Problem Set - Cardinality

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

Cardinality

	Define the following and give an example for each: (a)   cardinality, (b)   finite, (c)   infinite, (d)   countable, (e)   uncountable.
	Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left({\mathbb{Z}}_{\ge 0}\right)$.
	Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left(\mathbb{Z}\right)$.
	Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)=\mathrm{Card}\left(\mathbb{Q}\right)$.
	Show that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)\ne \mathrm{Card}\left(ℝ\right)$.
	Show that $\mathrm{Card}\left(ℂ\right)=\mathrm{Card}\left(ℝ\right)$.
	Let $S$ be a set. Show that $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(S\right)$.
	Show that if $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$ then $\mathrm{Card}\left(T\right)=\mathrm{Card}\left(S\right)$.
	Show that if $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$ and $\mathrm{Card}\left(T\right)=\mathrm{Card}\left(U\right)$ then $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(U\right)$.
	Define $\mathrm{Card}\left(S\right)\le \mathrm{Card}\left(T\right)$ if there exists an injective function $f:S\to T$. Show that if $\mathrm{Card}\left(S\right)\le \mathrm{Card}\left(T\right)$ and $\mathrm{Card}\left(T\right)\le \mathrm{Card}\left(S\right)$ then $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$.

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)