Let $A$ be a set represented by a regular expression $0^*1^*$ and let $B=\{0^n1^n\mid n\geq 0\}$.
It is known that $A$ is a regular language, while $B$ is a context-free language. I understand that $A$ is recognized by finite state automata, while $B$ is recognized by pushdown automata.
But, isn't $B$ a subset of $A$? For example, $01, 0011,000111,\ldots$ are all elements of $A$. My question is that how a language recognized by pushdown automata, not by finite state automata, can be a subset of regular language?