## Taster Sessions Update June 6 2021

Hi everyone,

Slides from all 6 of our recent Maths Taster Sessions are now available from the Taster Sessions web page, https://tinyurl.com/uonmathstaster
Meanwhile, we are continuing to post videos from these sessions on that page and on the School of Mathematical Sciences YouTube channel, https://tinyurl.com/uonmathsyt
(It takes me a while to correct the auto-generated captions!)

The latest video available is from Peter Neal’s Taster Lecture (11/05/21) on Probability. This video is available on YouTube at https://tinyurl.com/uonmaths110521

I am currently correcting the captions for Tom Wicks’s Taster Lecture on Problem Solving, so that should be ready soon.

## UoN Maths Taster Sessions May/June 2021

Our current series of Maths Taster Sessions at the University of Nottingham has now come to an end. We hope to be back in the autumn! But if you missed the talks, we are making slides and videos from the sessions available from our Taster Sessions web page at https://tinyurl.com/uonmathstaster, though the videos will have to wait until I have corrected the captions!

## Beyond Infinity? Several editions!

Hilbert’s Hotel has made many appearances in the literature and the media over the years. See, for example, the Wikipedia article https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel, which mentions quite a few examples in the body, and a few more in the External Links at the end. I have always enjoyed reading fiction where the Hotel turned up! However, authors haven’t always taken the opportunity to look in detail at more sets of numbers, such as the rational numbers (which do fit into Hilbert’s Hotel) and the real numbers (which don’t fit). Also, the case of (countably) infinitely many buses each with (countably) infinitely many passengers is relatively easy to handle using prime powers or prime factorization, and the story really shouldn’t stop there. In any case, I thought that this topic would make for a fun maths talk, and back in 2006 I finally got round to producing a Beamer presentation, with the title Beyond Infinity? It was a lot more work than I thought getting the “transitions” to play nicely! But I presented the talk in 2006 and 2007 at various University of Nottingham enrichment and recruitment events. I also recorded audio (15 and a half minutes), but not video.

In 2008, I combined the slides and audio using Windows Movie Maker (I have never had so many crashes!) to produce a video of the talk. At the time, 15+ minutes was too long, so I split it into Part I (https://www.youtube.com/watch?v=OdhD-cx0OHQ) and Part II (https://www.youtube.com/watch?v=BKQjL-nChv8). The slightly more advanced material about the real numbers not fitting into the hotel is in Part II. Once YouTube allowed videos to be longer, the University of Nottingham published the full video (https://www.youtube.com/watch?v=Tj6DwD6c4ro) on the official University YouTube channel.

I wasn’t completely happy with this edition. I felt that I could have said more in places, and that it might be good to have some webcam footage of me in the corner. Also, I decided that “Hotel Uncountable” wasn’t very good, and that “Hotel Continuum” would be much better. So in 2010, when I was presenting the talk again, I took the opportunity to record a new screencast together with webcam footage. The resulting video (https://www.youtube.com/watch?v=3FDXnChPVm8) was longer, at 24 and a half minutes. I think it was an improvement, but it has had fewer views. Perhaps that is because of the length? Maybe I should split it in two?

Last year I produced a new version for the University of Nottingham’s Virtually Nottingham online Open Days. My original slides did not conform to the latest University branding, but Helen Preston kindly created a new set of slides in PowerPoint using a colour scheme based on different shades of blue. Rather than recording a live presentation, this time I pre-recorded the audio to go with the PowerPoint slides. Given that I had given the talk live several times and recorded it before, I wasn’t expecting it to need many takes. But somehow pre-recording is different, and it took a lot longer than I thought before I was satisfied. You can find the resulting edition via https://www.nottingham.ac.uk/open-days/catch-up?tag=Mathematics or directly at https://www.nottingham.ac.uk/open-days/video.aspx?id=bb776f5c-c66d-4b7a-bf2f-14b2be38f6fc This is the shortest of the three editions (the talk itself is only about 12 minutes long, though it is followed by a longer Q&A session).

I still have a preference for live presentations myself, and for including webcam footage in the corner, but I’d be interested in comments if anyone can stand to watch at least some of each video!

Several of my colleagues have also presented this talk, and we have also run it as a longer interactive session (30-50 minutes) with puzzles for the pre-university students to think about, though it is probably best to refer them to the full video for the details of the last section.

