# Is the set of all strings over a finite alphabet finite? [closed]

Suppose $$Σ=\{0,1\}$$; then $$Σ^*$$ is the set of all strings over $$Σ$$.

Is $$Σ^*$$ over $$Σ$$ finte?

• did you look at Kleene star to see what $\Sigma^*$ actually means? – Ran G. Jun 16 '15 at 22:02
• Yes, I know what it means, but is | Σ* | finite or not ? – Mohammed Jun 16 '15 at 22:07
• can you specify some words that belong to $\Sigma^*$? how many of these can you specify (more than 2? more than 3? how about more than 10?) – Ran G. Jun 16 '15 at 22:10
• The answer is: Σ* does not have a length. The reason for it is that Σ* is a set of strings, an infinite set of strings, which are all of finite length. The notation |Σ*| is for the cardinality of Σ*, i.e. its number of elements, which as I said is infinite. – babou Jun 16 '15 at 22:18
• Thanks babou. Well, I guess I deserved that one for this (; – Ran G. Jun 16 '15 at 22:28

The star operator is a unary operator known as Kleene star (or Kleene closure) and the result of its application on $\Sigma$ (an arbitrary set of strings) is another set that contains all possible finite strings constructed using only strings from $\Sigma$.
• Actually, it is not true that Kleene star always produces an infinite set. counterexamples: $\emptyset^*$, and $\{\varepsilon\}^*$. – Ran G. Jun 16 '15 at 22:31