Describing the Language of a grammar in set theoretic notation where the length of strings need to be remembered

I am not well versed in this topic so please pardon any ambiguous notation. I am trying to describe the language of this grammar in set-theoretic notation.

The Grammar is given by:

$$S \rightarrow ASA|B$$

$$A \rightarrow a|b$$

$$B \rightarrow BC|\epsilon$$

$$C \rightarrow a|b|c$$

I think I can accurately describe this language in english:

If $$w=xyz$$ is a string and $$xyz$$ are substrings of $$w$$ then

$$|x| = |z|$$, and $$x$$ and $$z$$ are made up of symbols from $$\{a,b\}^*$$.

Furthermore, $$y$$ is made up of symbols from $$\{a,b,c\}^*$$

Mores simply, the strings are made up of two ends that are the same $$|length| \ge 0$$ and made up of symbols $$a,b$$. And the middle part of the string is made up of any symbols from $$a,b,c$$ and also has a $$|length| \ge 0$$

How do I more formally and accurately describe the language of the grammar $$L(G)$$? Is it possible to do this in a set-theoretic notation?

• Your question smells like XY problem. Why do you require "in set theoretic notation where the length of strings need to be remembered"? I would bet all you wanted was to describe the language generated by the given grammar. – Apass.Jack Oct 31 '18 at 4:09
• @Apass.Jack Didn't I already describe the language of the Grammar in my question? All I am asking is what a more formal description of the language would look like (what form would it take), and if that formal description would take a set-theoretic notation, or if it even can. – SeesSound Oct 31 '18 at 4:14
• The language of the give grammar can be described simply as "all words over the alphabet $\{a,b,c\}$", which is a formal description. I would say the set-theoretic notation of regular languages is the regular expression. That languages is $(a\mid b\mid c)^*$. Are those what you wanted? – Apass.Jack Oct 31 '18 at 4:29

Your grammar generates $$\Sigma^*$$, where $$\Sigma = \{a,b,c\}$$.
• I'm in disbelief that it is this simple. What about the parts of the grammar that are saying that the outsides are all $a's$ and $b's$, why are you ignoring those rules? Why wouldnt they just make the Grammar as: $S \rightarrow aS|bS|cS|\epsilon$? Is this an equivalent grammar? – SeesSound Oct 31 '18 at 4:41