# What is the regular expression describing this language?

The language is $\{w \mid \text{$w$is any string except$11$and$111$}\}$ where the alphabet is $\{0,1\}$.

Drawing the DFA recognizing $\{v \mid \text{$v$is either$11$or$111$}\}$, then switching the accept and non-accept states, and finally examining this resulting structure, I thought of it as being:

$(ϵ∪0Σ^*∪1∪10Σ^*∪110Σ^*∪111Σ^+)$

My reasoning was that it has to accept the empty string, or a 0 followed by whatever or just a $1$ or, if it begins with a $1$, a $10$ followed by whatever or, if it begins with $11$, then a $110$ followed by whatever or a $111$ followed by at least a $0$ or a $1$

Does this appear to be correct? If I'm wrong, could you please tell me why?

Thank you,

• You could have written it more compactly, e.g. $\epsilon + 1 + (0 + 1(\epsilon + 1 + 11)0)(1 + 0)^*$, but the idea is basically the same. Commented Sep 29, 2015 at 23:25
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– D.W.
Commented Sep 29, 2015 at 23:28

Basically here two things are happening.

1. Regular languages are closed under compliment operation
2. If L1 is accepted by DFA1 and DFA2 is obtained by interchanging the final states to non-final states and non-final states to final states of DFA1 then the language accepted by DFA2 is compliment of L1. That is, if
• DFA1 is a quintuple (Q, Σ, δ , q0, F)
And L1 is language accepted by DFA1
• DFA2 is (Q, Σ, δ , q0, (Q - F))
And L2 is language accepted by DFA2

Then L2 is compliment of L1.

In this case the languages
L1 = {w∣w is any string except 11 and 111} and L2 = {v∣v is either 11 or 111} are compliment o each other .
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