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I was studying Group Theory on my own from a mathematics journal and got confused at some point where it defines Cartesian products, from binary one, say ([itex]A × B[/itex]), to ntuples one, say ([itex]A_1 × A_2 × ... × A_n[/itex]). What confuses me when I tried to read it is that the definition made for infinite Cartesian product as shown below:
If I am correct that on the lefthand side [itex]\prod_{i\in{I}}X_i[/itex] by definition of Cartesian product corresponds to [itex]X_1 × X_2 × ...[/itex], which can be represented as in [itex](x_1,x_2,...)[/itex]. However, when I try to read and interpret the righthand side, I fail to create those tuples [itex](x_1,x_2,...)[/itex], but think of a set of [itex]\{f_i\}[/itex], which seemed to me absurd that how this set can turn into tuples.
I would like to know how to interpret all.
If I am correct that on the lefthand side [itex]\prod_{i\in{I}}X_i[/itex] by definition of Cartesian product corresponds to [itex]X_1 × X_2 × ...[/itex], which can be represented as in [itex](x_1,x_2,...)[/itex]. However, when I try to read and interpret the righthand side, I fail to create those tuples [itex](x_1,x_2,...)[/itex], but think of a set of [itex]\{f_i\}[/itex], which seemed to me absurd that how this set can turn into tuples.
I would like to know how to interpret all.
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