# Is this unambiguous grammar equivalent?

I have a simple context free grammar $(\{A\}, \{(,)\}, R, A)$, which consists of this one production rule:

$A \rightarrow AA\, \vert\, (A)\, \vert\, \epsilon$

I believe this is ambiguous! For example, I can represent "()" like this:

I'm sure there are an infinite number of these. I've attempted to make the grammar unambiguous whilst ensuring that the same language is accepted:

• $A \rightarrow AA\, \vert\, B$

• $B \rightarrow (C)$

• $C \rightarrow \epsilon$

How can I be sure that this captures the same language and is no longer ambiguous?

$$T\to T\;T$$
then either the grammar is ambiguous or the rule is useless. (Because $T\;T\;T$ can be reduced in two ways.)
$$S\to \epsilon\mid(\;S\;)\;S$$