Problem Set - Orders on $\mathbb{Z},\mathbb{Q},ℝ,$ and $ℂ$

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

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Orders on $\mathbb{Z},\mathbb{Q},ℝ,$ and $ℂ$

	Define the order $\ge$ on ${\mathbb{Z}}_{>0}$.
	Define the order $\ge$ on ${\mathbb{Z}}_{\ge 0}$.
	Define the order $\ge$ on $\mathbb{Z}$.
	Define the order $\ge$ on $\mathbb{Q}$.
	Show that $\frac{a}{b}\le \frac{c}{d}$ if and only if $ab{d}^2\le cd{b}^2$.
	Define the order $\ge$ on $ℝ$.
	Show that there is no order $\ge$ on $ℂ$ such that $ℂ$ is a totally ordered field.
	Show that if $x,y,z\in ℝ$ and $x\le y$ and $y\le z$ then $x\le z$.
	Show that if $x,y\in ℝ$ and $x\le y$ and $y\le x$ then $x=y$.
	Show that if $x,y,z\in ℝ$ and $x\le y$ then $x+z\le y+z$.
	Show that if $x,y\in ℝ$ and $x\ge 0$ and $y\ge 0$ then $xy\ge 0$.
	Show that if $x\in ℝ-\left\{0\right\}$ then $x^2>0$.
	Show that if $x,y\in ℝ$ and $0<x<y$ then $y^{-1}<x^{-1}$.
	(The Archimedean property of $ℝ$) Show that if $x,y\in ℝ$ and $x\in {ℝ}_{>0}$ then there exists $n\in {\mathbb{Z}}_{\ge 0}$ such that $nx>y$.
	Show that the Archimedean property is equivalent to ${\mathbb{Z}}_{>0}$ is an unbounded subset of $ℝ$.
	($\mathbb{Q}$ is dense $ℝ$) Show that if $x,y\in ℝ$ and $x<y$ then there exists $p\in \mathbb{Q}$ such that $x<p<y$.
	($ℝ-\mathbb{Q}$ is dense $ℝ$) Show that if $x,y\in ℝ$ and $x<y$ then there exists $p\in ℝ-\mathbb{Q}$ such that $x<p<y$.
	If $x,y\in ℝ$ and $x<y$ show that there exist infinitely many rational numbers between $x$ and $y$ as well as infinitely many irrational numbers.
	Let $x\in {ℝ}_{>0}$ and $n\in {\mathbb{Z}}_{>0}$. Then there exists a unique $y\in {ℝ}_{>0}$ such that $y^n=x$.
	Find the minimal $N\in {\mathbb{Z}}_{>0}$ such that $n<2^n$ for all $n\ge N$.
	Find the minimal $N\in {\mathbb{Z}}_{>0}$ such that $n!>2^n$ for all $n\ge N$.
	Find the minimal $N\in {\mathbb{Z}}_{>0}$ such that $2^n>2n^3$ for all $n\ge N$.
	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)   $A=\{p\in \mathbb{Q}|p^2<2\}$, (b)   $B=\{p\in \mathbb{Q}|p^2>2\}$, (c)   $E_1=\{r\in \mathbb{Q}|r<0\}$, (d)   $E_2=\{r\in \mathbb{Q}|r\le 0\}$, (e)   $E=\left\{\frac{1}{n}\right|n\in {\mathbb{Z}}_{>0}\}$, (f)   $[0,1)$, (g)   ${\mathbb{Z}}_{>0}$, (h)   $\{x\in \mathbb{Q}|x\le 0\hspace{0.17em}\text{or}\hspace{0.17em}(x>0\hspace{0.17em}\text{and}\hspace{0.17em}{x}^2>2)\}$, (i)   $\mathbb{Z}$, (j)   $[\sqrt{2},2]$, (k)   $(\sqrt{2},2)$, (l)   $\{x\in ℝ|x=\frac{{\left(-1\right)}^n}{n},\hspace{0.17em}n\in {\mathbb{Z}}_{>0}\}$, (m)   $\left\{\frac{1}{{\left(\right|n|+1)}^2}\right|n\in \mathbb{Z}\}$, (n)   $\{n+\frac{1}{n}|n\in {\mathbb{Z}}_{>0}\}$, (o)   $\{2^{-m}-3^n|m,n\in {\mathbb{Z}}_{\ge 0}\}$, (p)   $\{x\in ℝ|x^3-4x<0\}$, (q)   $\{1+x^2|x\in ℝ\}$,
	Let $S$ be a nonempty subset of $ℝ$. Show that $x=\sup S$ if and only if (a)   $x$ is an upper bound of $S$, and (b)   for every $\epsilon \in {ℝ}_{>0}$ there exists $y\in S$ such that $x-\epsilon <y\le x$.
	State and prove a characterization of $\inf S$ analogous to the characterization of $\sup S$ in the previous problem.
	Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that if $S$ is bounded then $c+S=\{c+s\text{<mi>\hspace{0.17em}</mi>}|\text{<mi>\hspace{0.17em}</mi>}s\in ℝ\}$ is bounded.
	Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that if $S$ is bounded then $cS=\left\{cs\text{<mi>\hspace{0.17em}</mi>}\right|\text{<mi>\hspace{0.17em}</mi>}s\in ℝ\}$ is bounded.
	Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that $\sup (c+S)=c+\sup S$.
	Let $c\in {ℝ}_{\ge 0}$ and let $S$ be a subset of $ℝ$. Show that $\sup \left(cS\right)=c\sup S$.
	Let $c\in ℝ$ and let $S$ be a subset of $ℝ$. Show that $\inf (c+S)=c+\inf S$.
	Let $c\in {ℝ}_{\le 0}$ and let $S$ be a subset of $ℝ$. Show that $\inf \left(cS\right)=c\inf S$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)