Intersection with Empty Set

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Theorem

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

$S \cap \O = \O$


Proof

\(\displaystyle S \cap \O\) \(\subseteq\) \(\displaystyle \O\) Intersection is Subset
\(\displaystyle \O\) \(\subseteq\) \(\displaystyle S \cap \O\) Empty Set is Subset of All Sets
\(\displaystyle \leadsto \ \ \) \(\displaystyle S \cap \O\) \(=\) \(\displaystyle \O\) Definition 2 of Set Equality

$\blacksquare$


Also see


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