Problem Set - Cardinality

## Problem Set - Cardinality

 Define the following and give an example for each: (a)   cardinality, (b)   finite, (c)   infinite, (d)   countable, (e)   uncountable. Show that $\mathrm{Card}\left({ℤ}_{>0}\right)=\mathrm{Card}\left({ℤ}_{\ge 0}\right)$. Show that $\mathrm{Card}\left({ℤ}_{>0}\right)=\mathrm{Card}\left(ℤ\right)$. Show that $\mathrm{Card}\left({ℤ}_{>0}\right)=\mathrm{Card}\left(ℚ\right)$. Show that $\mathrm{Card}\left({ℤ}_{>0}\right)\ne \mathrm{Card}\left(ℝ\right)$. 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 $\mathrm{Card}\left(S\right)=\mathrm{Card}\left(T\right)$.