For the alphabet $\sum = \{0,1\}$, let $A,B,C \subseteq \sum^*$ be the languages below.
$i. A = \{1, 0, 00, 11, 000, 111, 0000, 1111\}$
$ii. B = \{w \in \sum^*|||w|| \ge 2 \}$
$ii. C = \{w \in \sum^*|||w|| \le 2 \}$
||w|| is the length of the word w.
Determine the following languages of $\sum^*$:
(a.) $A \cap B$
For this one, I have: $w \in \sum^*$ such that the length of $w \ge 2$, thus $A \cap B = 00, 11, 000, 111, 0000, 1111$. The language of $\sum^* = \{00^n | 1 \le n \le 4\}\cap\{11^n|1\le n \le 4\}$
(b.) $B \cup C$
For this one, I have: $w\in \sum^*$ such that the length of $w \le 2$ ot the length of $w \ge 2$, then $B \cup C = \{ 00 | 11 \}.$ The language of $\sum^*=\{0^n | n=2\} \cup \{1^n |n=2\}.$
(c.) $A \Delta B$
For this one, I have: $A \Delta B = A \cup B$ (according to a proposition in my study guide). Thus I have, $A \Delta B = \{1,0,00,11,000,111,0000,1111 \}$ or $w \in \sum^*$ such that the length of $w \ge 2$ therefore, $A \Delta B =\{00,11,000,111,0000,1111 \}.$ The language of $\sum^* = \{ 0^n|1 \le n \le 4 \} \cap \{ 1^n | 1 \le n \le 4 \}$ or $\sum^*= \{ 00^n | 1 \le n \le 4 \}\cap \{11^n | 1 \le n \le 4\}.$
I think I am on the right track with these, but any help is appreciated.
Thanks
You seem to misinterpreting $\Sigma^*$ as its first given subset $A$.
$\Sigma^*$ denotes the set of all words on alphabet $\Sigma$.
(a) $A\cap B\ =\{00,11,000,111,0000,1111\}$ is correct. Nothing more is needed here.
(But if you insist, it is rather $\Sigma^*=\{00^n\,:\,1\le n\le 3\}\ \cup\{11^n\,:\,1\le n\le 3\}$.)
(b) $B\cup C$ will give all words, i.e. $B\cup C=\Sigma^*$.
(c) $A\Delta B$ usually denotes the symmetric difference, containing those elements which are either in $A$ or in $B$ but not in both, i.e. $A\Delta B\ =\ (A\cup B)\ \setminus\ (A\cap B)$. Can you describe it now?