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The closure of a set S, which is S*, is defined as S0 ∪ S1 ∪ S2 ∪ ...

So if S contained the English alphabet, S* = {empty string} ∪ {a,b,c,..} ∪ {aa, ab, ac, ... , ba, bb, bc, ... } ∪ {three character strings} ∪ ...

What is A* if A = {10}? I'm not sure how closure works for numbers because you can't just concatenate numbers like you can with strings.

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It has nothing to do with numbers. The notation $A=\{10\}$ means that $A$ consists of the single string $10$. Thus $A^\ast$ is the set of strings that consists of the null string (often called $\lambda$, but there are other names), together with $10$, $1010$, $101010$, $10101010$, and so on forever.

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    I see, but if you take {0,0,0} dotted with {10}* then would it be still be {0,0,0,0,...}? {0,0,0} is actual numbers in this case.2012-08-28
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    I am not sure what you mean by the *set* $\{0,0,0\}$ since the only element $0$ is mentioned three times. If $B$ is the set $\{0\}$, then $BA^\ast$ consists of the strings $0$, $010$, $01010$, and so on.2012-08-28
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    ^ sorry, the set is shown as {0^i|0<=i<=2} so i assumed it was three 0's. also does this mean that closure doesn't work for a set of numbers?2012-08-28
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    The closure operator $\ast$ works on sets of *strings*, also known as *words*. If we use binary notation, or decimal notation, then there is a close relationship between numbers and strings over a suitable alphabet.2012-08-28