Clearing a Confusion regarding the Proof of equal no of a's and b's not being a regular language

Question

I was wondering about its proof. The direct use of pumping lemma here is not a viability. So a certain teacher of mine proved this with the notion that $a^{n}b^{n}$ being a subset of this language $L=\{w\mid\#_{a}(w)=\#_{b}(w)\}$ and non-regular (simple proof using pumping lemma) implies that the language $L$ can therefore not be regular! The proof sounds alright initially but it got messy as I thought more on it! I read in the books that the subset property of a regular language is not closed under regular languages. So, how can the proof be correct because for all I know L is a regular language that a non-regular language $a^{n}b^{n}$ as its subset. How does it say anything more?

I do know ofcourse that $L$ here is not regular and can draw production rules for it(CFG)! But, I need to clear this confusion I have above! Some help me here.

Answer 1

Here is what your teacher meant. Let $M=\{a^nb^n:n\geq0\} $. We know $M$ is not regular. Since the intersection of regular languages is regular, if $L$ were regular then so would $L\cap a^*b^*=M$ be, reaching a contradiction. This shows that $L$ cannot be regular.

Answer 2

You are correct: the fact that $L$ is a subset of a non-regular language does not prove that $L$ is non-regular. The trivial counterexample to that claim is the empty language, which is regular and is a subset of any language. (Similarly, proving that $L$ is a subset of a regular language also doesn't prove that $L$ is regular, since every language is a subset of the regular language $\Sigma^*\!$.)

Actually, I don't really understand why anyone would try to prove it that way. The direct pumping-lemma proof that $\{a^nb^n\mid n\geq 0\}$ isn't regular is hardly any more difficult than proving that $\{w\mid \#_a(w)=\#_b(w)\}$ isn't regular. Note also that demonstrating that a language is context-free doesn't prove that it's non-regular, since every regular language is context-free.

It occurs to me that you may have misunderstood your teacher's proof. Since the string $a^kb^k$ is in both langauges, you can prove that both langauges are non-regular by applying the pumping lemma to that string.

Answer 3

We often use the Pumping Lemma for regular languages as the basis of a proof by contradiction.

The PL says that if $L$ is a regular language, then any sufficiently long word $w$ in $L$ can be written as $w=xyz$ (with certain restrictions on $x, y, z$) and can be pumped, namely that for all $i\ge 0$, we'll have $xy^iz\in L$. Now if that doesn't hold for a particular string in $L$ (i.e., for that particular word and for some particular $i$, the word $xy^iz\notin L$), then it must have been the case that our assumption, that $L$ was regular, must have been false.

In this case, pumping the particular string $a^pb^p$, where $p$ is the integer of the PL, gives you a string with more $a$s than $b$s, which can't possibly be in $L$, so $L$ couldn't possibly have been regular. In this case, we have a non-regular subset of a non-regular language: $E=\{a^nb^n\mid n\ge 0\}\subseteq L$. Roughly speaking, it's the way that $E$ behaves inside $L$ that says $L$ can't be regular.

On the other hand, if we had loosened the strictures of $L$ and instead had chosen another language $L_1=\{a, b\}^*$, then we'd have $E\subseteq L_1$ and now we'd have the non-regular $E$ as a subset of the regular $L_1$ (because the PL wouldn't yield a contradiction, no matter what string we tried, due to the way $L_1$ was defined).