# Conditions for an Language to be infinite

given 'r' , a regular expression that does not include λ or ∅, What are the Conditions of 'r' so that L(r) would be infinite?

Regular expressions correspond to a simple recursive definition, which starts with the symbols from some alphabet and extends them with applications of three operators: alternation, concatenation and Kleene closure.

So ask yourself: Which of these operators can produce an infinite set?

Formal definition:

Let $$\Sigma$$ be a finite set of symbols. Then the set of regular expressions $$R_\Sigma$$ is the smallest set of expressions which can be constructed using these rules:

• Every $$c \in \Sigma$$ is in $$R_\Sigma$$.

• $$\lambda$$ and $$\emptyset$$ are in $$R_\Sigma$$.

• If $$r$$ and $$s$$ are regular expressions over the alphabet $$\Sigma$$ then so are:

• $$(r+s)$$
• $$(rs)$$
• $$(r^*)$$

A standard proof technique for properties of regular expressions is to apply mathematical induction to this definition.