Real Analysis

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

Last update: 19 July 2014

Lecture 26

Sets

A set is a collection of elements.

Write $s\in S$ if $s$ is an element of the set $S\text{.}$

Let $S$ and $T$ be sets.

$T$ is a subset of $s$ if $T$ satisfies: if $t\in T$ then $t\in S\text{.}$

$T$ is equal to $S$ if $T\subseteq S$ and $S\subseteq T\text{.}$

The intersection of $S$ and $T$ is the set $$S\cap T=\left\{x\hspace{0.17em}\right|\hspace{0.17em}x\in S\hspace{0.17em}\text{and}\hspace{0.17em}x\in T\}\text{.}$$

The union of $S$ and $T$ is the set $$S\cup T=\left\{x\hspace{0.17em}\right|\hspace{0.17em}x\in S\hspace{0.17em}\text{or}\hspace{0.17em}x\in T\}\text{.}$$

The product of $S$ and $T$ is the set $$S\times T=\left\{(s,t)\hspace{0.17em}\right|\hspace{0.17em}s\in S,t\in T\}$$ of pairs with the first entry from $S$ and the second entry from $T\text{.}$

$$\begin{array}{ccc}{\displaystyle S}& =& {\displaystyle \{1,2,3,5,6,7\}}\\ {\displaystyle T}& =& {\displaystyle \{2,3,4,6,7,8\}}\end{array}$$ Then $$\begin{array}{ccc}{\displaystyle S\cap T}& =& {\displaystyle \{2,3,6,7\}}\\ {\displaystyle S\cup T}& =& {\displaystyle \{1,2,3,4,5,6,7,8\}}\\ {\displaystyle S\times T}& =& {\displaystyle \left\{\genfrac{}{}{0ex}{}{(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),\hfill }{\genfrac{}{}{0ex}{}{{\displaystyle (2,2),(2,3),(2,4),(2,6),(2,7),(2,8),}\hfill }{{\displaystyle (3,2),(3,3),\dots }\hfill }\hfill }\right\}\text{.}}\end{array}$$

Functions

A function $f:S\to T$ is an assignment of an element $f\left(s\right)\in T$ to each $s\in S\text{.}$

A function $f:S\to T$ is injective if it satisfies: if $s_1,s_2\in S$ and $f\left(s_1\right)=f\left(s_2\right)$ then $s_1=s_2\text{.}$

A function $f:S\to T$ is surjective if it satisfies: if $t\in T$ then there exists $s\in T$ such that $f\left(s\right)=t\text{.}$

A function $f:S\to T$ is bijective if it is injective and surjective.

$$\begin{array}{c} f: S ⟶ T 1 5 2 6 3 7 4 surjective not injective f: S ⟶ T 1 5 2 6 3 7 8 injective not surjective f: S ⟶ T 1 5 2 6 3 7 4 8 not a function \end{array}$$

Cardinality

Let $S$ and $T$ be sets.

$S$ and $T$ have the same cardinality if there exists a bijective function $f:S\to T\text{.}$ Write $\text{Card}\left(S\right)=\text{Card}\left(T\right)$ if there exists a bijective function $f:S\to T\text{.}$

Let $S$ be a set.

(a)	$S$ is finite if there exists $n\in {\mathbb{Z}}_{\ge 0}$ such that $\text{Card}\left(S\right)=\text{Card}\left(\{1,2,\dots ,n\}\right)\text{.}$
(b)	$S$ is infinite if $S$ is not finite.
(c)	$S$ is countable if $S$ is finite or $$\text{Card}\left(S\right)=\text{Card}\left(\mathbb{Z}\right)\text{.}$$
(d)	$S$ is uncountable if $S$ is not countable.

Write $\text{Card}\left(S\right)=n$ if $n\in {\mathbb{Z}}_{\ge 0}$ and $$\text{Card}\left(S\right)=\text{Card}\left(\{1,2,\dots ,n\}\right)\text{.}$$

Prove that $\text{Card}\left({\mathbb{Z}}_{>0}\right)=\text{Card}\left({\mathbb{Z}}_{\ge 0}\right)\text{.}$

		Proof.
	To show: There exists a bijective function $f:{\mathbb{Z}}_{>0}\to {\mathbb{Z}}_{\ge 0}\text{.}$ Let $$\begin{array}{cccc}f:& {\mathbb{Z}}_{>0}& ⟶& {\mathbb{Z}}_{\ge 0}\\ & 1& ⟼& 0\\ & 2& ⟼& 1\\ & 3& ⟼& 2\\ & ⋮& & ⋮\end{array}$$ $\square$

Prove that $\text{Card}\left({\mathbb{Z}}_{\ge 0}\right)=\text{Card}\left(\mathbb{Z}\right)\text{.}$

		Proof.
	To show: There exists a bijective function $f:{\mathbb{Z}}_{\ge 0}\to \mathbb{Z}\text{.}$ Let $$\begin{array}{cccc}f:& {\mathbb{Z}}_{\ge 0}& ⟶& \mathbb{Z}\\ & 0& ⟼& 0\\ & 1& ⟼& 1\\ & 2& ⟼& -1\\ & 3& ⟼& 2\\ & 4& ⟼& -2\\ & 5& ⟼& 3\\ & 6& ⟼& -3\\ & ⋮& ⟼& ⋮\end{array}$$ $\square$

Let ${(0,1]}_{\mathbb{Q}}=\{x\in \mathbb{Q}\hspace{0.17em}|\hspace{0.17em}0<x\le 1\}\text{.}$ Show that $\text{Card}\left({(0,1]}_{\mathbb{Q}}\right)=\text{Card}\left({\mathbb{Z}}_{>0}\right)\text{.}$

		Proof.
	List the expressions $\frac{a}{b}$ with $a\in {\mathbb{Z}}_{>0}$ and $b\in {\mathbb{Z}}_{>0}$ in the order $$(\frac{1}{1},\frac{1}{2},\frac{2}{2},\frac{1}{3},\frac{2}{3},\frac{3}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{4}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{5}{5},\dots )\text{.}$$ Take the subsequence of this sequence of reduced expression of elements in ${(0,1]}_{\mathbb{Q}},$ $$(\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\dots )\text{.}$$ This sequence is a bijective function $$\begin{array}{cccc}f:& {\mathbb{Z}}_{>0}& ⟶& {(0,1]}_{\mathbb{Q}}\\ & 1& ⟼& \frac{1}{1}\\ & 2& ⟼& \frac{1}{2}\\ & 3& ⟼& \frac{1}{3}\\ & 4& ⟼& \frac{2}{3}\\ & 5& ⟼& \frac{1}{4}\\ & 6& ⟼& \frac{3}{4}\\ & 7& ⟼& \frac{1}{5}\\ & ⋮& & ⋮\end{array}$$ $\square$

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100510Lect26.pdf and was given on 10 May 2010.

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