5.3 Workshop 3 (Construction)

In this workshop, work on solving all of the following seven problems. You are encouraged to discuss the problems with those in your group but DO NOT work on writing the proofs up together.

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Your assessed problem H03 is Problem 5. You must solve this problem, write up its proof and submit it on Gradescope as part of your portfolio assessment by the deadline stated below. Your write-up of the assessed proof must be your OWN work.
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More guidance on submitting and resubmitting your proofs can be found on IMU Learn $\rightarrow$ Assessments.
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You will be expected to talk and answer questions about your solutions to the remaining problems with your group in the Communication workshop next week.
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You are free to use any of the results in the summary notes in your proofs if they are relevant to your solution.

Important Dates

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DEADLINE FOR ASSESSED PROOF H03: 10:00AM Tuesday 21st October 2025
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H03 Feedback returned: 10:00AM Monday 27th October 2025
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H03 Resubmission deadline: 10:00AM Tuesday 4th November 2025
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H03 Final deadline: 10:00AM Friday 28th November 2025
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H03 RESIT DEADLINE: 10:00AM Friday 24th April 2026

5.3.1 Problems

1.
H03 Resit 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.
  
  NOTE: To pass the H03 Resit, you must prove that your choice of $Y$ makes $f:X\to Y$ injective and surjective.
2. (b)
  
  Find a formula for $f^{-1}:Y\rightarrow X$ .
  
  Show your working.
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.
3.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function with $A\subseteq\mathbb{R}$ and $B\subseteq\mathbb{R}$ . Prove that $f(A\cup B)=f(A)\cup f(B)$ .
4.

Prove that the composition of any function with an even function is even.
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 $A\subseteq(g\circ f)^{-1}(C)$ .
7.

Prove that any finite union of countable sets is countable.