Prove by weak or structural induction.

A string of balanced parentheses [ , ] is one of the following: (a) the empty string; (b) a string [S] where S is a string of balanced parentheses; or (c) a string S1S2 where S1 and S2 are both balanced strings of parentheses. Prove by structural induction (a) Every string of balanced parentheses has exactly the same number of open parentheses as close parentheses; (b) Every prefix of a string of balanced parentheses has at least as many open parentheses as it does close parentheses.