Proving that L is not regular using closure properties

Question

I need to show that the following language is not regular.

$$L = \{\ ab^jc^j\ |\ j \geq 0\ \}\ \cup\ \{\ a^ib^jc^k\ |\ i, j, k \geq 0 \ and\ i \neq 1\ \}$$

There is also a hint that it cannot be proven just with using the Pumping Lemma. Closure properties must also be used.

I'm having a hard time with this one. Any help would be appreciated.

Answer 1

Let $ L_1 = \{\ ab^jc^j\ |\ j \geq 0\ \} $ and $ L_2 = \{\ a^ib^jc^k\ |\ i, j, k \geq 0 \ and\ i \neq 1\ \} $. Then

$$ L = L_1 \cup L_2. $$

If we take $$ \begin{align} L\ \cap\ ab^*c^* &= (L_1\cup L_2) \cap ab^*c^* \\ &= (L_1\cap ab^*c^*)\ \cup\ (L_2 \cap ab^*c^*)\\ \end{align} $$

Since $L_1\cap ab^*c^* = L_1$ and $L_2 \cap ab^*c^* = \emptyset$, we have

$$ L\ \cap ab^*c^* = L_1 $$

Now, $ab^*c^*$ is regular, and $L_1$ is not regular (can be shown using the Pumping Lemma). If $L$ was regular, by the closure properties, $L_1$ would be to, which is a contradiction. Hence, $L$ is not regular.

Answer 2

Here is a further hint. $L$ consists of two parts. One of them as a language by itself is not regular. Can you see which part?

Can you carve out that part as the intersection of $L$ with another regular language?