## University of Nottingham Maths Taster Sessions: two announcements

This post has two parts:

1. Our next event

2. Slides and video on demand

Tags: #mathematics #maths #math #appliedmaths #university #nottingham #calculus #covid

1. Our next event

Our next event, which will be the last in our current series of Taster Sessions, will be a Taster Lecture on Applied Mathematics by Stephen Creagh, at 5PM on Thursday 3rd June 2021.

Applied mathematics is all about constructing models of the world around us, in contexts as diverse as mechanics, pandemic modelling or weather prediction, and then solving the resulting equations. Calculus is the central language of this modelling cycle. Typically, our mathematical models take the form of differential equations, in which the governing equations tell us about the rates of change (derivatives) of the quantities we are interested in.

In this taster lecture we will explore qualitative and graphical approaches to understanding the solutions of these models. These qualitative approaches often give really good insight into the problem while demanding much less effort than would be needed for complete solutions in the form of explicit formulas. For example, in modelling a pandemic, we might be much more interested in answering questions such as whether infection numbers will rise or fall, whether they will approach an equilibrium or keep changing steadily, and if rising, what parameters might be changed to make infections fall etc.

For full details of our Taster Sessions, see the page

https://tinyurl.com/uonmathstaster

To join one of our Teams Live events, simply visit that page at the right time and use the session’s Join Teams Live Event button.

2. Slides and video on demand

Slides from the first five Taster Sessions in the current series are now available (in PDF format) from the page https://tinyurl.com/uonmathstaster. Videos from our sessions will be available on demand from the same page once the captions are ready. So far you can watch videos from the first two sessions, which are the Taster Lecture on Calculus by Anna Kalogirou from 29th April 2021, and the popular maths talk Using maths in the fight against Covid-19 by Katie Severn from 6th May 2021. You can also find these videos on our YouTube channel, available at https://tinyurl.com/uonmathsyt

## University of Nottingham Maths Taster Sessions: Taster Lecture on Pure Mathematics

It’s my turn today!

Our next event will be a Taster Lecture on Pure Mathematics by Joel Feinstein, at 5PM on Wednesday 26th May 2021.

This session will give you a taste of what a university lecture on Pure Mathematics is like. In particular, we will look at the importance of definitions, proofs and examples when you want to be certain of the answers to mathematical questions. You will have a chance to think about some problems concerning prime numbers, and to vote on the possible answers. These days, prime numbers play a crucial role in keeping your online shopping and banking secure!

This talk is suitable for all students taking A-level mathematics.  In fact, anyone who is familiar with some algebra (as taught in GCSE mathematics) can understand and enjoy this talk.

After the main talk, I’ll also say something about Maths Courses at the University of Nottingham (https://tinyurl.com/mathscourseuon).

For full details of our Taster Sessions, see the page

https://tinyurl.com/uonmathstaster

where you can also sign up for our Maths Taster Sessions mailing list.

To join one of our Teams Live events, simply visit the Taster Sessions page https://tinyurl.com/uonmathstaster at the right time and use the session’s Join Teams Live Event button.

#mathematics #maths #math #puremaths #primenumber #university #nottingham

## University of Nottingham Maths Taster Sessions: three announcements

This post has three sections:

1. Our next event

2. Taster Lecture on Applied Mathematics

3. Slides and video on demand

1. Our next event

Our next event will be a Taster Lecture on Pure Mathematics by Joel Feinstein, at 5PM on Wednesday 26th May 2021.

This session will give you a taste of what a university lecture on Pure Mathematics is like. In particular, we will look at the importance of definitions, proofs and examples when you want to be certain of the answers to mathematical questions. You will have a chance to think about some problems concerning prime numbers, and to vote on the possible answers. These days, prime numbers play a crucial role in keeping your online shopping and banking secure!

This talk is suitable for all students taking A-level mathematics.  In fact, anyone who is familiar with some algebra (as taught in GCSE mathematics) can understand and enjoy this talk.

For full details of our Taster Sessions, see the page

https://tinyurl.com/uonmathstaster

To join one of our Teams Live events, simply visit the page above at the right time and use the session’s Join Teams Live Event button.

2. Taster Lecture on Applied Mathematics

In a new addition to our series, there will be a Taster Lecture on Applied Mathematics by Stephen Creagh, at 5PM on Thursday 3rd June 2021.

Applied mathematics is all about constructing models of the world around us, in contexts as diverse as mechanics, pandemic modelling or weather prediction, and then solving the resulting equations. Calculus is the central language of this modelling cycle. Typically, our mathematical models take the form of differential equations, in which the governing equations tell us about the rates of change (derivatives) of the quantities we are interested in.

