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'''Definition.'''  The ''<math>n\!</math>-fold product'' (''cartesian product'') of the sets <math>X_1, \ldots, X_n\!</math> is notated and defined as follows:

The "n fold product" ("cartesian product") of the sets X1, ... , Xn is notated and defined as follows:

Xi Xi  = X1 x ... x Xn  = {<x1, ... , xn> : xi C Xi for all i}.

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| <math>\prod_{i=1}^n X_i ~=~ X_1 \times \ldots \times X_n ~=~ \{ (x_1, \ldots, x_n) : x_i \in X_i \}.\!</math>

As an alternative definition, the n tuples of Xi Xi can be regarded as sequences of elements that come from the successive Xi, and thus as the various functions of a certain sort that map [n] into the sum of the Xi, namely, as the sort of functions f : [n]  > Ui Xi that obey the condition f(i) C i^Xi.

As an alternative definition, the n tuples of Xi Xi can be regarded as sequences of elements that come from the successive Xi, and thus as the various functions of a certain sort that map [n] into the sum of the Xi, namely, as the sort of functions f : [n]  > Ui Xi that obey the condition f(i) C i^Xi.