# Kleene Star of a Set

The question is relates to languages in automata theory. Give an example of a language W such that W∗ has exactly one more element than W.

My guess is that the answer to this problem is the empty set (∅). Since the star of the empty set is {e}, that makes it one element bigger.

Am I correct? My other guess is that it is an infinite set, but I have no idea how to represent an infinite set on paper.

Thanks very much.

## 1 Answer

Your answer is correct. For an infinite example, try the set $A=\{a\}^*\setminus\{\epsilon\}$, sometimes written $\{a\}^+$: this is the set of all non-empty strings of $a$’s.

This example and yours are just instances of a more general observation: if $W$ is any language such that $\epsilon\notin W$ and $W$ is closed under concatenation, then $W^*=W\cup\{\epsilon\}$.

• @user1305850: You’re very welcome. – Brian M. Scott Sep 19 '13 at 2:56