Intersection with Empty Set

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Theorem

The intersection of any set with the empty set is itself the empty set:

$S \cap \varnothing = \varnothing$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S \cap \varnothing$	$\subseteq$	$\displaystyle \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Intersection Subset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \varnothing$	$\subseteq$	$\displaystyle S \cap \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Empty Set Subset of All
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle S \cap \varnothing$	$=$	$\displaystyle \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Equality

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S \cap \varnothing\)	\(\subseteq\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Intersection Subset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \varnothing\)	\(\subseteq\)	\(\displaystyle S \cap \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Empty Set Subset of All
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle S \cap \varnothing\)	\(=\)	\(\displaystyle \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Equality