Michael Sipser Edition 3 Exercise 2 Question 18 (Page No. 156)

Question

1. Let $C$ be a context-free language and $R$ be a regular language$.$ Prove that the language $C\cap R$ is context-free.
2. Let $A = \{w\mid w\in \{a, b, c\}^{*}$  $\text{and}$  $w$  $\text{contains equal numbers of}$  $a’s, b’s,$ $\text{and}$ $c’s\}.$ Use $\text{part (a)}$ to show that $A$ is not a CFL$.$

Answer 1

CFL $\cap$ RL $\rightarrow$ CFL

let B=a*b*c*

if A is CFL then A $\cap$ B should be CFL

A $\cap$ B=$a^nb^nc^n$ which is not CFL

hence, A is not CFL