In this taster lecture we will explore qualitative and graphical approaches to understanding the solutions of these models. These qualitative approaches often give really good insight into the problem while demanding much less effort than would be needed for complete solutions in the form of explicit formulas. For example, in modelling a pandemic, we might be much more interested in answering questions such as whether infection numbers will rise or fall, whether they will approach an equilibrium or keep changing steadily, and if rising, what parameters might be changed to make infections fall etc.

3. Slides and video on demand

Once we have completed minor edits, video from the sessions will be available on demand from the page https://tinyurl.com/uonmathstaster. Slides from the sessions will also be available. Currently video and slides are available from the Taster Lecture on Calculus by Anna Kalogirou from 29th April 2021.

## Core Topics in University Mathematics

This is a set of videos that some of my colleagues and I recorded back in the years 2011-2014. (This project was initiated by Steve Cox: thanks Steve!)

There are twenty videos there. They are primarily aimed at first-year maths undergraduates, but may also be of interest to pre-university maths students. They cover some of the topics which students often find hard, but which are fundamental to success in university mathematics.

The topics covered include: proof by induction; the exponential function; properties of functions (think of a function); vector spaces; proof by contradiction; curve sketching; properties of sequences and series.

Study with us: https://tinyurl.com/mathscourseuon
Maths Taster Sessions: https://tinyurl.com/uonmathstaster

## University of Nottingham Maths Taster Sessions: Taster Lecture on Problem Solving

Our next Teams Live event will be a Taster Lecture on Problem Solving by Tom Wicks, at 5PM on Wednesday 19th May 2021.

In this session, you will learn why problem solving is an essential skill for a mathematics degree and is valued highly by employers. The lecture will provide essential tips and techniques on using mathematics to solve problems effectively. The lecture will include plenty of interactive activities, where you will have the opportunity to have a go at solving some university-style problems yourself.

For full details of our Taster Sessions, see the page https://tinyurl.com/uonmathstaster

To join one of our Teams Live events, simply visit the page above at the right time and use the session’s Join Teams Live Event button.

## University of Nottingham Maths Taster Session Recordings

We are now beginning to release recordings of our Maths Taster Sessions on YouTube. These will appear in the UoN (University of Nottingham) School of Mathematical Sciences YouTube channel, https://tinyurl.com/uonmathsyt

The first talk, Anna Kalogirou’s Taster Lecture on Calculus (29/04/21) is available directly at https://tinyurl.com/uonmaths290421

Unfortunately the audio in this video was affected by global issues with Microsoft Teams at the time, but captions (subtitles) are available. (We have corrected the auto-generated captions, so they should be pretty accurate.)

## University of Nottingham Maths Taster Sessions: Taster Lecture on Probability

Our next event will be a Taster Lecture on Probability by Peter Neal, at 6PM (note later time than usual) today, Tuesday 11th May 2021.

This session will give you a taste of what a university lecture on probability is like. We will consider a range of problems and the techniques used to solve them. In particular, we will look at expectation and conditional probability, and we will see how useful these are. You will have the opportunity to test your intuition on some probability questions and to vote on the answers.

For full details of our Taster Sessions, see the page

https://tinyurl.com/uonmathstaster

To join one of our Teams Live events, simply visit the page above at the right time and use the session’s Join Teams Live Event button.

## University of Nottingham Maths Taster Sessions: Using maths in the fight against Covid-19

For more details on our Taster Sessions, see Maths Taster Sessions at the University of Nottingham and https://tinyurl.com/uonmathstaster

Our next Taster Session is a Popular maths talk by Katie Severn on Using maths in the fight against Covid-19.

Thursday 6th May 2021, 5PM BST.

Over the past year we have seen many predictions about how Covid-19 will affect us. Some of these theories have been based on pure speculation whilst others involve significant mathematical modelling. Are you curious about how these modelled predictions can be used? During this lecture you will discover some of the approaches that have been taken to model the pandemic.

You’ll learn why our ability to mathematically model epidemics is so critical for today and for future generations. There will be plenty of opportunities to test your knowledge with several scenarios to think about too. You will also find out how you can learn more about mathematical modelling at the University of Nottingham as part of a maths degree.

Join Teams Live Event

## Maths Taster Sessions at the University of Nottingham

We are running a series of free online maths Taster sessions. These will include Taster Lectures, and Popular Maths Talks.

