Let the alphabet $C$ be $\{a,b,c\}.$ Let the language $x$ be the set of all strings over $C$ with at

least two occurrences of $b.$ Let $Y$ be the language of all strings over $C$ that never have two

occurrences of $c$ in a row. Let $Z$ be the language of all strings over $C$ in which every $c$ is

followed by an $a.($Recall that any string with no $c\text{'}$s is thus in $Z.)$

$($a$)$ List the three$-$letter strings in each of $x,Y,$ and $Z.$ The easiest way to do this may

be to first list all $27$ strings in $C^3$ and then see which ones meet the given conditions.

$($b$)$ List the four$-$letter strings that are both in $x$ and in $Y,$ those that are both in $x$ and

in $Z,$ those that are both in $Y$ and in $Z,$ and those that are in all three sets. How

many total strings are in $C^4?$

$($c$)$ Are any of $x,Y,$ or $Z$ subsets of any of the others?

$($d$)$ Suppose $u$ and $v$ are two strings in $x.$ Do we know that the strings $u^R,v^R,uv,$ and

$vu$ are all in $x?$ Either explain why this is always true, or give an example where it is

not.

$($e$)$ Repeat the previous question for the languages $Y$ and $Z.$