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I'm trying to understand the cartesian product of a family.

I understand if $X = \{1,2,3\}$ and $Y = \{4,5,6\}$ then the cartesian product of these two sets is $\{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)\}$

If ($X_{i}$} is a family of sets where $i \in I$, the Cartesian product of the family is, by definition, the set of all families $x_{i}$ with $x_{i} \in X_{i}$ where each $i \in I$

Say I = $\{1,2,3\}$ and $X_{i} = \{4,5,6\}$ How can you have a cartesian product of 1 set?

Thanks in advance

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  • $\begingroup$ $X_1, X, X_i$ ? $\endgroup$ – Mauro ALLEGRANZA Sep 26 '18 at 8:00
  • $\begingroup$ See the following post about Cartesian products of families. $\endgroup$ – Mauro ALLEGRANZA Sep 26 '18 at 8:00
  • $\begingroup$ We have an index set $I$ and and a "family" of sets $\{ X_i \}$. The cartesian product of the family is the set of all families $\{ x_i \}$ with $x_i ∈ X_i$ for each $i \in I$. $\endgroup$ – Mauro ALLEGRANZA Sep 26 '18 at 8:03
  • $\begingroup$ If $I$ is a $3$ elements set, each element of the cartesian product will be a $3$-uple : $(a_1,a_2,a_3)$ where $a_i \in X_i$. $\endgroup$ – Mauro ALLEGRANZA Sep 26 '18 at 8:05
  • $\begingroup$ So the cartesian product is I times X? $\endgroup$ – Paul Sep 26 '18 at 8:13
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Let $X=\{4,5,6\}$ and $X_1=X_2=X_3=X$. Then

$X_1 \times X_2 \times X_3=\{(a,b,c): a,b,c \in X\}$.

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  • $\begingroup$ I'm confused. Why X1=X2=X3=X? $\endgroup$ – Paul Sep 26 '18 at 8:02
  • $\begingroup$ @Paul: Because you said $X_i = \{ 4,5,6\}$ for all $i$. $\endgroup$ – user14972 Sep 26 '18 at 8:02
  • $\begingroup$ so is the cartesian product something like {(4,4,4),(,4,4,5),(4,4,6),(4,5,4),(4,5,5).....? $\endgroup$ – Paul Sep 26 '18 at 8:43
  • $\begingroup$ Yes. The cartesian product has $27$ elements. $\endgroup$ – Fred Sep 26 '18 at 8:46
  • $\begingroup$ Okay thanks for your help, but what would I be in your example? {1,2,3}? $\endgroup$ – Paul Sep 26 '18 at 8:56

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