Question

So, I have this description for a string: S consists of all strings of a’s, b’s and c’s where an increasing number of a’s come before an increasing number of b’s and c’s or c’s and b’s. does this make sense for this recursive definition of the set S over {a,b,c}: 1. a is in S and 2. if x in S then ax, xb, xc are in S?

Answer 1

It makes sense but is incomplete. The string in question follows this regex:

a+[bc]*

So, you need to be clear that there is always at least one a, and the b‘s and c‘s can intermix.