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.