# What does $S = \{1,2,3 \}^*$ mean?

Apparently, if one adds an asterisk to the right side of a set definition, it means the set to the left can be built out of elements in the set to the right.

How is this so? What does the asterisk officially mean?

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## 3 Answers

This notation is most common in discrete mathematics. In that context the set $S$ is considered to be an alphabet and $S^*$ just means the set of all finite strings that can be formed with letters from the alphabet. Variants are $S^n$ for the set of all strings of length $n$, and $S^{\le n}$ for the set of all strings with length no more than $n$. These latter two variants are widely used in set theory as well.

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1
And it should probably be explicitly noted that this includes the empty string, usually denoted either by $\lambda$ or by $\varepsilon$. – Brian M. Scott Mar 2 '12 at 11:39
Yes, the string of length 0. – Patrick Mar 2 '12 at 14:11

It means whatever it says it means in the place where you found it. Without knowing that context, it could mean anything.

One common meaning is something like what you've said. It could mean all finite strings of symbols made up from the symbols 1, 2, and 3.

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+1 for the first line alone! – user21436 Mar 2 '12 at 8:49

It's called the "Kleene star": see http://en.wikipedia.org/wiki/Kleene_star

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