For $\text{A, B} \subseteq \Sigma^*,$ define
$A/B = \{x \in \Sigma^* | \exists y \in B , xy \in A \}$
If L is a CFL and R is regular, then L/R is
- Regular
- CFL but not regular
- Recursive but not CFL
- None of the above
To solve this problem I am trying in this way –
Let us consider that $L$ is $a^nb^n$ and $R$ is $b^*$,
$A/B = \frac{a^nb^n}{b^*} \\= \frac{a^nb^n}{ \{\epsilon , b, b^2 , \dots \}} \\ = \{a^nb^n, a^nb^{n-1}, a^nb^{n-2} , \dots\}$
So it contains strings like $a^nb^n, a^nb^{n-1}, \dots $ which we know that are not regular but they are CFL.
Hence, $L/R$ is CFL but not Regular.
Please advise me that am I thinking in correct way or not?