
%%%%%
%\includegraphics[width=2.05in,height=1.39in]{welcome.bmp}
%\input epsf
%\epsfverbosetrue
%\def\pict#1#2#3{\epsfxsize=#1in \epsfysize=#2in \epsffile{#3.eps}}
\documentclass[12pt]{article}
\usepackage{amssymb,amsmath,amscd}
%\usepackage{graphicx}
\setlength{\oddsidemargin}{-.5in}
\setlength{\evensidemargin}{-.5in}
\setlength{\textwidth}{7.25in}
\topmargin  -.7in
\textheight 9.45in
\newcommand{\bigfrac}[2]{\frac{\textstyle #1}{\textstyle #2}}
\newcommand{\starttest}{\begin{enumerate}}
\newcommand{\nextpage}{\vspace*{\fill}\newpage}

\newcommand{\mat}[2]{\left( \begin{array}{*{#1}{c}}#2\end{array}\right)}
\newcommand{\matr}[2]{\left( \begin{array}{*{#1}{r}}#2\end{array}\right)}
\newcommand{\dat}[2]{\left| \begin{array}{*{#1}{c}}#2\end{array}\right|}
\newcommand{\datr}[2]{\left| \begin{array}{*{#1}{r}}#2\end{array}\right|}
\begin{document}

%\thispagestyle{empty}
\begin{center}
{\bf $\stackrel{\mbox{MA330}}{Sathaye}$ \hfill {\bf Class discussion of Math. overview} \hfill \ \   }
\end{center}

I list below main points of various topics discussed in class. More parts would be added as needed.
\centerline{\bf Spring 2017}
\begin{enumerate}
\item Fundamental parts of Mathematics: numbers, logic.
\item Discussion about the concept of ``truth'' in Mathematics. A proper mathematical statement only claims that a certain assertion is true, provided the set axioms hold.
\item Examples of axioms. Euclid's 5th postulate and non euclidean  geometries.
\item Which axiom systems are ``valid''? Ans: If they have a ``model''.
\item Well order and the principle of induction. (Homework exercises will appear on the web page.) 
\end{enumerate}

\centerline{Topics from previous semester.}
\begin{enumerate}
\item Numbers and shapes form the Mathematical language. Numbers have two main models: Greeks: lengths of line segments, Indians: counting money (or wealth). The Indian model leads to easy understanding of zero and negatives.

These distinct models perhaps lead to Calculus and Algebra respectively.

\item Modern construction of number systems.
\begin{itemize}
\item We discussed Peano axioms and construction of natural numbers starting with just an empty set. I recommend googling for ``Peano Axioms'' for a deeper understanding.
\item Then we discussed a way to ``build'' the integers by considering equivalence classes of pairs of natural numbers. I illustrated how the same consstruction leads to other constructions of more numbers.

\item Thus we constructed rationals, algebraic numbers and so on.
A general construction of adjoining a ``root '' of any polynomial due to Kronecker.

We discussed other possible extensions of number systems which lose many desirable properties. These are the quaternions and Cayley-Dickson algebras.

\item The next discussion is that of real numbers. Their description is, in some sense easy but as a set it is too big!  

The set of all real numbers can be described as decimal expaansions
$\{ a_0\dot a_1\, a_2\, \cdots ,a_n\, \cdots \}$
where all $a_i$ are integers, the numbers from $a_1$ onwards are in the set $\{ 0,1,2,3,4,5,6,7,8,9 \}$.

In addition, to avoid two different expansions giving the ``same'' number, (for example $0.100\cdots 0 \cdots = 0.099\cdots 9 \cdots$) we require that the sequence shall not have all  digits as $9$ from some position onwards.

\item We discussed the concept of cardinality $c(A)$ of a set $A$ due to Cantor. For a finite set $A$ it denotes the number of elements in $A$.
However, for infinite sets the definition is rather subtle. 

\item Thus for any two sets $A,B$ we say that $c(A)\leq c(B)$ if we can find an injective map from $A$ into $B$, or alternatively a surjective map from $B$ onto $A$.

It is a theorem that if  $c(A)\leq c(B)$ and $c(B)\leq c(A)$ both hold, then there exists a bijective map between $A$ and $B$ and we declare that $c(A)=c(B)$.

Further, we declare $c(A)<c(B)$ when $c(A)\leq c(B)$ but $c(A)\neq c(B)$.

\item {\bf Cantor's Theorem}
Given any set $A$, Cantor proved that $c(A) < c(\mathcal{P}(A))$ where 
$\mathcal{P}(A)$ denotes the power set (i.e. the set of all subsets) of $A$.

\item We call a set $A$ countably infinite if $c(A)=c(\mathbb{N})$ where 
$\mathbb{N}$ denotes the set of natural numbers.
We then explicitly argued that $c(\mathbb{R})>  c(\mathbb{N})$ where $\mathbb{R}$ is the set of reals.

\item We discussed several known and unknown results in this connection.



\end{itemize}

    
\end{enumerate}




\rightline{$ \Box $ More details to come ...}
\vspace*{\fill}
\end{document}