• Taster Lectures give you a taste of what a maths lecture at the University of Nottingham is really like, often based on content that we teach our first-year students. These are mainly suitable for Year 12/13 A level maths students.
• Popular Maths Talks give you the opportunity to hear one of our university teachers talk about an exciting mathematical topic, and how it links to the maths we teach at Nottingham. This could range from how to use mathematics in the fight against Covid-19, to climate change predictions. These are still most suitable for Year 12/13 A level maths students. Younger pupils such as Year 10-11 GCSE maths students who are considering taking maths A level will also find them of interest.

For more details, see https://tinyurl.com/uonmathstaster or you can use the QR code below

The first Taster Session will be a taster lecture on Calculus on Thursday April 29 2021 at 5PM BST.

### Speaker: Anna Kalogirou

This session will give you a taste of what a university lecture on Calculus is like. In particular, it will provide an introduction to the topic of Ordinary Differential Equations (ODEs). The lecture will also cover why such equations are useful and will present examples of well-known ODEs that can be used to model real-life problems, such as the dynamics of an epidemic.

To enjoy and understand this talk fully, students should have some basic familiarity with the notion of differentiation, and with the exponential and logarithmic functions (exp and log).

Join Teams Live Event

## Microsoft Teams and audio devices

I just had a bizarre experience with Microsoft Teams, which I think I have finally figured out.

I had been in a Teams meeting for 50 minutes, when suddenly I couldn’t hear anyone speak. But I still got notification beeps whenever anyone posted in the chat, and my speakers were working fine.

I tried the obvious things, like leaving the meeting and rejoining the meeting, and restarting the laptop (a Surface Pro 6). But nothing helped. Eventually I realised that the audio settings in Teams had changed, but I didn’t recognize the device named there, Logitech something-or-other. This made no sense to me: there were no Logitech devices physically attached to my Surface. And anyway, why would Teams suddenly change its audio settings in the middle of a meeting? Still, everything worked fine again once I changed the Teams audio settings back to use the actual speakers attached to the Surface Dock.

I thought this might be one of those unsolvable IT mysteries. But since it was rather worrying, I decided to investigate a bit. My first guess was that there was some keyboard shortcut that affected Teams Settings that I might have triggered accidentally. A Google Search for that didn’t turn up what I was looking for. So … what exactly was that device again? It was a Logitech BT stereo Adapter. That really should have cleared it up for me, but maybe I’m tired! I didn’t remember ever attaching a device involving both Logitech and BT. Still, maybe looking up this device would reveal something.

Hah! A year or so ago I had bought a BlueTooth device to attach to my old HiFi system to allow me to play audio through the sort-of-OK speakers. That was the BT involved here. But why did my Surface suddenly connect to the device? And why did Teams change its settings without me doing anything? Well, here is what I think happened.

The Logitech device gets its power from a 4-way extension lead downstairs in the living room, which also has some of our decorative lights attached. Although the sockets have separate switches, it is tricky to remember which is which, and the easiest thing is to turn them all on. This causes the Logitech BlueTooth device to power up. Now this particular device has a curious feature (I remember warnings in the reviews) that disconnecting from it doesn’t work: it always tries to reconnect to any device it can find that it was previously paired with. That is usually one of our mobile phones, but I have occasionally used it with my Surface when I wanted to check the quality of an audio recording. I think I last did that in June 2020. Anyway, today someone caused the device to turn on in the middle of my meeting. It looked around and found my Surface Pro with BlueTooth enabled, and reconnected (after a 9 month gap). The surface audio settings didn’t change: I still got alert beeps through the speakers, and when I checked the audio levels they were fine. But Teams must have detected the new audio device and assumed that, since I had connected a new audio device, I must want to use that instead of my actual computer speakers. So, without telling me, Teams switched its audio output to the device downstairs (attached to the aux input of a HiFi that was on standby). That’s why I suddenly couldn’t hear anyone talking, but they could still hear me, and I still received the audible alerts whenever anyone posted in the chat.

Be warned!

I do still want to use my BlueTooth keyboard. But for now I have “removed” all BlueTooth audio devices from the Surface. I might consider reconnecting to the BlueTooth earphones, but only once I have tested to make sure that they don’t behave the same way as the Logitech BT Stereo Adapter!

