# Set theory - Cartesian product of family

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?

• $X_1, X, X_i$ ? – Mauro ALLEGRANZA Sep 26 '18 at 8:00
• See the following post about Cartesian products of families. – Mauro ALLEGRANZA Sep 26 '18 at 8:00
• 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 $i \in I$. – Mauro ALLEGRANZA Sep 26 '18 at 8:03
• 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 $a_i \in X_i$. – Mauro ALLEGRANZA Sep 26 '18 at 8:05
• So the cartesian product is I times X? – Paul Sep 26 '18 at 8:13

Let $$X=\{4,5,6\}$$ and $$X_1=X_2=X_3=X$$. Then
$$X_1 \times X_2 \times X_3=\{(a,b,c): a,b,c \in X\}$$.
• @Paul: Because you said $X_i = \{ 4,5,6\}$ for all $i$. – user14972 Sep 26 '18 at 8:02
• Yes. The cartesian product has $27$ elements. – Fred Sep 26 '18 at 8:46