Section $1\mathrm{.}2$ Three Basic Concepts $($Pages $28-30)$

$(2$ points$)$ Section $1\mathrm{.}2$ Question $1$:

How many substrings aab are in $w{w}^{R}w,$ where $w=$aabbab?

$(1$ point$)$ Section $1\mathrm{.}2$ Question $8$:

$****$ Are there languages for which $\frac{?}{\text{b}\text{a}\text{r}}(L^{**})=(\frac{?}{\text{b}\text{a}\text{r}}(L){)}^{**}?$

$(5$ points$)$ Section $1\mathrm{.}2$ Question $14($Grammars$)$:

Find grammars for $=\{a,b\}$ that generate the sets of

$($a$)****$ all strings with exactly two $a\text{'}$s$.$

$($b$)$ all strings with at least two $a\text{'}$s$.$

$($c$)$ all strings with no more than three $a\text{'}$s$.$

$(2$ points$)$ Section $1\mathrm{.}2$ Question $15($Grammars$)$:

Give a simple description of the language generated by the grammar

with productions

$S$aaA

$AbS$

$S.$