Let $\backslash$Sigma $=\{$a$,$b$\}.$

Define:

L$2=(\backslash$Sigma $=2)*$

L$3=(\backslash$Sigma $=3)*$

$3($a$)$

Give a complete description of

$\backslash$Sigma $=2$

$\backslash$Sigma $=3$

and an informal description of

L$2=(\backslash$Sigma $=2)*$

L$3=(\backslash$Sigma $=3)*$

$3($b$)$

Prove that for all w in L$2,$ length$($w$)=(2)0.$

Question$3($a$)$

Give a complete description of

$\backslash$Sigma $=2$

$\backslash$Sigma $=3$

and an informal description of

L$2=(\backslash$Sigma $=2)*$

L$3=(\backslash$Sigma $=3)*$

Question$3($b$)$

Prove that for all w in L$2,$ length$($w$)=(2)0.$

$3($c$)$

Show that $\backslash$Sigma $=2$ and $\backslash$Sigma $=3$ give a counterexample to the proposition that for all languages X$,$ Y $\backslash$Sigma $*$:

$($X$\backslash$cap Y$)*=$ X$*\backslash$cap Y$*$

$3($d$)$

Prove that

L$2\backslash$cap L$3=(\backslash$Sigma $=6)*$

$3($e$)$

Using the observation that every natural number n$>=2$ is either even or $3$ more than a non$-$negative even number, prove that:

L$2$L$3=\backslash$Sigma $*\backslash \{$a$,$b$\}$