We define the set of balanced parentheses S recursively as follows: Basis: the string () is in S Recursive rules: (a) Vx E S((x) E S) (b) VX E SVY E S(xy E S) where xy means the concatenation of x and y Using structural induction, prove that for any string a in S, the number of left parentheses in x is equal to the number of right parentheses in x.