## Relations – Full justification

Here is another question and answer from my first-year Pure Piazza forum this year.
Question: When a question asks whether a relation is either reflexive, symmetric or transitive and requires full justification for your answer, if for example it’s not reflexive, is it enough just to give one counter example of it not being reflexive? Does a counter example count as enough for full justification?
Hi,
Suppose that $R$ is a relation on a set $S$.
Then, in full, $R$ is reflexive if and only if the following condition (E1) holds:
(E1)      $\forall x \in S, x R x$.
To prove such a “for all” statement true, you need to prove it for all $x \in S$. Such a proof begins with “Let $x \in S$” or (if you think $S$ might be empty) “Suppose that $x \in S$.”
However, if you want to show that $R$ isn’t reflexive, then you just need one counterexample (that’s how you disprove a “for all” statement). So you just need to find one specific $x \in S$ such that $x{\mkern 3 mu\not\mkern -5 mu R\mkern 2 mu}x$. Of course, with “full justification” you have to show that your example really does have the properties you claim, though sometimes you can say (if it’s true!) that the relevant property of your specific example is “clear”.

• You may have similar questions about symmetry and transitivity? If so, let me know!

[Note: for symmetry and transitivity the negations are harder. This is because “not if” is tricky! “Does not imply”, $\not \Rightarrow$, is not too bad, but needs careful interpretation. In all cases, you are looking for a specific counterexample to disprove the relevant implications. See Lecture Engagement and some of my other posts for more on this.]
Fortunately (E1) is fairly easy to negate mechanically, to get
$\neg$(E1)    $\exists x \in S : x {\mkern 3 mu\not\mkern -5 mu R\mkern 2 mu} x$.
This is a “there exists” statement, so to prove $\neg$(E1) you only need to find one value of $x$ which works. So this time we would be looking for one specific example of $x \in S$ satisfying the condition $x{\mkern 3 mu\not\mkern -5 mu R\mkern 2 mu}x$ in order to prove that $\neg$(E1) is true. Of course this example is nothing but a counterexample to (E1).
We often say “proof by example is not valid”. But what we really mean is that you can’t (usually) prove a for all statement true by checking a few examples.
[The exception is when your set only has finitely many elements, when you really can check all the cases separately. This is essentially case by case analysis, or proof by exhaustion (it has lots of different names)]
But “proof by example” is completely correct as a way to prove a there exists statement. You only need one valid example!

It gets more complicated if you have a statement with more than one quantifier, like the true statement
$\forall n \in \mathbb{Z}, \exists m \in \mathbb{Z}: m
To prove this, you need to prove that the “for all” statement is true for all $n$, so you start with
Let $n \in \mathbb{Z}$.
(We know that $\mathbb{Z}$ isn’t empty!)
Now you have to deduce (from the fact that $n \in \mathbb{Z}$) that the “there exists” statement is true. This only needs one example, and the example is allowed to depend on $n$! In fact, here the example really does depend on $n$: the smaller $n$ is, the smaller $m$ has to be. This is like $\varepsilon$ and $\delta$ in the definition of continuity, and $\varepsilon$ and $N$ (or $n_0$) in the definition of convergence of sequences.
Fortunately, it is easy to find a suitable example of $m$ in terms of $n$. The example $m=n-1$ is perhaps the most obvious example that proves the “there exists” statement true. (This $m$ “clearly has the required properties”, namely $m \in \mathbb{Z}$ and $m.)
Now I always say that a specific example is the most convincing, and that there shouldn’t be any variables left in your answer. But when your example depends on $n$, $n-1$ is really about as specific as you can get!

One way to think about this is that $n$ is treated as a constant in the “inner statement” $(\exists m \in \mathbb{Z}: m that we are investigating. So $m=n-1$ really is just one example that proves this “there exists” statement true.

Why do I call it the “inner statement”? Because you can add in some optional brackets to the original statement if you want, to obtain:.

$\forall n \in \mathbb{Z}, (\exists m \in \mathbb{Z}: m

So this is a bit like working with a double sum (see @59). Instead of an “inner sum” you have an “inner statement” in which the outer variable is treated as constant.

This also relates back to Question 4(a) on the FPM Assessed Coursework, where (with one approach) you needed to prove the true statement
$\forall x \in \mathbb{Q}^c, \exists y \in \mathbb{Q}^c: xy \in \mathbb{Q}.$
Again you need a proof valid for all irrational numbers $x$, so you have to start with “Let $x \in \mathbb{Q}^c$“.
Then you have to investigate the “inner statement”. This is a “there exists” statement, so you need to find an example $y$ which has the desired properties and, as usual, $y$ has to depend on $x$. This isn’t as easy as the example $m=n-1$ above. But, once you realise that this is what you need to do (and you notice that $x \neq 0$, because $x$ is irrational) you may have the idea that you have to set $y=q/x$ for some rational number $q$, and, to make it as specific as possible, you should choose a specific rational number $q$. At the very least, you must specify that $q \neq 0$, because otherwise $y$ won’t be irrational! The most obvious thing to try is probably $q=1$, in which case you try $y=1/x$. Here $xy=1 \in \mathbb{Q}$ is clear, but the fact that $1/x$ is irrational is not standard, and needs a proof.

