Closure of regular languages

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

Closure of a set under an operation

We say that a set M is closed under an operation ⊗ if

$$\mathrm{\forall }x,y\in M:x\otimes y\in M$$

An example is addition on the set of positive integers. The sum of any two positive integers is again a positive integer. On the other hand, the set of positive integers is not closed under subtraction, because, for example, 7 − 15 = −8 and the number −8 is not a positive integer.

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