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 27

Let $S$ be a set.

A relation on $S$ is a subset $\mathrm{\Gamma }$ of $S\times S\text{.}$

$$\begin{array}{ccc}{\displaystyle S}& =& {\displaystyle \{\alpha ,\beta ,\gamma \},}\\ {\displaystyle S\times S}& =& {\displaystyle \left\{\genfrac{}{}{0ex}{}{(\alpha ,\alpha ),(\alpha ,\beta ),(\alpha ,\gamma )\hfill }{{\displaystyle \genfrac{}{}{0ex}{}{{\displaystyle (\beta ,\alpha ),(\beta ,\beta ),(\beta ,\gamma )}\hfill }{{\displaystyle (\gamma ,\alpha ),(\gamma ,\beta ),(\gamma ,\gamma )}}}}\right\},}\\ {\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \{(\beta ,\beta ),(\beta ,\gamma ),(\gamma ,\alpha )\}\text{.}}\end{array}$$

A partial order on $S$ is a relation $\mathrm{\Gamma }$ on $S$ such that

(a)	if $x,y,z\in S$ and $(x,y)\in \mathrm{\Gamma }$ and $(y,z)\in \mathrm{\Gamma }$ then $(x,z)\in \mathrm{\Gamma },$
(b)	if $x,y\in S$ and $(x,y)\in \mathrm{\Gamma }$ and $(y,x)\in \mathrm{\Gamma }$ then $x=y\text{.}$

A total order on $S$ is a relation $\mathrm{\Gamma }$ on $S$ such that

(a)	if $x,y,z\in S$ and $(x,y)\in \mathrm{\Gamma }$ and $(y,z)\in \mathrm{\Gamma }$ then $(x,z)\in \mathrm{\Gamma },$
(b)	if $x,y\in S$ and $(x,y)\in \mathrm{\Gamma }$ and $(y,x)\in \mathrm{\Gamma }$ then $x=y,$
(c)	if $x,y\in S$ then $(x,y)\in \mathrm{\Gamma }$ or $(y,x)\in \mathrm{\Gamma }\text{.}$

If $\mathrm{\Gamma }$ is a partial order on $S$ write $x\le y$ if $(x,y)$ in $\mathrm{\Gamma }\text{.}$

(a partial order that is not a total order.) Let $E=\{\alpha ,\beta ,\gamma \}$ and let $$S=\left\{\text{subsets of}\hspace{0.17em}E\right\}=\left\{\genfrac{}{}{0ex}{}{\varnothing ,\left\{\alpha \right\},\left\{\beta \right\},\left\{\gamma \right\}}{{\displaystyle \genfrac{}{}{0ex}{}{{\displaystyle \{\alpha ,\beta \},\{\alpha ,\gamma \},\{\beta ,\gamma \}}}{{\displaystyle \{\alpha ,\beta ,\gamma \}}}}}\right\}\text{.}$$ Let $\mathrm{\Gamma }$ be the relation on $S$ given by $$\mathrm{\Gamma }=\left\{(A,B)\hspace{0.17em}\right|\hspace{0.17em}A\subseteq B\}\text{.}$$ In other words, inclusion is a partial order on $S$ since

(a)	if $A,B,C\in S$ and $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C,$ and
(b)	if $A,B,\in S$ and $A\subseteq B$ and $B\subseteq A$ then $A=B\text{.}$

$$\begin{array}{c} {α,β,γ} {α,β} {α,γ} {β,γ} {α} {β} {γ} ∅ \end{array}$$ Inclusion is not a total order since $X=\left\{\alpha \right\}$ and $Y=\left\{\beta \right\}$ are in $S$ and $X⊈Y$ and $Y⊈X\text{.}$

Let $S$ be a set with a partial order $\mathrm{\Gamma }\text{.}$ Write $x\le y$ if $(x,y)\in \mathrm{\Gamma }\text{.}$

Let $A$ be a subset of $S\text{.}$

An upper bound of $A$ is an element $b\in S$ such that if $a\in A$ then $a\le b\text{.}$

