Is Σ* the same as saying L*

Question

I'm getting confused over the two. Are they the same? The complement of a language L is Σ* - L. Is saying L* - L wrong?

Answer 1

By definition, $L^* = \bigcup_{n \geq 0} L^n$. Also, $\Sigma$ is the language of all words of length $1$ (equivalently, $\Sigma$ is the alphabet). So $\Sigma^* = \bigcup_{n \geq 0} \Sigma^n$. There is no particular reason to assume that $\Sigma^* = L^*$. For example, suppose that $\Sigma = \{a,b\}$ while $L = \{a\}$. Then $\Sigma^*$ consists of all words over $a,b$, while $L^*$ consists of all words over $a$.

In fact, it is not hard to show that $L^* = \Sigma^*$ iff $\Sigma \subseteq L$ (exercise).