What is the language of Sigma^n? Confused about meanining
By definition $\Sigma^n$ is simply $\underbrace{\Sigma\times \Sigma \times \cdots \times\Sigma}_{n \ \text{times}} = \{(\sigma_1, \sigma_2, \ldots, \sigma_n): \forall \ i\in [n], \sigma_i \in \Sigma \}$ which is the set of ordered tuples of length $n$ over $\Sigma$. Usually, when we refer to a tuple $w = (\sigma_1, \sigma_2, \ldots, \sigma_n)$, we omit the commas &quot;,&quot; and the parenthesis &quot;(&quot; and &quot;)&quot;, so the tuple $w$ is written as $\sigma_1\cdot \sigma_2\cdots \sigma_n$ to which we refer as a word of length $n$ over $\Sigma$. In other words, $\Sigma^n$ is the set of all words of length $n$ over $\Sigma$.
Formally, the map that maps a tuple in $\Sigma^n$ to the word that we get by omitting the commas and the parenthesis from the tuples, is a bijection from $\Sigma^n$ to the words of length $n$ over $\Sigma$.