Best wishes,
Dr Feinstein

## Permutations and supports

Here are a post and a follow-up comment of mine from my first-year Pure Piazza forum, in response to a question asking about the notation $\mathrm{supp}(\sigma)$ (the support of a permutation $\sigma$).

#### My original response

The support of the permutation is the set of elements where the permutation has any effect.
Points outside the support are left where they are by the permutation.
Points in the support are all moved around (to other points in the support), and none of them stay where they were. So all the action takes place in the support.
The identity permutation has empty support.
Two permutations have disjoint support if … well, if their supports are disjoint! You can just say that such permutations are disjoint for short.
When you write a permutation as a product of disjoint cycles (including 1-cycles if you want), then the support of the whole permutation is the union of the supports of the disjoint cycles. For most cycles the support is the set of elements mentioned in the cycle, but 1-cycles have empty support! (A 1-cycle is just the identity permutation.)
For example, consider the permutation $\rho$ of the $7$-set $\{1,2,3,4,5,6,7\}$ written, as a product of disjoint cycles (including some $1$-cycles), as
$\rho=(1\,5\,7)(2\,6)(3)(4)\,.$
(The $1$-cycles are optional here and have no effect). Then
$\mathrm{supp}(\rho)= \{1,5,7\} \cup \{2,6\}$ = $\{1,2,5,6,7\}$.
This only works for products of disjoint cycles, though. For example, you can quickly check that $(1\,2) (3\,2\,1) = (1\,3)$. In this example, $2$ is actually in the support of both cycles, but not in the support of the product of these cycles.

#### Follow-up post

Here (below, after an introduction) is a nice exercise related to the above.

Introduction concerning transpositions

Note that a transposition is another name for a $2$-cycle. These just swap two different elements of the set you are working with (which must have at least two elements). Note that if you multiply a transposition by itself you get the identity permutation (the second application puts the two elements back where they started). So if $\tau$ is a transposition we can say that $\tau^2$ is the identity permutation. (There are also some other permutations with this property which are not transpositions. What are they?)
To be clear, for any permutation $\tau$, $\tau^2$ means that you multiply the permutation $\tau$ by itself, using the usual multiplication of permutations, which is the same as composition of functions. So for any permutation $\tau$, we have
$\tau^2=\tau\tau=\tau \circ \tau$.
If $\tau$ is a transposition, then $\tau^2$ is the identity permutation. (But the converse is not true, unless your set has fewer than $4$ elements and you assume that $\tau$ is not the identity permutation!)

Exercise

Let $X$ be a non-empty set. (If you want, you can assume that $X$ is finite, or even that $X=\{1,2,\dots,n\}$ for some natural number $n$ if you like, but it isn’t necessary.) Here is a fact about permutations of $X$. (Where we talk about transpositions below, we mean permutations of $X$ that are $2$-cycles.)
Let $\rho$ be a permutation of $X$ and suppose that supp$(\rho)$ is a non-empty, finite set. (So $\rho$ is not the identity permutation, but $\rho$ only moves finitely many elements.) Note here that supp$(\rho)$ must have at least two elements. (Why?)
Let $k$ be the number of elements in supp$(\rho)$.
Prove that $\rho$ is a product of some finite number of transpositions, and that it is possible to do this using strictly fewer than $k$ transpositions in the product.
(You won’t need to use the same transposition more than once, but if you do it counts more than once.)

Hint: Induction on $k$, or design an algorithm based on the following idea.
Suppose you have some bottles in front of you numbered $1$ to $10$, but they are in the “wrong” order. How can you put them into the right order (left to right, $1$ to $10$) if all you are allowed to do is swap two bottles at a time? How would you guarantee to do it with at most $9$ swaps, even if all of the bottles are in the wrong places? Can you think of several different strategies?
Warning! if you can do a particular case with exactly $8$ swaps then it turns out that you can’t do that case with exactly $9$ swaps, even if you tried completely different swaps! (But that’s another story, about “odd” and “even” permutations.)

Have a go!