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 grammar generates $\Sigma^*$, where $\Sigma = \{a,b,c\}$.