# Is it possible to create a regular language from an non regular language? (details inside)

I am wondering, is it is possible to create a regular language from a non regular language if we add or remove finite number of words from it?

say L is irregular, if we add or remove finite number of words can we create a regular language?

i might be mistaken, but since all regular languages are finite - if we add a finite amount to a non regular language - it still stays non regular, but if we substract, let's say a finite amount from infinity, it is still infinity.

so is it safe to say that in both cases a regular language cannot not be obtained by adding/substracting a finite amount of words?

i was told to ask this question here rather then in softwareengineering.

We have a language $$L$$ which is non-regular. We subtract a finite subset $$S \subset L$$ of words in $$L$$ to get the rest: $$R = L \setminus S$$.
Assume that $$R$$ is regular. Since regular languages are closed under union, and finite sets of words are trivially regular, we can construct a language $$L' = R \cup S$$ which is also regular.
Since $$S \subset L$$, we have $$(L \setminus S) \cup S = L$$, thus $$L' = L$$. But $$L'$$ is regular while $$L$$ is not - a contradiction. Thus our assumption that $$R$$ is regular was wrong.