closure as a proof tool
Closure properties are not just for building recognizers — they are an indirect way to PROVE a language is not regular without touching the pumping lemma at all. The trick is a kind of logical judo: assume your suspect language is regular, then combine it with known-regular ingredients using operations that preserve regularity (intersection, complement, homomorphism, reversal, difference). If that legal combination lands on a language you ALREADY know is not regular, you have a contradiction — so the suspect was not regular to begin with.
The recipe runs like this. Suppose L is regular (for contradiction). Pick a regular language R you can write down easily. Since the regular languages are closed under intersection, L ∩ R would also be regular. Now choose R so that L ∩ R equals a famous non-regular language — most often a^n b^n. But a^n b^n is not regular, so L ∩ R cannot be regular, which clashes with the closure conclusion. The only assumption we made was 'L is regular', so that assumption is false.
This style is often cleaner than the pumping lemma because it offloads the hard work onto one canonical non-regular language whose status you already trust. It also generalizes: closure under homomorphism, inverse homomorphism, and reversal each give different angles, so a language that resists one closure argument may fall to another. The price is that you must KNOW some base non-regular language and find the right regular partner to combine with.
Claim: L = strings over {a,b} with equally many a's and b's is not regular. Suppose it were. Intersect with the regular language a* b* (all a's then all b's). The result is exactly { a^n b^n : n ≥ 0 }, which is not regular — contradiction. So L is not regular.
Combine the suspect with a regular language to expose a known non-regular language.
This proves NON-regularity, never regularity — closure under an operation cannot certify that a single given language is regular. And the contradiction is only as solid as your base fact (e.g. that a^n b^n is non-regular), which itself must be proved by the pumping lemma or Myhill-Nerode.