How to show that given language is unambiguous

Problem Detail: 

Given following grammar:

$$ \begin{align} S \rightarrow &A1B \\ A \rightarrow & 0A \mid \varepsilon \\ B \rightarrow & 0B \mid 1B \mid \varepsilon \\ \end{align} $$

How can I show that this grammar is unambiguous? I need to find a grammar for the same language that is ambiguous, and demonstrate it.

I know if I was asked to prove that the language is ambigious then I should find two different parse trees for same string, but I don't know what to do.

Asked By : quartaela

Answered By : Hendrik Jan

To show a grammar is unambiguous you have to argue that for each string in the language there is only one derivation tree.

In this particular case you can observe that $A$ only generates $0$'s, so the $1$ generated by the start symbol $S$ must be the first $1$ in the string.

Any grammar can be made ambiguous by adding chain productions like $S\to S$.

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Question Source : http://cs.stackexchange.com/questions/7518