Cartesian Product is Empty iff Factor is Empty

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Theorem

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


Thus:

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


Proof

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


So by the Rule of Transposition:

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

$\blacksquare$


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