Example of language that does not contain substring $11$ Let $\Sigma=\{1,0\}$
I just thought of sample strings that can work, such as, $\epsilon, 1, 11, 10, 0, 01$, and came up with:
$$R=1^*(\epsilon+01^*)$$
Is this correct? And is this the correct way to do it? I don't particularly like these questions where they ask you $x$ is not a substring. How do we even do those?
 A: The title of your question is misleading. I suppose you are interesting in the language of all words not containing $11$ as a substring. More generally, given a word $u$, the set of words containing $u$ as a substring is the language
$$
L(u) = \Sigma^*u\Sigma^*
$$
Therefore, the set of words not containing $u$ as a substring is the complement of $L(u)$. If you want a regular expression for this language, you can proceed as follows


*

*First compute the minimal DFA of $L(u)$ (this automaton has $|u| + 1$ states).

*Compute the minimal DFA of its complement (just swap the final states and the non final ones).

*Compute a regular expression from the resulting DFA.


For your example, $u = 11$, the minimal automaton of $L(11)$ is

and the minimal automaton of its complement is

It is now easy to get a regular expression for the language accepted by this automaton: $(0 + 10)^*(\varepsilon + 1)$.
A: As the substring $11$ is forbidden, any $1$ must be followed by a $0$ or nothing. So a good candidate is
$$(0+10)^*(\epsilon+1)$$ a little more symmetrically written $$((\epsilon+1)0)^*(\epsilon+1).$$
There are many more ways to avoid $11$, such as with the trivial $0^*$.
