Hi, my question is, in these questions, what are u and w exactly. Ie, what are the values they can take. I can't draw the transition diagram without knowing them.
From what I understand, the $\Sigma^*$ symbol is the set of all subsets of $\Sigma$={a, b} which I think would be {Ø, a, b, ab}. So if u and w are members of $\Sigma^*$, does that mean they can take any of those values?
So for example, in a) could the possible strings be... a, aa, ab, aab?
Any help would be appreciated because I cant draw the transition diagrams without knowing this.
(To make the symbols $\Sigma$ and $\Sigma^*$, write
$\Sigma$and$\Sigma^*$, respectively.)Your understanding of $\Sigma^*$ is incorrect. If $\Sigma$ is a set of symbols, $\Sigma^*$ is the set of all possible finite strings composed of symbols from $\Sigma$. Thus, if $\Sigma=\{a,b\}$,
$$\Sigma^*=\{\epsilon,a,b,aa,ab,ba,bb,aaa,aab,aba,baa,abb,bab,bba,bbb,\ldots\}\;.$$
Here $\epsilon$ is a conventional symbol for the empty string, the string of no symbols at all; some books use $\lambda$ instead. As you can see, $\Sigma^*$ is an infinite set. This is true even if $\Sigma$ has only one element: if $\Sigma=\{1\}$, for instance, $$\Sigma^*=\{\epsilon,1,11,111,1111,11111,\ldots\}\;.$$
The set $\{aw:w\in\Sigma^*\}$ in (a) is therefore the set of all strings of $a$’s and $b$’s that start with $a$: $a,aa,ab,aaa,aab,aba,abb,\ldots$. The set $\{awb:w\in\Sigma^*\}$ in (b) is the set of all strings of $a$’s and $b$’s that start with $a$ and end with $b$. The set $\{uabaw:u,w\in\Sigma^*\}$ in (c) is the set of all strings of $a$’s and $b$’s that contain the substring $aba$ somewhere. And the set in (d) is pretty self-explanatory.