# Why descriptive power of context-free grammar is greater than regular expression?

Let the set of languages that can be represented by regular languages be $$R$$. And set of languages that can be generated by a context free grammar be $$G$$. Then surely $$R \subset G$$. For the proof that every regular language has a context free grammar that generates it, I encourage you to look up Michael Sipser's book on complexity. And for the second part, there are many languages which have a context free grammar generating them but are not regular. For example $$L=\{ww^{reverse}\}$$, where $$w^{reverse}$$ is reverse of the string $$w$$. How do we know $$L$$ has a context free grammar ( you can easily come up with a push down automaton for it ), look up Sipsers. And look up pumping lemma in the book to prove $$L$$ is not regular.
To get an intuition, any regular language can be represented by a finite state automaton ($$FSA$$) and vice versa. And any context free grammar can be represented by a pushdown automaton ($$PDA$$) and vice versa (again, read the proof from Sipser). And in a $$PDA$$, in addition to the states (like in $$FSA$$), you also have an infinite stack. Thus intuitively context free grammars have "power" greater than that of $$FSA$$.