620-295 Real Analysis with Applications

Assignment 2: Due 5pm on 21 August

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

Due 5pm on 21 August in the appropriate assignment box on the ground floor of Richard Berry.

1. Let $f:S\to T$ be a function. Show that the inverse function to $f$ exists if and only if $f$ is bijective.
   
   
2. Add up the positive integers from 1 to 100. Then add up the squares $1^2$ to ${100}^2$.
   
   
3. Let $S$ be a set with an associative operation with identity. Show that the identity is unique. (This tells us that any commutative monoid has only one heart.)
   
   
4. Let $S$ be a set with an associative operation with identity. Let $s\in S$ and assume that $s$ has an inverse in $S$. Show that the inverse of $s$ is unique. (This tells us that any element of an abelian group has only one mate.)
   
   
5. Let $S$ be a ring. Show that if $s\in S$ then $s\cdot 0=0$.
   
   
6. Prove that ${\displaystyle \sum _{k=1}^n{k}^2}=\frac{1}{6}n(n+1)(2n+1)$.
   
   
7. Define the following and give an example for each:

   (a)   order,

   (b)   maximum,

   (c)   minimum,

   (d)   upper bound,

   (e)   lower bound,

   (f)   bounded above,

   (g)   bounded below,

   (j)   supremum,

   (k)   infimum,

   
   
8. Prove that if $n\in {\mathbb{Z}}_{>0}$ then $x-y$ is a factor of $x^n-y^n$.
   
   
9. For each of the following subsets of $ℝ$ find the maximum, the minimum, an upper bound, a lower bound, the supremum, and the infimum:

   (a)   $\{2^{-m}-3^n|m,n\in {\mathbb{Z}}_{\ge 0}\}$,

   (b)   $\{x\in ℝ|x^3-4x<0\}$,

   (c)   $\{1+x^2|x\in ℝ\}$,

   
   
10. What is the triangle inequality and how do you justify it?