A string over the alphabet {[, ]} is called a string of balanced parentheses if two conditions hold: (i) every [ is later closed by a ]; and (ii) every ] closes a previous [. (You must close...

A string over the alphabet {[, ]} is called a string of balanced parentheses if two conditions hold: (i) every [ is later closed by a ]; and (ii) every ] closes a previous [. (You must close everything, and you never close something you didn’t open.) Let Bn ⊆ {[, ]} n denote the set of strings of balanced parentheses that contain n symbols.

Show that |B_n| ≤ 2ⁿ
: define a one-to-one function f : B_n
→ {0, 1} n and use the Mapping Rule.