In my book ( a generic introduction to higher math textbook ), we are given the following:
$B^{\{1,2,...,n\}}$ is the set of all n-sequences.
I'm taking this to mean $B$ is a set, an n-sequence is a sequence with $n$ elements, and $B^{\{1,2,...,n\}}$ is the set of all n-sequences composed of elements from $B$.
So, the superscript $^{\{1,2,...,n\}}$ is used to communicate three things: it is raised and therefore expresses some operation on $B$, this operation produces sequences ( but, for some reason, the superscript uses set brackets to suggest this fact ), and thirdly, everything between the opening and closing brackets, $^{1,2,...n}$, is merely there to indicate the length of the sets. ( But, if that is the case, wouldn't it be possible to express that idea by simply writing $^{n}$ rather than $^{1,2,...n}$ ? )
Clearly, I'm not understanding the intent of this notation.
Can anyone help explain it clearly?
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$\begingroup$In general, if $A$ and $B$ are any sets, $A^B$ denotes the set of all functions from $B$ to $A$. Here, $B^{\{1,2,...,n\}}$ denotes the set of all functions from $\{1,2,...,n\}$ to $B$. This shouldn't be too weird; a sequence of length $n$ of elements from $B$ is essentially a way to assign an element of $B$ to each $k$ such that $1\leq k \leq n$.
This notation is used because, in the finite case, if $A$ has $n$ elements and $B$ has $m$ elements, then there are $n^m$ functions from $B$ to $A$. Since for each of the $m$ elements of $B$ we have $n$ choices of what to define as the image of that element, there are $n^m$ total ways to "build" a function from $B$ to $A$.
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