For example, [[]][] is a string of balanced parentheses, using Rule 3 on [[]] and []. (Note that [] is a string of balanced parentheses using Rule 2 on the empty string (Rule 1), and therefore [[]] is...

For example, [[]][] is a string of balanced parentheses, using Rule 3 on [[]] and []. (Note that [] is a string of balanced parentheses using Rule 2 on the empty string (Rule 1), and therefore [[]] is by using Rule 2 on [].)

1.Prove by structural induction that every string of balanced parentheses according to this definition has exactly the same number of open parentheses as close parentheses.

2. Prove by structural induction that any prefix of a string of balanced parentheses according to this definition has at least as many open parentheses as it does close parentheses.