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I understand how in 1.23 Axler states that Fs is the set of all functions that map from the set S to F.

He later states that we can think of Fn as F{1,2,3,....,n}

Although an explanation in the book is provided, I'm still confused as to how we can represent it that way.

For example, according to the previous statement, R4 can be written as R{1,2,3,4}. However wouldn't that say R4 is the set of all functions that map from {1,2,3,4} to a real number R? Wouldn't R4 just be the all the sets that have 4 elements from R or am I misunderstanding something?

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    $\begingroup$ In standard notation, $$\mathbb{R}^4 = \{ (a,b,c,d) : a,b,c,d \in \mathbb{R}\}$$ Can you see how this bijects naturally with the functions $\{1,2,3,4\} \to \mathbb{R}$? For culture, the notation originates with combinatorics, as the number of functions $F \to G$ for finite sets $F,G$ is precisely $|G|^{|F|}$. $\endgroup$
    – William
    Jan 19 at 3:52
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    $\begingroup$ Not quite sure that I am understanding your confusion, but (much as William is doing), write $S=\left\{1,2,3,4\right\}$. Suppose that $f$ is a real valued function on $S$. Then one gets an element of ${\mathbb R}^4$ by $(f(1), f(2), f(3), f(4) )$. Conversely, given an element $(x_1,x_2,x_3,x_4) \in {\mathbb R}^4$ one gets a real valued function $f$ on $S$ by the recipe $f(k) = x_k$ . Hope that helps! $\endgroup$
    – peter a g
    Jan 19 at 3:56

1 Answer 1

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$\mathbb{R}^4$ is not the set of all sets that contain $4$ elements from $\mathbb{R}$. For instance, $\{1,5,10,12\}=\{12,5,1,10\}$, but for elements of $\mathbb{R}^4$, $(1,5,10,12)\ne (12,5,1,10)$. The distinction is that in sets order of elements make no difference, but elements of $\mathbb{R}^4$ are $4$-tuples where order matters.

An $n$-tuple can be thought of as a function that maps each index $i=1~..~n$ to its corresponding value. For instance $(1,5,10,12)$ can be thought of as the function $f:\{1,2,3,4\}\to\mathbb{R}$ given by $$1\mapsto 1,~~2\mapsto 5,~~3\mapsto 10,~~4\mapsto 12.$$

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  • $\begingroup$ Ah I see my misunderstanding. Thank you! I was somewhat confused as R^{1,2,3,4} would be a set of functions no? Or do I just consider 1 function from that set? $\endgroup$ Jan 19 at 4:06
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    $\begingroup$ Yes, $\mathbb{R}^{\{1,2,3,4\}}$ would be the set of all functions from $\{1,2,3,4\}$ to $\mathbb{R}$. Any specific element of $\mathbb{R}^4$ can be thought of as a single function, and the whole of $\mathbb{R}^4$ can be thought of as the set of all such functions, just the same way as we think of it as the set of all possible $4$-tuples. (Keep in mind that in practice you probably want to keep thinking of $\mathbb{R}^4$ as the set of all $4$-tuples. When one imagines an element of $\mathbb{R}^4$ one usually imagines a $4$-tuple, not a function. Of course, the two views are equivalent.) $\endgroup$
    – Snaw
    Jan 19 at 4:10

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