This is a very basic question. Let $\Sigma=\{a,b,c\}$ be an alphabet, and $n=2$ be an integer number, then what is $\Sigma^n$?

I am confused if $a,b,c \in \Sigma^2$.


Usually $\Sigma^n$ denotes the cartesian product. However in this context I suppose it stands for the set of words of length $n$ (which is very similar to a cartesian product, just without the brackets).

The formal definition of $\Sigma^n$ is the following: $$\Sigma^n = \{x_1x_2...x_n : x_i\in \Sigma\}$$

For instance if $\Sigma = \{a,b,c\}$ then $\Sigma^1 =\Sigma$. While $\Sigma^2$ are words of length two, so $\Sigma^2 = \{aa,ab,ac,ba,bb,bc,ca,cb,cc\}$. $\Sigma^3$ denotes all the words of length three and so on...


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