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Come up with a regular expression for the following language $$ \{w \mid w \text{ doesn’t contain the substring } 110\} $$ The book provides a more complex answer, but I think that using the complement is easier. Do you think this works?:
$\Sigma^*-\Sigma^*110\Sigma^*$

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    subtraction is not defined. see en.wikipedia.org/wiki/Regular_expression#Formal_language_the‌​ory – 2017-02-09
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    @dave I think the author wanted to express this in terms of set difference. – 2017-02-13

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Your expression of the language as the complement of the language $\Sigma^*110\Sigma^*$ is correct. However, it does not give a regular expression for the language. A regular expression can only use letters, union, product and star operator (and the complement is not allowed). The fact that the complement of a language $L$ given by a regular expression can also be expressed by a regular expression is a nontrivial result, due to Kleene. The standard three step algorithm to find a regular expression for the complement of $L$ is the following:

Step 1. Find a deterministic complete automaton $\mathcal{A}$ accepting $L$.

Step 2. Find a deterministic automaton $\mathcal{B}$ accepting the complement $L$. This is the easy step: it suffices to modify $\mathcal{A}$ by replacing its set of final states by its complement.

Step 3. Convert the automaton $\mathcal{B}$ into a regular expression.