# Cartesian Product of Subsets/Family of Subsets

## 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.

$\blacksquare$