Closure of regular languages

A language L is regular precisely if there exists an automaton A that accepts this language, i.e. L(A) = L. We prove that regular languages are closed under the operations of union, intersection, concatenation, and closure.

Closure on a set with respect to an operation

We say that the set M is closed under the operation $\otimes$ if it holds that

$$ \forall x, y \in M: x \otimes y \in M $$

An example is the addition operation on the set of natural numbers. The sum of any two natural numbers is again a natural number. Conversely, the set of natural numbers is not closed to subtraction because, for example, 7 − 15 = −8 and the number −8 are not natural numbers.

We can prove against which operations the set of regular languages is closed. All the proofs will be constructive, i.e. we will construct an automaton that accepts the resulting language.