620-295 Real Analysis with applications

Assignment 1: Due 7 August 2009

Lecturer: Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 25 July 2009

1. Define the following sets and give examples of elements of each:

   (a)   the set of rational numbers,

   (b)   the set of real numbers,

   (c)   the set of complex numbers.

2. Let $\frac{a}{b},\frac{c}{d},\frac{e}{f}\in \mathbb{Q}$. Show that $\frac{a}{b}+\left(\frac{c}{d}+\frac{e}{f}\right)=\left(\frac{a}{b}+\frac{c}{d}\right)+\frac{e}{f}$.
   
   
3. State and prove the Pythagorean Theorem.
   
   
4. Compute and graph the following:

   (a)   $\frac{-15+i}{4+2i}$,

   (b)   ${\left({27}^{\mathrm{1/3}}\right)}^4$,

   (c)   ${27}^{(4+\mathrm{1/3})}$.

5. Let $z=x+iy$ with $x,y\in ℝ$. Compute and graph ${\displaystyle \left|\frac{(3+4i)(-1+2i)}{(-1-i)(3-i)}\right|}$.
6. Define the following and give examples:

   (a)   injective,

   (b)   surjective,

   (c)   composition of functions,

   (d)   abelian group.

   
   
7. Let $D:\mathbb{Q}\left[x\right]⟶\mathbb{Q}\left[x\right]$ be a function such that

   (a) If $f,g\in \mathbb{Q}\left[x\right]$ then $D(f+g)=D\left(f\right)+D\left(g\right)$

   (b) If $c\in \mathbb{Q}$ and $f\in \mathbb{Q}\left[x\right]$ then $D\left(cf\right)=cD\left(f\right)$,

   (c) If $f,g\in \mathbb{Q}\left[x\right]$ then $D\left(fg\right)=fD\left(g\right)+D\left(f\right)g$,   and

   (d) $D\left(x\right)=1$.

   Compute $D\left(x^n\right)$, for $n\in {\mathbb{Z}}_{\ge 0}$.
   
   
8. Write $\frac{1-x^n}{1-x}$ as an element of $\mathbb{Q}\left[x\right]$.