Answer :

Final answer:

If L1 and L2 are regular languages, then the difference L1 - L2 is also a regular language. This can be proved by expressing L1 - L2 as an intersection of L1 with the complement of L2. The closure properties of regular languages further confirm that the resulting L1 - L2 is a regular language.

Explanation:

You have asked to prove that if L1 and L2 are both regular languages, then the difference of these languages, L1 - L2, is also a regular language. Regular languages hold closure properties, which means if you perform operations like union, intersection, complementation or difference on regular languages, the result will still be a regular language.

For our specific scenario, we use two known facts: the first is that the intersection of two regular languages is a regular language, and the second is that the complement of a regular language is a regular language. We can construct L1 - L2 as L1 ∩ ~L2 (where ~L2 is the complement of L2). Thus L1 - L2 is the intersection of L1 with the complement of L2, and hence, by the closure properties of regular languages, L1 - L2 is a regular language as well.

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