even$/$odd$/$substring:

x in $\{$a$,$ b$\}$ x has an even number of a$$s$,$ or x has an odd number of b$$s$,$ or

x contains both the substrings babb and aabaa

first appears more:

$\{$x in $\{0,1\}||$x$|>=3$ and the first symbol of x appears at least three times total in x$\}$

repeat near end: $\{$x in $\{0,1\}|$ x$[|$x$|5]=$ x$[|$x$|3]\}$

Assume we start indexing at $1,$ so that x$[|$x$|]$ is the last symbol in x$,$ and x$[1]$ is the first.

email: $\{$x in $\backslash$Sigma $|$ x is a syntactically valid email address$\}$

Definition of $$syntactically valid email address$$: Let $\backslash$Sigma $=\{.,$ @$,$ a$,$ b $\}$ contain the

alphabetic symbols a and b$,1$ as well as the symbols for period $.$ and $$at$$ @$.$ Syntactically

valid emails are of the form username@host.domain where username and host are nonempty

and may contain alphabetic symbols or $.,$ but never two $.$s in a row, nor can either of them

begin or end with a $.,$ and domain must be of length $2$ or $3$ and contain only alphabetic

symbols. For example, aaba@aaabb.aba and ab$.$ba@ab$.$abb.ba are valid email addresses,

but aaabb.aba is not $($no @ symbol$),$ nor is $.$ba@ab$.$abb.ba $($username starts with a $.),$ nor is

$1$It$$s not that hard to make a regex that actually uses the full alphanumeric alphabet here, but historically we$$ve

found that many students$$ solutions are correct but use so many subexpressions that they crash the simulator. Using

only two alphabetic symbols a and b reduces this problem, even though it makes the examples more artificial$-$looking.

$2$