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As an alternative definition, the <math>n\!</math>-tuples of <math>\prod_{i=1}^n X_i\!</math> can be regarded as sequences of elements from the successive <math>X_i\!</math> and thus as functions that map <math>[n]\!</math> into the sum of the <math>X_i,\!</math> namely, as the functions <math>f : [n] \to \bigcup_{i=1}^n X_i\!</math> that obey the condition <math>f(i) \in i \widehat{~} X_i.\!</math>

As an alternative definition, the <math>n\!</math>-tuples of <math>\prod_{i=1}^n X_i\!</math> can be regarded as sequences of elements from the successive <math>X_i\!</math> and thus as functions that map <math>[n]\!</math> into the sum of the <math>X_i,\!</math> namely, as the functions <math>f : [n] \to \coprod_{i=1}^n X_i\!</math> that obey the condition <math>f(i) \in i \widehat{~} X_i.\!</math>

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| <math>\prod_{i=1}^n X_i ~=~ X_1 \times \ldots \times X_n ~=~ \{ f : [n] \to \bigcup_{i=1}^n X_i ~|~ f(i) \in X_i \}.\!</math>

| <math>\prod_{i=1}^n X_i ~=~ X_1 \times \ldots \times X_n ~=~ \{ f : [n] \to \coprod_{i=1}^n X_i ~|~ f(i) \in X_i \}.\!</math>

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