Cartesian Product of Subsets/Family of Subsets

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.

Let $S = \ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.


Let $\family {T_i}_{i \mathop \in I}$ be a family of sets.

Let $T = \ds \prod_{i \mathop \in I} T_i$ be the Cartesian product of $\family {T_i}_{i \mathop \in I}$.


Then:

$\paren {\forall i \in I: T_i \subseteq S_i} \implies T \subseteq S$.


Nonempty Subsets

Let $T_i \ne \O$ for all $i \in I$.


Then:

$T \subseteq S \iff \forall i \in I: T_i \subseteq S_i$.


Proof

Let $T_i \subseteq S_i$ for all $i \in I$.

Then:

\(\ds \family {x_i}\) \(\in\) \(\ds T\)
\(\ds \leadsto \ \ \) \(\ds \forall i \in I: \, \) \(\ds x_i\) \(\in\) \(\ds T_i\) Definition of Cartesian Product of Family
\(\ds \leadsto \ \ \) \(\ds \forall i \in I: \, \) \(\ds x_i\) \(\in\) \(\ds S_i\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds \family {x_i}\) \(\in\) \(\ds S\) Definition of Cartesian Product of Family

Thus $T \subseteq S$ by the definition of a subset.

$\blacksquare$