$8\mathrm{.}10\mathrm{.}8$: Recursively computing the set of all binary strings of a fixed length.

help$\_$outline

$($a$)$

Give a recursive algorithm that takes as input a non$-$negative integer n and returns a set containing all binary strings of length n$.$ Here are the operations on strings and sets you can use:

Initialize an empty set S$.$

Add a string x to S$.$

y$$:$=0$x$.$This operation adds a $0$ to the beginning of string x and assigns the result to string y$.$

y$$:$=1$x$.$ This operation adds a $1$ to the beginning of string x and assigns the result to string y$.$

Return$($ S $)$

A looping structure that performs an operation on every string in a set S:

for every x in S$,$

$//$ perform some sequence of steps with string x$.$