exist a coarser relation on $X$ ? Prove your claim.
5.

Recall that congruence $\pmod{m}$ is an equivalence relation on $\mathbb{Z}$ . When is congruence $\pmod{m}$ a finer equivalence relation than congruence $\pmod{n}$ ?