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.
$\blacksquare$