difference between the empty set, or empty language, and the set, or language, that 

contains the empty string $\{\text{""}\}$: the former has no elements, the latter has one 

element.

As seen there are languages which contain infinitely many strings, like the set of all strings.

The ``natural'' languages English, French and so on contain many but only finitely many 

strings (the ones listed in a good dictionary). It might be therefore surprising that the

language consisting of all email addresses is infinite if we assume it is defined by the

regular expression\footnote{See \url{http://goo.gl/5LoVX7}}

\[

([\text{\it{}a-z0-9\_.-}]^+)@([\text{\it a-z0-9.-}]^+).([\text{\it a-z.}]^{\{2,6\}})

\]

\noindent

The reason is that for example before the $@$-sign there can be any string you want if it 

is made up from letters, digits, underscore, dot and hyphen---there are infinitely many

of those. Similarly the string after the $@$-sign can be any string. This does not mean 

that every string is an email address. For example

\[

\text{\it foo}@\text{\it bar}.\text{\it c}

\]

\noindent

is not, since the top-level-domains must be of length of at least two. Note that there is

the convention that uppercase letters are treated in email-addresses as if they were

lower-case.

\bigskip

\noindent

\emph{Regular expressions} are meant for conveniently describing languages...at least languages

we are interested in in Computer Science.  For example there is no convenient regular expression

for describing the English language short of enumerating all English words/strings like in a dictionary. 

But they seem useful for describing all permitted email addresses, as seen above. 

Regular expressions are given by the following grammar:

\begin{center}

\begin{tabular}{r@{\hspace{1mm}}r@{\hspace{1mm}}l@{\hspace{13mm}}l}

  $r$ & $::=$ &   $\varnothing$         & null\\

        & $\mid$ & $\epsilon$              & empty string / "" / []\\

        & $\mid$ & $c$                         & character\\

        & $\mid$ & $r_1 \cdot r_2$      & sequence\\

        & $\mid$ & $r_1 + r_2$            & alternative / choice\\

        & $\mid$ & $r^*$                      & star (zero or more)\\

  \end{tabular}

\end{center}

\noindent

There are some subtleties you should be aware of. First, we will use parentheses to disambiguate

regular expressions. For example we will write $(r_1 + r_2)^*$, which is different from $r_1 + (r_2)^*$.

The former means roughly zero or more times $r_1$ or $r_2$, while the latter means $r_1$ or zero or more times

$r_2$. We should also write $(r_1 + r_2) + r_3$ which is a regular expression different from $r_1 + (r_2 + r_3)$,

but in case of $+$ and $\cdot$ we actually do not care and just write $r_1 + r_2 + r_3$, or $r_1 \cdot r_2 \cdot r_3$,

respectively. The reasons for this will become clear shortly. In the literature you will often find that the choice

$r_1 + r_2$  is written as $r_1\mid{}r_2$