5.4 Workshop 4 (Communication)

In this workshop, you’ll practice verbally sharing your proofs to the problems you worked on in the Construction workshop with those in your group.

NOTE: This forms some of your preparation for the dialogic assessment so it is very important that you engage in this workshop.

Task 5.19 (10 min).

Warm Up

1.

Allocate a number to each of the remaining non-assessed problems from the Construction workshop for this block.
2.

Create a random list (using e.g. https://www.random.org/lists/) to allocate each member of your group a random number from 1 to 6 inclusive. No number should be allocated to more than one person. The number you get indicates the problem whose proof you’ll be explaining.

NOTE: If you have fewer than 6 people in your group, make sure you allocate ALL 6 proofs amongst those in your group. This will mean some people will be talking about more than one proof.
3.
For each person in your group, allocate another person to be their “Encourager”. When a speaker is talking about their proof, their encourager must:
- •
  
  highlight at least one positive aspect of the speaker’s proof/explanation.
- •
  
  ask at least one question.
- •
  
  invite questions from the other listeners.
- •
  
  keep an eye on time. As a group, you should spend approx. 15 mins on each proof before moving onto the next speaker.
4.

Remind yourselves of the “Guidance for the Communication Workshops” on the IMU Learn course.
5.

Use this time to familiarise yourself with the proof you’ll be explaining to the group. What are the key steps? How will you structure your explanation?
6.

If there’s time, practice explaining your proof to the person next to you.

Task 5.20 (90 min).

Communication Time!

1.

Decide amongst yourselves which order each of you will explain your proofs in.
2.

Take it in turns to talk about your proofs.
3.

If you’ve run out of questions to ask the speaker before the 15 mins is up, check through the suggested questions - have you asked at least one from each category? If so, ask your tutor for some questions about this proof.

Suggestions for Questions to ask

•
Questions regarding the mathematical objects that appear in the proof and their definitions
- –
  
  Can you remind me of the definition of…?
- –
  
  What does that theorem state?
- –
  
  Can you explain why that theorem can be used in this context?
•
Questions regarding the logical links between parts of the proof
- –
  
  Can you explain why you can assume that?
- –
  
  Can you give a bit more explanation about why that step follows?
•
Questions regarding the key ideas of the proof
- –
  
  Where in your proof do you use this property/assumption?
- –
  
  What are the key steps in this proof?
•
Questions regarding the proof and its links to the main ideas seen in the course.
- –
  
  Are there other proofs in the course that use a similar proof technique/idea? Can you briefly describe them?
- –
  
  If we changed this [object, assumption, conclusion], how would your proof change? Would the statement still be valid?

Extending Questions

1.
Let $X=\mathbb{R}\setminus\{4\}$ , and let $f:X\rightarrow\mathbb{R}$ be defined by

$f(x)=\frac{5x}{x-4}.$
1. (a)
  
  Find a non-empty subset $Y\subseteq\mathbb{R}$ so that $f:X\rightarrow Y$ is both injective and surjective.
2. (b)
  
  Find a formula for $f^{-1}:Y\rightarrow X$ .
1. (i)
  
  Is the original function $f:X\to\mathbb{R}$ bijective? Can you prove your claim?
2. (ii)
  
  Are there any other subsets of $\mathbb{R}$ that $Y$ could be?
2.
Define $f:\mathbb{R}\rightarrow\mathcal{P(\mathbb{R})}$ by the formula

$f(x):=\{y\in\mathbb{R}|y^{2}<x\}.$

(The co-domain $\mathcal{P(\mathbb{R})}$ is the power set of $\mathbb{R}$ .)
1. (a)
  
  Find $f(3)$ .
2. (b)
  
  Is $f$ injective? Is $f$ surjective? Prove your claims.
1. (i)
  
  Can you make $f$ injective or surjective just by changing the domain and/or codomain?
3.
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function with $A\subseteq\mathbb{R}$ and $B\subseteq\mathbb{R}$ . Prove that $f(A\cup B)=f(A)\cup f(B)$ .
1. (a)
  
  Is this statement true if you replace the union with the intersection? If not, is there an additional requirement you can place on $f$ that would make $f(A\cap B)=f(A)\cap f(B)$ ?
2. (b)
  
  Is it true that $f^{-1}(f(A\cup B))=f^{-1}(f(A))\cup f^{-1}(f(B))$ ? What if $f$ is not bijective?
4.

Prove that the composition of any function with an even function is even.
1. (a)
  
  Is this claim true if we replace “even” with “odd”?
2. (b)
  
  What if we replace “even” with “periodic”? Is the claim still true? If so, what period does the composition have?
5.

H03: Let $f:A\to B$ , $g:B\to C$ be functions. Prove that if $g\circ f:A\to C$ is bijective then $f$ is injective and $g$ is surjective.
6.
Let $f:X\to Y$ and $g:Y\to Z$ be functions, where $X$ , $Y$ and $Z$ are sets and $A\subseteq X$ and $C\subseteq Z$ with $f(A)\subseteq g^{-1}(C)$ . Prove that $A\subseteq(g\circ f)^{-1}(C)$ .
1. (a)
  
  Is $A=(g\circ f)^{-1}(C)$ ? If not, what additional assumption do we need to make this equality hold?
7.
Prove that any finite union of countable sets is countable.
1. (a)
  
  Can you formally define the bijections you’ve described in your proof?
2. (b)
  
  Can we do the induction with only one base case $P(1)$ ? Should $P(1)$ even be in the induction proof?