Cartesian Product is Empty iff Factor is Empty

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Theorem

$S \times T = \O \iff S = \O \lor T = \O$


Thus:

$S \times \O = \O = \O \times T$


Family of Sets

Let $I$ be an indexing set.

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

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

Then:

$S = \O$ if and only if $S_i = \O$ for some $i \in I$


Proof

\(\ds S \times T\) \(\ne\) \(\ds \O\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists \tuple {s, t}\) \(\in\) \(\ds S \times T\) Definition of Empty Set
\(\ds \leadstoandfrom \ \ \) \(\ds \exists s \in S\) \(\land\) \(\ds \exists t \in T\) Definition of Cartesian Product
\(\ds \leadstoandfrom \ \ \) \(\ds S \ne \O\) \(\land\) \(\ds T \ne \O\) Definition of Empty Set
\(\ds \leadstoandfrom \ \ \) \(\ds \neg \leftparen {S = \O}\) \(\lor\) \(\ds \rightparen {T = \O}\) De Morgan's Laws: Conjunction of Negations


So by the Rule of Transposition:

$S = \O \lor T = \O \iff S \times T = \O$

$\blacksquare$


Sources

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