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Consider the following set constructed with a regular Kleene-star operation: $$ \{10^n \mid n\geq1\}^* $$

Would something like $10100$ be in this set? I know $1010,100100100,1000,$ etc would be, but I'm not sure if you can change $n$ per repetition and have it be something like $10100, 101001010010$, etc.

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Short answer: Yes. $10100$ is included.

Long answer:

Let's go through how this particular set is constructed. First, the expression $$ 10^n $$ (in this context) denotes a single string consisting of a $1$ followed by $n$ $0$s (so the string depends on one variable $n$). Then, the set $$ \{10^n \mid n \ge 1\} $$ denotes the set of strings $\{10, 100, 1000, 10000, \ldots\}$. This set can also be denoted using the Kleene-star operation as $100^*$ or $10^+$. Note that this set does not depend on $n$ -- rather, $n$ is just a dummy variable used to describe the elements of the set.

Finally, the set $$ \{10^n \mid n \ge 1\}^* $$ then denotes a concatenated sequence of elements of the original set, like $10100$, $1000$, $1010000010$, etc. When you use the $*$ operation, the string used each time does not have to be the same (that is, you can change $n$ for each repetition). So you can do $10$ first, then $100$, then $10$ again, etc.

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