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\handout{symbols}{Notation and properties of numbers}{September 8, 1994}

\noindent
{\large \bf Symbols:}\\
\begin{itemize}
\item
$\nat = \{ 0, 1, 2, 3,\ldots\}$; the natural numbers.

\item
$\ints = \{\ldots,-4, -3,-2,-1,0,1,2,3,4,\ldots \}$; the integers.

\item
$\rationals = \{ a/b : a\in\ints, b\in\ints, b\neq 0\}$;
the rational numbers.

\item
$\reals$; the real numbers.

\item
$\reals - \rationals$; the irrational numbers.

\item
$\complex = \{ a+bi : a\in\reals, b\in\reals, \mbox{where } i^2 = -1\}$.

\item $\{2,3,5,7,11,13,\dots\}$; the prime numbers---the subset of
$\nat$ containing every number with exactly two divisors.

\item
$\{4,6,8,9,10,12,14,15,\ldots\}=\nat - \{\mbox{primes}\}$; 
the composite numbers.
%	the subset of $\nat$ containing every number that is divisible by some 
%	number other than itself and 1.

\end{itemize}

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\noindent
{\large \bf Properties of the integers and the reals:}\\
%The notation ``$\forall a$'' means ``for all $a$''.
\begin{itemize}
\item
Each is closed under addition: 
\begin{itemize} \vspace*{-.1in}
\item If $a\in\ints$ and $b\in\ints$, then $(a+b)\in \ints$.
\item If $a\in\reals$ and $b\in\reals$, then $(a+b)\in\reals$.
\end{itemize}

\item
Addition is associative: $(a+b)+c = a+(b+c)$.

\item
Addition is commutative:  $a+b = b+a$.

\item
There is an additive identity:  
there exists a real number (an integer), namely $0$, 
such that $0+a=a$ for all $a$.

\item
Each element has an additive inverse: 
\begin{itemize} \vspace*{-.1in}
\item For each $a\in\ints$, there is an integer, written $-a$, 
	such that $a + (-a) = 0$.
\item For each $a\in\reals$, there is a real number, written $-a$, 
	such that $a + (-a) = 0$.
\end{itemize}

\item
Each is closed under multiplication:
\begin{itemize} \vspace*{-.1in}
\item If $a\in\ints$ and $b\in\ints$, then $(ab)\in \ints$.
\item If $a\in\reals$ and $b\in\reals$, then $(ab)\in\reals$.
\end{itemize}

\item
Multiplication is associative:  $(ab)c = a(bc)$.

\item
Multiplication is commutative:  $ab = ba$.

\item
There is an multiplicative identity:  there exists a real number (an integer),
namely $1$, such that for all $a$, $1a=a$.

\item
Each nonzero real number element has a multiplicative inverse in the reals: 
for each $a\in\reals$, $a\neq 0$, there is a real 
number, written $a^{-1}$, such that $a a^{-1} = 1$.\\
(Note that the nonzero integers do not each have a multiplicative inverse
in the integers.)

\item 
Cancellation:  if $ac = bc$ and $c\neq 0$, then $a=b$.

\item
Multiplication distributes over addition:  $a(b+c) = ab + ac$.

\item
Trichotomy of $<$:  Either $a<b$, or $a=b$, or $b<a$.


\item
Well-Ordering Principle:
If $A\subseteq\nat$ and $A$ is not empty, then $A$ contains a smallest element.

\item
Weak Induction:
Let $P(n)$ be a statement about $n$ that is either true or false.
If 
\begin{enumerate}
\item $P(0)$ is true,  and 
\item For all $n\in\nat$, if $P(n)$ is true then $P(n+1)$ is true,
\end{enumerate}
then $P(n)$ is true for all $n\in\nat$.

\item
Strong Induction:
If 
\begin{enumerate}
\item $P(0)$ is true,  and 
\item For all $n\in\nat$, if $P(0), P(1), \ldots, P(n)$ are all true then 
$P(n+1)$ is true,
\end{enumerate}
then $P(n)$ is true for all $n\in\nat$.

\end{itemize}


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