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$\mathop X\nolimits_1 ,\mathop X\nolimits_2 ,\mathop X\nolimits_3 ,...\mathop X\nolimits_n $ be the sets then the set of tuples
$({x_1}, \ldots ,{x_n}) \in \mathop \prod \limits_{i = 1}^n {X_i}$ such that ${x_i} \in {X_i}$ is the product of given sets.

which is very straightforward and clear to me but I am having diffculty understanding the standard generalization of this concept i.e $$\mathop \prod \limits_{i \in I} {X_i} = \{ f:I \to \bigcup\limits_{i \in I} {{X_i}:f(i) \in {X_i}} \} $$ let me walk you through what I do understand and what I am having problem with so for this we will only consider two sets ${X_1}$ and ${X_2}$ let ${X_1}$ = { 2 , 3 } and ${X_2}$ ={ 5 , 7 }

now the product ${X_1}$$ \times $${X_2}$={ (2,5),(2,7)(3,5)(3,7) } and we have the indexing set I = { 1, 2 }

now what i don't understand is

"Another way of thinking about this is ${X_1}$$ \times $${X_2}$ as the collection of all functions from $I \to {X_0} \cup {X_1}$ such that $f(1) \in {X_1}$ and $f(2) \in {X_2}$"

1 give those functions from { 1 , 2 } $\to$ { 2 , 3 , 5 , 7 } and explain how the two definitions are same or related (straightforward one and the standard generalization )

2 give one example of product but using 3 sets containing 2 elements each using the function concept of standard generalization

3 since it is called "standard generalization" what does it explain in a better way that the straightforward definition can't

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    The first couple in $X_1 \times X_2$ is $(2,5)$; it corresponds to $f_1 = \{ (1,2), (2,5) \}$. The second couple $(2,7)$ to $f_2 = \{ (1,2), (2,7) \}$, and so on. – 2017-01-31
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    For an $n$-uple $(x_1, x_2, \ldots, x_n)$ we will have : $\{ (1,x_1), (2, x_2), \ldots, (n, x_n) \}$. – 2017-01-31
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    @MauroALLEGRANZA thanks but can you please answer all three in detail – 2017-01-31
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    See [$n$-ary product](https://en.wikipedia.org/wiki/Cartesian_product#n-ary_product). – 2017-01-31
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    @MauroALLEGRANZA also how does the correspondence help i mean it looks like we just defined a function that relates the two sets it does not tell me how the product of those sets looks like – 2017-01-31
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    what extra information do i have when i have those functions how do they help – 2017-01-31
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    The issue is with "generalization" : it is not easy to "formalize" the dots into $X_1 \times X_2 \times \ldots \times X_n$. – 2017-01-31
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    oh ok i think I am getting it now – 2017-01-31
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    The product of two sets is simply a set of couples; in the alternative def, the product is a set of functions : one for each couple, where the components of the "original" couple are the corresponding "values" of the function. – 2017-01-31
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    @MauroALLEGRANZA one thing i still can't understand is that in the standard generalization you are equating the product of sets (which is a set of $n$-tuple elements where n is number of elements in indexing set $I$ ) to a set of functions (n-tuple elements = function ?) – 2017-01-31
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/52828/discussion-between-user306284-and-mauro-allegranza). – 2017-01-31

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