Cartesian Product Null

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Theorem

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

Thus:

$S \times \varnothing = \varnothing = \varnothing \times T$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S \times T$	$\ne$	$\displaystyle \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \exists \left({s, t}\right)$	$\in$	$\displaystyle S \times T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Empty Set
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \exists s \in S$	$\land$	$\displaystyle \exists t \in T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Cartesian Product
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle S \ne \varnothing$	$\land$	$\displaystyle T \ne \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Empty Set
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle \neg (S = \varnothing$	$\lor$	$\displaystyle T = \varnothing)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	De Morgan's Laws

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

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \times T\)	\(\ne\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \exists \left({s, t}\right)\)	\(\in\)	\(\displaystyle S \times T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Empty Set
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \exists s \in S\)	\(\land\)	\(\displaystyle \exists t \in T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Cartesian Product
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle S \ne \varnothing\)	\(\land\)	\(\displaystyle T \ne \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Empty Set
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle \neg (S = \varnothing\)	\(\lor\)	\(\displaystyle T = \varnothing)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	De Morgan's Laws