# Regular set corresponding to regular language

I have just started learning regular expressions and I don't have anybody around me to help me building conceptions. So I rely on online mediums. My question is whether every regular set corresponds a language.

I had a regular expression $$(0+10^*)$$ and I was trying to find the regular sets. My books suggests that $$L=\{0, 1, 10, 100, 1000, 10000,\ldots\}$$. But I was thinking why they did not consider the strings starting with $$0$$ and then only one $$1$$ and zero or more number of occurrences of $$0$$, like $$010, 0100, 01000$$ etc.

1. Aren't these strings belong to $$L$$?
2. Is this $$L$$ a language? (According to me, it is a language because it is the set of strings over $$\{0,1\}$$.

A regular language $$L$$, is a set over the alphabet $$\Sigma$$ (that is, $$L\subseteq \Sigma^*$$) that has a regular expression that generates it. Therefore, by definition - any language of a regular expression is a regular language.
Regarding the language in question, $$01$$ for example is not generated by $$(0+10^*)$$, since it is not $$0$$ and also is not generated by $$10^*$$ (since it doesn't start with $$1$$). The same goes for any string starting with $$0$$, and has more than one letter in it.