What is the meaning of $\{1,\dots,k \}^n$?

Question

I'm trying to figure out what the set $\{1,\dots,k \}^n$ is.

I know that the number of $n$-tuples formed from the numbers $1,\dots,k$ is the cardinality of that set, but I'm not sure what the elements are.

Is this a standard notation?

Answer 1

The elements of the set are of the form $(a_1, \dots, a_n)$ for $a_i\in\{1, \dots, k\}$. The notation is standard.

Answer 2

Well you stated it yourself. The elements are $n$-tuples with entries from $\lbrace 1, \dots, k \rbrace$. That is also why you can compute the cardinality that way. Just to give an example, if $k = 2$ and $n=3$, this set contains exactly the following elements:

$(1,1,1)$, $(2,2,2)$, $(1,2,2)$, $(2,1,2)$, $(2,2,1)$, $(2,1,1)$, $(1,2,1)$ and $(1,1,2)$.