Question

(5) Describe the strings in the set S of strings over the alphabet Σ = a, b, c defined recursively by (1) c E S and (2) if x є S then za E S and zb є S and cr є S. Hint: Your description should be a sentence that provides an euasy test to check if a given string is in the set or not. An example of such a description is: S consists of all strings of a's, b's, and c's, with more a's than b's. That isn't a correct description since cab is in S and doesn't have more a's than b's, and also baac isn't in S, but does have more a's than b's. So that attempted description is really terrible. The best way to do this problem is to use the rules to build a bunch of strings in S until a suitable description becomes obvious

Answer 1

Given that the (1) hence that the a belongsto s for condition (2) if x belongsto s since a x, b x b, x c are each elements of s so as it is a recursive relation we can see it anything string its starts with a and can end with a, b, c but it ends with a only when x = a^+, is x = a, aa, aaa also a belongsto S finally so s + {a^+} union { strings beginning with a and ending either b or c}