A lower bound of $A$ is an element $\ell \in S$ such that if $a\in A$ then $\ell \le a\text{.}$

A maximum of $A$ is an element $M\in A$ such that there does not exist $a\in A$ such that $a\ge M$ (i.e. if $A\in A$ then $M\nleqq a\text{).}$

A minimum of $A$ is an element $m\in A$ such that if $a\in A$ then $m\ngeqq a\text{.}$

A supremum of $A$ is an element $s\in S$ such that

(a)	$s$ is an upper bound of $A,$ and
(b)	if $b$ is an upper bound of $A$ then $b\ge s\text{.}$

An infimum of $A$ is an element $i\in S$ such that

(a)	$i$ is a lower bound of $A,$ and
(b)	if $\ell$ is a lower bound of $A$ then $\ell \le i\text{.}$

If $A=\{\{\alpha ,\beta \},\{\beta ,\gamma \},\left\{\beta \right\}\}$ then $\{\alpha ,\beta \}$ and $\{\beta ,\gamma \}$ are both maximums of $A$ and $\text{sup}\hspace{0.17em}A=\{\alpha ,\beta ,\gamma \}\text{.}$

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

		Proof.
	Proof by contradiction. Assume $$\begin{array}{cccc}f:& {\mathbb{Z}}_{>0}& ⟶& {(0,1]}_{ℝ}\\ & k& ⟼& r_k\end{array}$$ is a bijection $$\begin{array}{ccc}{\displaystyle r_1}& =& {\displaystyle 0\text{.}{r}_{11}{r}_{12}{r}_{13}{r}_{14}\dots }\\ {\displaystyle r_2}& =& {\displaystyle 0\text{.}{r}_{21}{r}_{22}{r}_{23}{r}_{24}\dots }\\ {\displaystyle r_3}& =& {\displaystyle 0\text{.}{r}_{31}{r}_{32}{r}_{33}{r}_{34}\dots }\\ {\displaystyle r_4}& =& {\displaystyle 0\text{.}{r}_{41}{r}_{42}{r}_{43}{r}_{44}\dots }\\ & & {\displaystyle ⋮}\end{array}$$ Let $s=0\text{.}{s}_1{s}_2{s}_3{s}_4{s}_5{s}_6$ with $$s_1\ne r_{11},s_2\ne r_{22},s_3\ne r_{33},\dots$$ Then $s\in {(0,1]}_{ℝ}$ and does not appear in the sequence $(r_1,r_2,r_3,\dots )\text{.}$ So $f$ is not surjective. This is a contradiction to $f$ being bijective. So $\text{Card}\left({\mathbb{Z}}_{>0}\right)\ne \text{Card}\left({(0,1]}_{ℝ}\right)\text{.}$ $\square$

Let $S$ be a set and let $\mathrm{\Gamma }$ be a partial order on $S\text{.}$ Write $$x\le y\text{if}(x,y)\in \mathrm{\Gamma }\text{.}$$ Let $a,b\in S\text{.}$ Then let $$\begin{array}{ccc}{\displaystyle [a,b]}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a\le x\le b\},}\\ {\displaystyle (a,b]}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a<x\hspace{0.17em}\text{and}\hspace{0.17em}x\le b\},}\\ {\displaystyle [a,b)}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a\le x\hspace{0.17em}\text{and}\hspace{0.17em}x<b\},}\\ {\displaystyle (a,b)}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a<x\hspace{0.17em}\text{and}\hspace{0.17em}x<b\},}\\ {\displaystyle [a,\infty )}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a\le x\},}\\ {\displaystyle (a,\infty )}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}a<x\}\text{.}}\\ {\displaystyle (-\infty ,a]}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}x\le a\},}\\ {\displaystyle (-\infty ,a)}& =& {\displaystyle \{x\in S\hspace{0.17em}|\hspace{0.17em}x<a\}}\end{array}$$ where $x<y$ means $x\le y$ and $x\ne y\text{.}$ These sets are intervals in $S\text{.}$

Notes and References

These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100512Lect27.pdf and was given on 12 May 2010.

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