# Languages math problem

1. Jan 27, 2010

### ayusuf

1. The problem statement, all variables and given/known data
If L is a language over $$\Sigma$$ then lim n -> inf of Ln = $$\Sigma$$* iff ($$\Sigma$$$$\cup$${$$\lambda$$})$$\subseteq$$ L

Also Lk = {x1x2...xk | x1, x2, ...xk $$\in$$ L}

2. Relevant equations

3. The attempt at a solution
I started by saying there is an w is an element of sigma then it is also an element of L so I might use induction but I really don't even know if I started right. Thanks.

2. Jan 28, 2010

### ystael

Re: Languages

What is the definition of limit you use in the expression $$\lim_{n\to\infty} L^n$$?

If $$L$$ does not contain the empty string $$\epsilon$$, then it is not true that $$L \subset L^2 \subset L^3 \subset \cdots$$, because the shortest string in $$L^n$$ has length $$n$$ times the length of the shortest string in $$L$$. In this case, you therefore cannot use $$\lim_{n\to\infty} L^n = \bigcup_{n=0}^\infty L^n$$ as a definition. You can even construct a case (say, $$L = \Sigma$$) where the $$L^n$$ are pairwise disjoint!