$(16$ points, $8$ points each$)$ Do part $($a$)$ and either $($b$)$ or $($c$)$

Recursive Definitions: For the following descriptions of sets, write a recursive definition for the set.

Be sure to include the basis step and the recursive step for each part. Your recursive step MUST

build new elements from existing elements of the set. Also, provide a brief explanation for why your

recursive definition works for the set described.

$($a$)$ The set $N$ of all binary strings $($including the empty string$),$ with an equal number of $0$ s and $1s,$

with all of the ones preceding all of the zeroes. For example, $1100$ and $1111100000$ are in $N,$ but

$0101$ and $110$ are NOT.