# Does every regular expression describe only 1 language?

If we have a regular expression $$R$$, will $$R$$ describe only regular language $$L$$, but that language $$L$$ can have multiple different regular expressions such as $$Q,W,A,S,D \ etc..$$ describing it

Also, $$R$$ can be equivalent, in terms of describing $$L$$, to infinite regular expressions including $$Q,W,A,S,D \ etec..$$

Is my understanding correct?

• How could a regular expression describe more than one language? Jan 10 at 9:45

If $$e$$ is a regular expression such that $$\mathcal{L}(e) = L$$, then so is $$e+\emptyset$$, $$e+\emptyset+\emptyset$$, …
• A regular expression is a sequence of symbols. Two regular expressions with different sequences of symbols cannot be equal. However, the language they describe can be equal. That's why I am talking about the interpretation of a regular expression, denoted $\mathcal{L}(e)$. Though one can sometimes write something like $(a+b)^* = (a^*b^*)^*$, this is an abuse of notation that, in reality, means $\mathcal{L}((a+b)^*) = \mathcal{L}((a^*b^*)^*)$. Jan 10 at 12:49