### Concepts of Mathematics (21-127) — Feedback on Homework 9

Homework 9 was due on Wednesday 9th April 2014. I graded Q2,5 and the grader graded Q1,3,4. All questions are marked out of 6.

**Question 1.** Most errors in this question arose from people giving sloppy proofs of surjectivity. For (a) surjectivity means that for any $n \in \mathbb{Z}$ the equation $4x+5y=n$ has a solution; this follows from Bézout's identity since $4$ and $5$ are relatively prime. For (b) you had to write every integer $n \in \mathbb{Z}$ in the form $x^2-y$ for some $x,y \in \mathbb{N}$; the idea here is to let $y$ be such that $n+y$ is a perfect square, and let $x$ be its square root; then $x^2=y=n$.

**Question 2.** Some people asserted $2^{x-1}(2y-1)=2^{x'-1}(2y'-1) \Rightarrow x=x',\ y=y'$ either without justification or with odd made-up rules. This follows from the fundamental theorem of arithmetic: since $2y-1$ and $2y'-1$ are odd, no $2$s appear in their prime factorisations, meaning that $x-1=x'-1$ (by FTA) and hence $x=x'$. Dividing through by $2^x$ then yields $2y-1=2y'-1$, and hence $y=y'$. You could also have split into cases $x>x'$ and $x

**Question 3.** This was mostly done well. Please avoid sloppy wording and stick to proper terminology and definitions, none of this 'hits everything' stuff. I may say it out loud in recitation, but I try to avoid writing it in proofs!

**Question 4.** By far the biggest error here was only doing half of proving inverseness. A function $H$ is an inverse for a function $h$ if and only if **both** $H \circ h = \mathrm{id}$ **and** $h \circ H = \mathrm{id}$. Many people only showed one of these.

**Question 5.** This question was mostly done well. The hard part was juggling the definitions of inverse and preimage at the same time. Correct solutions were usually very short.

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